# Wilson Loop Expectations for Non-Abelian Gauge Fields Coupled to a Higgs
Boson at Low and High Disorder

Arka Adhikari
Stanford University
<EMAIL_ADDRESS>
###### Abstract

We consider computations of Wilson loop expectations to leading order at large
$\beta$ in the case where a non-abelian gauge field interacts with a Higgs
boson. By identifying the main order contributions from minimal vortices, we
can express the Wilson loop expectations via an explicit Poisson random
variable. This paper treats multiple cases of interests, including the Higgs
boson at low and high disorder, and finds efficient polymer expansion like
computations for each of these regimes.

###### Contents

  1. 1 Introduction
    1. 1.1 Background and History
    2. 1.2 Organization of the Paper
    3. 1.3 Preliminary Notation and Discussion
      1. 1.3.1 Previous Work: Wilson Loops in Abelian Lattice Higgs Model
    4. 1.4 Extension of the Analysis of Wilson Loop Expectation
      1. 1.4.1 The Excitations of the Higgs field
    5. 1.5 The Low Disorder Regime: High $\kappa$
    6. 1.6 The High Disorder Regime: $\kappa$ Low
      1. 1.6.1 Informal Discussion
  2. 2  The Low Disorder Regime: A Toy Model
    1. 2.1 Introduction and Informal Discussion
      1. 2.1.1 Higgs boson Configuration
      2. 2.1.2 Wilson Loop Action
    2. 2.2 Rigorous Definitions
    3. 2.3 Analysis of the Wilson Loop Action
      1. 2.3.1  Trivial Configurations
      2. 2.3.2 The main term contribution
      3. 2.3.3 Bounding the probability of large vortices
    4. 2.4 Approximating $M_{\gamma}$ as a Poisson Random Variable
  3. 3 The Non-Abelian Case
    1. 3.1 Model and Some Preliminary Discussion
      1. 3.1.1 Definitions
      2. 3.1.2 Removing Trivial Gauge Invariances
    2. 3.2 The case of Low Disorder in the Higgs Field
  4. 4 The High Disorder Regime: Small $\kappa$
    1. 4.1 Expansion in Random Currents
    2. 4.2 Knot Expansion
    3. 4.3 Identifying the Main Order Excitations
    4. 4.4 Approximation by a Poisson Random Variable
  5. 5 Decorrelation Estimates and a More Precise Expansion
    1. 5.1 Minimal Vortices and Decorrelation
    2. 5.2 Approximation by Poisson Random Variables
  6. A Details on Vertex Decompositions

##  1 Introduction

###  1.1 Background and History

Quantum field theory and the Standard Model are one of the greatest successes
of modern physics; they are able to compute the behavior of the smallest
particles to remarkable accuracy. However, these physical theories have yet to
be given a rigorous formulation due to multiple difficulties in trying to
define the Hamiltonians used on a continuous space. Glimm and Jaffe [11]
attempted to define quantum field theories by first defining a Euclidean
lattice gauge theory and then applying a Wick rotation to the proper complex
space. Seiler [14] pointed out that there were issues when trying to follow
this path in general; Seiler instead proposed that the basic object one should
study are appropriate random functions on an appropriate space of closed
curves.

One promising strategy to do this was via the method of lattice gauge
theories. Inspired by ideas from statistical physics, Wilson [17] proposed
lattice gauge theories as a means of computing some quantities of interest in
lattice gauge theory; more specifically, he wanted to explain quark
confinement. Lattice gauge theories are essentially quantum field theories
defined on a discrete lattice; on this discrete space, the Hamiltonians can be
well-defined. The ultimate hope is that one could take the limit of quantities
defined on the lattice as the lattice size of the lattice goes to $\infty$ and
the lattice spacing goes to $0$ to define a continuous theory.

This approach is promising, but yet there is still no proof that in the
interesting physical dimension of $4$ there is a way to take the lattice
spacing to $0$ and still obtain obtain a meaningful probability distribution
in the end. However, it is still a very interesting question to determine
whether there are important physical quantities of interest such that there is
a meaningful limit when computed on lattice gauge theories as the size of the
lattice goes to $\infty$. The most important quantity of interest are Wilson
loop observables. These Wilson loop expectation will be formally defined at
the end of subsection 1.3, but we can describe the physical meaning of these
values here. In the Standard Model, many larger subatomic particles, called
baryons, are theorized to be composed of smaller constituents called quarks.
An unusual fact about quarks is that individual quarks are not found alone in
nature. It is argued that in real world scenarios, the quarks are tightly
bound to each other. Wilson in [17] argues that if Wilson loop expectations
satisfy appropriate decay conditions, namely the Area Law, then this would be
sufficient to establish quark binding.

Though the most physically relevant case of lattice gauge theories occur when
studying the group $SU(4)$, there is substantial literature in both the math
and physics literature studying lattice gauge theories on finite groups [2, 3,
7, 12, 13, 15, 16]. In more recent history, there were multiple works
computing the values of Wilson loop expectation for various discrete groups.
In particular the papers [5] and [4] compute Wilson loop expectations in the
case of pure gauge field. The paper [10] computes Wilson loop expectations for
a pure $U(1)$ gauge group. However, in the Standard Model, it is expected that
the gauge field interacts with multiple other particles of interest; the
interaction with other particles makes the analysis substantially more
completed. This model considers the computation of Wilson loop expectations
when our Hamiltonian includes an interaction with a Higgs boson. Under certain
conditions, we would expect that the analysis in this work could be adapted to
the case of multiparticle interactions without too much difficulty. We remark
here that the work [9] treats the problem of computing Wilson loop
expectations with a Higgs boson, but only in the case that the gauge group is
abelian and there is low disorder in the Higgs field. This work substantially
generalizes the analysis to non-abelian groups and to low disorder as well,
amongst other generalizations. In the proceeding subsections, we give further
mathematical background on the problem of computing Wilson loop expectations
as well as our main results.

###  1.2 Organization of the Paper

In this short subsection, we describe the organization of the paper. In the
rest of the introduction, we present the notation we will use as well as
further details about the problem of computing Wilson loop expectations on
models containing a Higgs boson. We discuss some of the history behind the
problem and introduce our two major new regimes, the low disorder and the high
disorder regime.

These two regimes have different polymer expansions an each have their own
section. Section 2 analyzes the low disorder regime with a toy model, where
$G$ and $H$ are both abelian $Z_{2}$ groups. Within this section, we introduce
the model in Subsection 2.1, introduce our polymer expansion in subsection
2.2, perform our identification of the main order contributions in Subsection
2.3, and finally compare our model to a computation involving a Poisson random
variable in Subsection 2.4.

Wondrously, there are no substantial differences in the low disorder regime
between the abelian and non-abelian cases. Thus, Section 3 has the
simultaneous job of introducing the notation of the non-abelian Higgs boson
model in subsection 3.1 and perform the brief modifications necessary to
analyze the low disorder regime in Subsection 3.2.

In Section 4, we give the more involved analysis of the lower disorder regime.
The first two Subsections 4.1 and 4.2 introduce elements of our polymer like
expansion. Subsection 4.3 identifies the main order contribution and
Subsection 4.4 performs a comparison to a Poisson random variable.

In Section 5, we perform a far more delicate analysis of the error terms in
Section 4 involving particular properties of the minimal vortices that form
the crux of our main-order contribution in Section 4. There are delicate
decorrelation estimates established in Subsection 5.1 which are helpful in
improving the error term and then improving the approximation by Poisson
random variables in Subsection 5.2.

###  1.3 Preliminary Notation and Discussion

Lattice gauge theories, as their name implies, are studied in some lattice
$\Lambda_{N}:=[-N,N]^{d}$ which is a sublattice of $\mathbb{Z}^{d}$. As a
graph, the structure is inherited as a subgraph of $\mathbb{Z}^{d}$. We let
$V_{N}$ denote the set of vertices of $\Lambda_{N}$ and we let $E_{N}$ be the
set of oriented edges of $\Lambda_{N}$; the orientation distinguishes the edge
$e=(x,y)$ from $-e=(y,x)$. Also important in lattice gauge theories are the
set of plaquettes $P_{N}$. If we let $x_{1},x_{2},x_{3},x_{4}$ be a set of
four vertices such that $x_{i+1}$ is adjacent to $x_{i}$ (with $x_{5}=x_{1}$),
then the square bounded by these four vertices will be called a plaquette.
Plaquettes have a natural orientated inherited from the order in which we
traverse the boundary vertices. (Thus, if we let $p$ be the plaquette with
boundary vertices traversed in the order $x_{1},x_{2},x_{3}$ and then $x_{4}$,
then $-p$ is the plaquette with boundary vertices traversed in the order
$x_{4},x_{3},x_{2}$ and then $x_{1}$.)

A gauge configuration $\sigma$ with gauge group $G$ on a lattice,
$\Lambda_{N}\subset\mathbb{Z}^{d}$ is an assignment $\sigma:E_{N}\to G$ such
that $\sigma_{-e}=(\sigma_{e})^{-1}$. The gauge group $G$ also comes with a
unitary representation $\rho:G\to U_{D}$ to the appropriate space of $D$ by
$D$ unitary matrices.

A Higgs boson configuration is the assignment $\phi:V_{N}\to\mathbb{C}$ of a
complex valued function at each vertex. Later in this introduction, we will
provide some restrictions on the values that $\phi$ could take. Associated to
the Higgs boson is a covariant derivative that mixes the action of Higgs boson
and the gauge field. Let $e$ be the edge that connects the vertex $x$ to the
vertex $y$. Then, we have,

$D_{e}\phi=\phi_{x}\text{Tr}[\rho(\sigma_{e})]-\phi_{y}.$ (1)

With this, we have the following interaction term between the Higgs boson and
the gauge field,

$S^{D}(\sigma,\phi):=\frac{1}{2}\sum_{e\in E_{N}}|D_{e}\phi|^{2}=2d\sum_{x\in
V_{N}}|\phi_{x}|^{2}-\sum_{e=(x,y)\in E_{N}}\phi_{x}\text{
Tr}[\rho(\sigma_{e})]\overline{\phi}_{y}.$ (2)

A gauge field configuration also has a self-energy, corresponding to the
curvature form, or Wilson action functional. Associated to any oriented
plaquette $p$, whose boundary vertices are $x_{1},x_{2},x_{3}$ and $x_{4}$ (
with edges $e_{i}=(x_{i},x_{i+1})$), we can define the action $\psi_{p}$ along
the plaquette to be
$\text{Tr}[\rho(\sigma_{e_{1}})\rho(\sigma_{e_{2}})\rho(\sigma_{e_{3}})\rho(\sigma_{e_{4}})]$.
Note that due to the cyclic property of the trace, it is indeed the case that
the value of $\psi_{p}$ does not depend in particular on which vertex we start
the ordering of the boundary vertices( though the actions will differ if we
consider $\psi_{p}$ or $\psi_{-p}$). The Wilson action function will be the
sum of the actions $\psi_{p}$ over all oriented plaquettes $\psi_{p}$.

$S^{W}(\sigma):=-\sum_{p\in P_{N}}\psi_{p}(\sigma).$ (3)

For an abelian group and a 1-dimensional representation $\rho$, we can
represent this in a much more direct form as follows:

$S^{W}(\sigma):=-\sum_{p\in P_{N}}\rho((\text{d}\sigma)_{p}).$ (4)

If we consider the oriented plaquette $p$ with oriented boundary edges
$e_{1}$,$e_{2}$,$e_{3}$ and $e_{4}$, then $(\text{d}\sigma)_{p}$ is the well-
defined element $\sum_{i}\sigma_{e_{i}}$. We needed to introduce more general
notation for the non-abelian case since we are no longer able to assign a
canonical element to each plaquette.

The Higgs boson also has a self- interaction given by the following form,

$V(\phi):=\sum_{x\in V_{N}}(|\phi_{x}|^{2}-1)^{2}.$ (5)

Consider the following interaction Hamiltonian between the Higgs boson and a
gauge field,

$H_{N,\beta,\kappa,\zeta}(\sigma,\phi):=-\beta S^{W}(\sigma)-\kappa
S^{D}(\sigma,\phi)-\zeta V(\phi).$ (6)

We consider the formal limit when we take $\zeta\to\infty$. In this limit, the
values of $|\phi_{x}|$ are confined to $1$, so $\phi_{x}$ can only take values
along the unit circle. We will let $\phi_{x}$ take values in some finite
subgroup, $H$, of the multiplicative group of the unit circle. The group $H$
is isomorphic to $Z_{k}$( the additive group of integers mod $k$) for some
$k$; namely, $\phi_{x}$ can only take values of the form
$e^{2\pi\text{i}\frac{j}{k}}$ for some $0\leq j<k$. We also remark that after
fixing $|\phi|=1$, we have that $\bar{\phi}=\phi^{-1}$. This replacement will
be used many times when expressing the Hamiltonian.

We consider the following measure of gauge and Higgs boson configurations on
the lattice $\Lambda_{N}$.

$\mu(\sigma,\phi)=\frac{1}{Z}\exp[H_{N,\beta,\kappa}(\sigma,\phi)],$ (7)

where $Z$ is the partition function coming from the sum of
$\exp[H_{N,\beta,\kappa}(\sigma,\phi)]$ over all configurations of $\sigma$
and $\phi$.

On loop $\gamma$, the Wilson loop action $W_{\gamma}(\sigma,\phi)$ is the
product $\text{Tr}[\prod_{e\in\gamma}\rho(\sigma_{e})]$ of the $\sigma$’s
along $\gamma$. As before, the cyclic property of the trace ensures that this
quantity is well-defined.

####  1.3.1 Previous Work: Wilson Loops in Abelian Lattice Higgs Model

In this previous work, the authors considered the case that $G$ was abelian,
$\rho$ was a one-dimensional representation, and $H$ is contained inside the
image of $\rho(G)$(which will be a subgroup of the unit circle). This
simplification allows one to effectively gauge out the effect of the Higgs
boson.

Before describing this construction, we remark that we can always assume that
the only element $g\in G$ such that $\rho(g)=1$ is the identity element. If
this were not true, then we can remove a trivial gauge invariance that does
not change the Hamiltonian action, but modifies the group $G$ so that there is
only the only element with $\rho(g)=1$ is the identity. The construction is as
follows: first let $G^{\prime}$ be the subgroup of elements $g^{\prime}$ in
$G$ consisting of those elements with $\rho(g^{\prime})=G$. Since multiplying
a configuration $\sigma_{e}$ by some element $g^{\prime}\in G^{\prime}$ at any
edge does not change the Hamiltonian, we see that this is a trivial gauge
invariance. We might as well consider the quotient group $G/G^{\prime}$ as our
gauge group. This $G/G^{\prime}$ has the property that the only element $g$
with $\rho(g)=1$ is the identity element.

Now let us return to our simplification of the Higgs boson action. We can make
the following map that preserves the action, but simplifies the Higgs boson
term. Let $\eta_{x}$ be the term such that $\rho(\eta_{x})=\phi_{x}$ (this is
unique since $\rho(g)=1$ only if $g=1$), then we see that under the map
$(\phi_{x},\sigma_{e})\to(1,\eta_{x}^{-1}\sigma_{e}\eta_{y})$ where $(x,y)$
are the endpoints corresponding to the edge $e$. We let the map
$\sigma_{e}\to\eta_{x}^{-1}\sigma_{e}\eta_{y}$ be $F_{\phi}$. For any choice
of $\phi$, this $F_{\phi}$ is a bijection between the set of gauge field
configurations $\sigma_{e}$ and itself.

This argument shows that when considering the expectation of a gauge invariant
function, such as a Wilson loop expectation, one can merely consider the
expectation with respect to the net Hamiltonian,

$H_{N,\beta,\kappa}=-\kappa\sum_{e\in E_{N}}\rho(\sigma_{e})-\beta\sum_{p\in
P_{N}}\rho((\text{d}\sigma)_{p}).$ (8)

This reduces the question of the Higgs boson to a slightly modified question
about pure gauge configurations. To some extent, the paper [9] only dealt with
case under the assumption that $\kappa$ is sufficiently large.

###  1.4 Extension of the Analysis of Wilson Loop Expectation

One can imagine multiple changes to the model that make the model proposed in
(8) substantially more difficult to analyze.

One question that one could ask is the following:

What happens if the group $H$ for the Higgs boson is not contained in the
image $\rho(G)$ of the gauge group?

The above question is well-defined for abelian groups with a 1-dimensional
representation. For more general groups and representations, the analogue is
follows. Let $\tilde{G}$ be the group of elements whose image under the
representation $\rho$ is a multiple of the identity, i.e. $g\in\tilde{G}$ if
$\rho(g)=cI$, where $c$ is some constant. We let $X$ be the group of these
coefficients $c$ from elements of $\tilde{G}$; since $\rho$ is a unitary
representation, $X$ is a subgroup of the unit circle. The analogue we have to
consider is that $H$ is not a subgroup of these elements $X$.

If $H$ were not contained in $X$, one immediately sees that one cannot
immediately gauge out the effects of the Higgs boson. If one now has to
consider a non-trivial Higgs boson group, one must also need to consider the
excitations of the Higgs field.

For example, if the Higgs field is constant, one automatically knows that the
lowest energy states are those that set $\sigma_{e}=1$ and any other
configuration on edges can be reasonably understood to be an exponentially
suppressed excitation.

However, if we do not start with a constant Higgs field, it may be possible
that we could get a lower energy configuration by setting $\sigma_{e}\neq 1$
instead of $\sigma_{e}=1$.

####  1.4.1 The Excitations of the Higgs field

To describe this phenomenon, we will first give a simple calculation. Consider
the case that $G=Z_{3}$ while $H=\\{e^{2\pi\text{i}\frac{k}{9}}\\}$ for $k$
between $0$ and $9$ and $\beta=0$ (or at least $\beta\ll\kappa$). Observe that
$H$ is isomorphic to $Z_{9}$.

In this case, one can apply a gauge transformation similar to one applied in
the previous subsection. However, $\rho(\sigma_{e})$ can only take values
$\\{1,e^{2\pi\text{i}\frac{3}{9}},e^{2\pi\text{i}\frac{6}{9}}\\}$ rather than
all of $H$. Thus, the furthest we are able to simplify the values of
$\phi_{x}$ under a gauge transformation are to
$\\{1,e^{2\pi\text{i}\frac{1}{9}},e^{2\pi\text{i}\frac{2}{9}}\\}$. Let
$\theta_{x}=\\{1,e^{2\pi\text{i}\frac{1}{9}},e^{2\pi\text{i}\frac{2}{9}}\\}$
be the simplified value of $\phi_{x}$ under the gauge transformation.

Now consider an edge $e=(x,y)$ such that
$\theta_{x}=e^{2\pi\text{i}\frac{2}{9}}$ and $\theta_{y}=1$. We see that the
maximum value of
$\exp[\theta_{x}\rho(\sigma_{e})\bar{\theta_{y}}+\bar{\theta_{x}}\rho(\sigma_{e}^{-1})\theta_{y}]$
is $\exp[2\cos(2\pi\frac{8}{9})]$ which occurs when
$\sigma_{e}=e^{2\pi\text{i}\frac{6}{9}}$. Observe that the maximum is not
$\exp[2]$, as one would get if $\theta_{x}=\theta_{y}$.

What this illustrates is the fact that if one fixes the configuration of the
Higgs boson at every lattice point, one could possibly have substantial
changes to the lowest energy configuration. Thus, in order to analyze the
Higgs boson model in general, one would necessarily need a method to analyze
the excitations of the Higgs boson beyond just reducing it to a case of a
modified pure gauge field. We will show that various scales of $\kappa$ will
change the behavior of the excitations of the Higgs boson. One will have to
treat these different regimes through different types of analysis.

###  1.5 The Low Disorder Regime: High $\kappa$

In the case that $\kappa$ is large, much like the case dealt with in [9], one
finds that the large value of $\kappa$ prevents excitations of $\sigma_{e}\neq
1$ for many edges. They can show that in this regime, one can reduce the
computation of Wilson loop expectations to the computation of the contribution
from minimal vortices, e.g. excitations that consist of single edge with
$\sigma_{e}\neq 1$ with all other edges in a neighborhood around it set to
$\sigma_{e}=1$.

When we introduce a Higgs boson field, we can assert a slightly similar
statement. Though it is not the case that, given a fixed Higgs boson field
configuration, the lowest energy configuration fixes all values of $\sigma$ to
$\sigma_{e}=1$, we do see that an edge $e=(x,y)$ with $\sigma_{e}\neq 1$ and
arbitrary Higgs boson values $\phi_{x}$,$\phi_{y}$ at the boundary vertices
will certainly have less energy at that edge than a configuration that sets
$\phi_{x}=\phi_{y}$ and $\sigma_{e}=1$. Thus, if one is willing to modify the
Higgs field in addition to the gauge field configuration, one may be able to
treat edges with $\sigma_{e}\neq 1$ as an excitation.

This is exactly what is done in Section 2. In this section, beyond just
generalizing the analysis of [9] to a more general Higgs boson group, we
simplify the analysis of [9] by providing a true polymer expansion. One
further consequence of this polymer expansion is that it helps express the
Wilson loop expectation as a computation involving an explicit Poisson random
variable. Roughly speaking, if we define the support of a configuration
$(\sigma,\phi)$ of a joint gauge field and Higgs field configuration as those
plaquettes that bound excited edges $e$ with $\sigma_{e}\neq 1$ or
$\phi_{x}\neq\phi_{y}$, then we derive the desired properties behind a true
polymer expansion. The details of the polymer expansion are given in
Definition 1. The main principle behind a polymer expansion is that it is easy
to analyze probabilities if one can always perform the following procedure. If
a configuration has support $P_{1}\cup P_{2}$, where $P_{1}$ and $P_{2}$ are
disjoint, then configuration can be bijectively split into a configuration
supported on $P_{1}$ and another supported on $P_{2}$.

There is a rather straightforward way to do this splitting for the gauge field
configurations $\sigma$, but one has to exert effort in order to split the
Higgs field configurations into the disjoint supports. Section 2 performs this
analysis in a toy case that $G=Z_{2}$ and $H=Z_{2}$, where the splitting is
easier to describe. The analysis for more general gauge group $G$ and Higgs
boson group $H$ is similar, but more cumbersome in the description of the
division.

We informally describe our main theorem in this section as follows.

######  Informal Theorem 1 (Informal Version of Theorems 1 and 3 of Section 2
).

For sufficiently large $\beta$ and $\kappa$, the main order contributions to
the Wilson loop observables are by the number of minimal vortices (roughly,
those plaquette exciations centered around a single edge with $\sigma_{e}\neq
1$) that are centered around edges of $\gamma$. Furthermore, the number of
minimal vortex excitations along the edges of $\gamma$ can be treated as
roughly a Poisson random variable with parameter $O(|\gamma|)$.

Define the quantity

$A_{\beta,\kappa}:=\frac{\sum_{g\neq
1}\rho(g)\exp[12\beta\text{Re}[(\rho(g)-\rho(1))]]\exp[2\kappa\text{Re}[\rho(g)-\rho(1)]]}{\sum_{g\neq
1}\exp[12\beta\text{Re}[(\rho(g)-\rho(1))]]\exp[2\kappa\text{Re}[\rho(g)-\rho(1)]]}$

$\mathbb{E}[W_{\gamma}]\approx\mathbb{E}[A_{\beta,\kappa}^{X}],$ (9)

where $X$ is a Poisson random variable with expectation
$|\gamma|\exp[12\beta\text{Re}[\rho(g)-\rho(1)]\exp[2\kappa\text{Re}[\rho(g)-\rho(1)]]$.

Wondrously, our polymer expansion is robust enough to treat the case of non-
abelian $G$ with few changes to the proof.

###  1.6 The High Disorder Regime: $\kappa$ Low

A difficulty in the case of small $\kappa$ is one can no longer consider edges
with $\sigma_{e}\neq 1$ as excitations of the configuration $(\sigma,\phi)$.
Instead, the predominant suppression to the probability is caused by excited
plaquettes with $\rho((\text{d}\sigma)_{p})\neq 0$. Due to this fact, we see
that it would be better to separate the gauge field excitations as follows. We
first create an auxiliary field $\eta:V_{N}\to G$ satisfying the property that
$\tilde{\sigma}_{e}=\eta_{x}\sigma_{e}\eta_{y}^{-1}$ has as ‘few’ nontrivial
edges with $\tilde{\sigma}_{e}\neq 1$ as possible. The effects of the Higgs
boson interaction on the gauge field configuration can be understood as a
fluctuation of the auxiliary field $\eta_{x}$. What follows is an informal
discussion of the procedure being described.

####  1.6.1 Informal Discussion

To do this, we introduce a spanning tree $T$ with base point $b$ of the
lattice $\Lambda$. Our goal is to find the auxiliary field $\eta:V_{N}\to G$
such that the field $\tilde{\sigma}_{e}=\eta_{x}^{-1}\sigma_{e}\eta_{y}$ for
$e=(x,y)$ satisfies some particular properties. The property we try to ensure
is the following: for each edge $e\in T$ we assign the value
$\tilde{\sigma}_{e}=1$.

The auxiliary field $\eta$ can be constructed inductively as follows. We first
set $\eta_{b}=1$. Now, for any other vertex $v$ on the lattice $\Lambda_{N}$,
let $p=(b=v_{0}),v_{1},v_{2},\ldots,(v_{n}=v)$ be the path on the spanning
tree $T$ connecting $b$ to $v$. Assume that we have already assigned the
values $\eta_{v_{0}}$ to $\eta_{v_{n-1}}$. Then, we choose
$\eta_{v_{n}}:=\sigma_{e}^{-1}\eta_{v_{n-1}}$, where $e$ is the edge
$(v_{n-1},v_{n})$. Clearly, with this choice of $\eta_{v_{m}}$, we have
$\tilde{\sigma}_{e}=1$ for $e=(v_{n-1},v_{n})$. Since $T$ is a spanning tree,
this fixes all values of $\eta$ on $V_{N}$. Finally, we can define
$\tilde{\sigma}$ as $\tilde{\sigma}_{e}=\eta_{x}^{-1}\sigma_{e}\eta_{y}$ for
$e=(x,y)$. We remark that in the course of the proof, $T$ will not be fixed
a-priori and will be chosen as is most appropriate for that part of the proof.

Let us give some heuristics for the partition function and other associated
quantities for this new Hamiltonian with the auxiliary field $\eta$. In the
computation of the partition function to highest order in $\beta$ (for $\beta$
sufficiently large and regardless of the specific value of $\kappa$), we can
ignore the fluctuation of $\tilde{\sigma}_{e}$ and set $\tilde{\sigma}_{e}=1$
for all edges $e$. The partition function can be treated as a sum over all
possible configurations of $\eta_{v}$ and $\phi_{v}$ for the Higgs boson.

Namely,
$Z\approx\sum_{\eta_{v},\phi_{v}}\exp[\sum_{e=(v,w)}\kappa\phi_{v}\text{Tr}[\rho(\eta_{v}\eta_{w}^{-1})]\phi^{-1}_{w}]$.
The configuration of the values of $\tilde{\sigma}_{e}$ are fixed to $1$,
since these excitations would be lower order in $\beta$.

When computing the probability of seeing a single excited minimal vortex
$P(e)$ in the Wilson action, the probability can be computed as a fraction
with some numerator and $Z$ as the denominator. To leading order, we see that
the numerator can roughly be expressed as a sum over configurations with
$\tilde{\sigma}_{e}^{\prime}=1$ for $e^{\prime}$ not the center of the single
excited vortex and $\sigma_{e}\neq 1$ for $e=(a,b)$ the center of the single
excited vortex. We see that it would be important to specifically consider the
values of $\eta_{a},\eta_{b},\phi_{a},\phi_{b}$.

We see that one way to write the numerator as,

$\displaystyle\sum_{\tilde{\sigma}_{e}\neq
1}\exp[12\beta(\text{Tr}[\rho(\tilde{\sigma}_{e})]-\text{Tr}[\rho(1)])]$ (10)
$\displaystyle\sum_{\hat{\eta}_{a},\hat{\eta}_{b},\hat{\phi}_{a},\hat{\phi}_{b}}\exp[\kappa\text{Re}[\hat{\phi}_{a}\text{Tr}[\rho(\hat{\eta}_{a}\tilde{\sigma}_{e}\hat{\eta}_{b}^{-1})]\hat{\phi}^{-1}_{b}-\hat{\phi}_{a}\text{Tr}[\rho(\hat{\eta}_{a}\hat{\eta}_{b}^{-1})]\hat{\phi}^{-1}_{b}]]$
$\displaystyle\sum_{R}\exp[\sum_{e=(v,w)}\kappa\phi_{v}\text{Tr}[\rho(\eta_{w}\eta_{w}^{-1})]\phi^{-1}_{w}],$

where $\sum_{R}$ denotes the sum over all elements not specified by
$\eta_{a},\eta_{b},\phi_{a},\phi_{b}$. Additionally, for the quantity inside
the second sum, we will use the convention $\phi_{a}=\phi_{a}$ and so on. The
reason we introduce this notation is to specifically indicate that we fix the
values of $\phi_{a}$ to $\hat{\phi}_{a}$ and so on.

Now, the ratio

$\frac{\sum_{R}\exp[\sum_{e=(v,w)}\kappa\phi_{v}\text{Tr}[\rho(\eta_{w}\eta_{w}^{-1})]\phi_{w}^{-1}]}{\sum_{\eta_{v},\phi_{v}}\exp[\sum_{e=(v,w)}\kappa\phi_{v}\text{Tr}[\rho(\eta_{v}\eta_{w}^{-1})]\phi_{w}^{-1}]}$
(11)

is the probability that under the Ising type interaction
$\phi_{v}\text{Tr}[\rho(\eta_{v}\eta_{w}^{-1})]\phi_{w}^{-1}$ on edges, we
will get
$\eta_{a}=\hat{\eta_{a}},\eta_{b}=\hat{\eta_{b}},\phi_{a}=\hat{\phi}_{a},\phi_{b}=\hat{\phi}_{b}$
for some specified values of
$\hat{\phi_{a}},\hat{\phi_{b}},\hat{\eta}_{a},\hat{\eta}_{b}$. These can be
considered to be some type of magnetization.

We see at low $\kappa$, we see that understanding the behavior of the $\eta$’s
and the $\phi$ fluctuations are non-trivial questions. Furthermore, we see
that the introduction of the $\eta$ and $\phi$ fields can lead to highly non-
trivial correlations.

From the example above, we can imagine what would happen if we consider a more
general family of excited plaquettes. For example, let us consider two excited
minimal vortices centered at edge $e_{1}=(a_{1},b_{1})$ and
$e_{2}=(a_{2},b_{2})$. To do this, we have to adjust the computation of the
numerator to specifically fix the values of $\eta_{a_{i},b_{i}}$ and
$\phi_{a_{i},b_{i}}$.

Namely, we see that we can better express the numerator as,

$\displaystyle\sum_{\tilde{\sigma_{e_{i}}}}\prod_{i}\exp[12\beta(\text{Re}[\rho(\tilde{\sigma_{e_{i}}})]-\text{Re}[\rho(1)])]$
(12)
$\displaystyle\sum_{\hat{\eta}_{a_{i},b_{i}},\hat{\phi}_{a_{i},b_{i}}}\prod_{i}\exp[\kappa\text{Re}[\hat{\phi}_{a_{i}}\text{Tr}[\rho(\hat{\eta}_{a_{i}}\tilde{\sigma}_{e_{i}}\hat{\eta}^{-1}_{b_{i}})]\hat{\phi}_{b_{i}}-\hat{\phi}_{a_{i}}\text{Tr}[\rho(\hat{\eta}_{a_{i}}\hat{\eta}^{-1}_{b_{i}})]\hat{\phi}_{b_{i}}]]$
$\displaystyle\sum_{R}\exp[\sum_{e=(v,w)}\kappa\phi_{v}\text{Tr}[\rho(\eta_{w}\eta_{w}^{-1})]\phi_{w}],$

where as before $\sum_{R}$ is the sum over variables that are not
$\hat{\eta}_{a_{i},b_{i}}$, $\hat{\phi}_{a_{i},b_{i}}$ while $\phi_{a_{i}}$ is
set equal to $\hat{\phi_{v}}$ in the second sum.

The important ratio we have to consider in this example is,

$\frac{\sum_{R}\exp[\sum_{e=(v,w)}\kappa\phi_{v}\text{Tr}[\rho(\eta_{w}\eta_{w}^{-1})]\phi_{w}]}{\sum_{\phi_{v},\eta_{v}}\exp[\sum_{e=(v,w)}\kappa\phi_{v}\text{Tr}[\rho(\eta_{w}\eta_{w}^{-1})]\phi_{w}]}.$
(13)

We see that the computation involves understanding the correlation between
$\hat{\eta}_{a_{i}},\hat{\eta}_{b_{i}}$ and
$\hat{\phi}_{a_{i}},\hat{\phi}_{b_{i}}$ at different sites. Though at small
$\kappa$ we would expect exponential decay of correlations, it is the very
presence of these correlations that makes it difficult to define a polymer
expansion in the case of small $\kappa$ purely from knowing the values of
$\sigma$ and $\phi$, even in the abelian case.

For simplicity of notation, let us consider the abelian case. The main issue
can be observed by looking at the following example. Consider a gauge field
configuration $\sigma_{1}$ such that the set of excited plaquettes , those
with $\text{d}(\sigma)_{p}\neq 0$ is a small set with size $k$, but the set of
edges with $\sigma_{e}$ with $\sigma_{e}\neq 0$ is much larger, say of
$O(k^{4})$. This is an obstruction to a polymer expansion.

Naively, what this means is that if we find another configuration $\sigma_{2}$
whose support contains one of these $O(k^{4})$ edges, then there will be
correlations between observing both $\sigma_{1}$ and $\sigma_{2}$. Namely,
$\mathbb{P}(\sigma_{1}\sigma_{2})$ is far different from
$\mathbb{P}(\sigma_{1})\mathbb{P}(\sigma_{2})$. This is in contrast to the
case of a pure abelian gauge field, where configurations can easily be split
as the set of excited plaquettes associated to $\sigma_{1}$ and $\sigma_{2}$
are disjoint.

For small $\kappa$, we found an auxiliary Hamiltonian based on the random
currents representation of the Ising model [8] that exactly allows us to
characterize when the effects of $\kappa$ cause two disjoint configurations to
be correlated with each other. The random current expansion introduces a new
field $I(e)$ for each edge $e$ and couples our original Hamiltonian,
$H_{N}(\sigma,\phi)$ to a new Hamiltonian $\mathcal{H}(\sigma,\phi,I)$ ,
equation (72), in the new variables $I$. One can define a polymer expansion
using these new variables, as in Definition 7. Unfortunately, even in the
abelian case, the introduction of the Higgs Field leads to knotting problems,
as in the non-abelian pure gauge field case [4]. Through this new polymer
expansion, we show that the Wilson loop expectation can be reduced to
understanding the effects of minimal vortices lying along the edges of
$\gamma$ in section 4. There is a simple argument with a worse error rate
showing that the number of minimal vortices along $\gamma$ can be treated as a
Poisson random variable in the end of the same section 4. However, by
carefully understanding the decorrelation between the contribution of the
Wilson loop action from different minimal vortices, we can improve our error
analysis. Roughly speaking, this involves relating quantities of the form (13)
to products of those of the form (11). This rather delicate task was performed
in Section 5.

We have the following informal version of our main result,

######  Informal Theorem 2 (Based on Theorems 4 and 5 of Section 4 and Theorem
9 of Section 5).

For $\beta$ sufficiently large and $\kappa$ sufficiently small, the main order
contribution to Wilson loop expectations come from minimal vortices centered
around the edges of $\gamma$ .

One can compute the Wilson loop expectation as,

$\mathbb{E}[W_{\gamma}]\approx\mathbb{E}[\text{Tr}[\mathcal{D}_{\beta,\kappa}^{X}]],$
(14)

where $\mathcal{D}_{\beta,\kappa}$ is a matrix defined in Theorem 7 and $X$ is
a Poisson Random Variable whose expecation is $O(|\gamma|)$.

##  2  The Low Disorder Regime: A Toy Model

###  2.1 Introduction and Informal Discussion

As we have discussed in the introduction, when the gauge group is not the same
as the symmetry group of the Higgs field, the fluctuations of the Higgs boson
will affect the lowest energy configurations of the gauge field. To illustrate
the main ideas in the low disorder regime(large $\kappa$), we consider the
following slightly simplified model.

On the lattice $\Lambda_{N}=[-N,N]^{4}$, we allow the Higgs boson $\phi_{p}$
at each lattice point $p\in\Lambda_{N}$ to take 2 values, either $+1$ or $-1$
(which we will later refer to $+$ and $-$, respectively, and will be called
charges). The gauge group $G$ of the gauge field $\sigma$ is $Z_{2}$; as such,
we will talk about setting edges $e$ to $\sigma_{e}=1$ or $\sigma_{e}=-1$. We
will also let $\rho$ be a 1-dimensional representation of $Z_{2}$.

We remark that since for all elements $g\ \in Z_{2}$, we have $g=g^{-1}$, we
do not need to concern ourselves with the orientations of edges or of
plaquettes. Thus, we do not need to take into account the orientation of
edges. Furthermore, since Higgs boson values $\phi$ are assigned to vertices
and gauge field values $\sigma$ are assigned to edges, we may use language as
‘assign + to a vertex v’ to unambigously mean assigning the Higgs boson value
$\phi_{v}=+$ at the vertex $v$ and ‘assign -1 to an edge e’ to mean assigning
the gauge field value $\sigma_{e},\sigma_{-e}=-1$ to either orientation of the
edge $e$.

####  2.1.1 Higgs boson Configuration

We start by describing, in words, the type of interaction between the gauge
group and the Higgs field. Consider an edge $e\in E_{N}$ of the lattice. If it
connects two points of the same charge, then assigning $\\{+1\\}$ to $e$ will
give energy $E_{1}$ and assigning $\\{-1\\}$ to $e$ will give energy
$E_{2}<E_{1}$. If instead, the edge $e$ connects two vertices of opposite
charges, then assigning $\\{-1\\}$ as the gauge field to edge $e$ will give
energy $E_{3}$ and assigning $\\{1\\}$ to the edge $e$ will give energy
$E_{4}<E_{3}$. The energies are ordered as follows

$E_{1}>E_{3},E_{2},E_{4}.$

We will have a standard Wilson loop energy of the form
$\beta\rho((\text{d}\sigma)_{p})$, where for the abelian group $Z_{2}$ we can
understand $(\text{d}\sigma)$ as the sum of all the gauge group elements along
the edges that bound $p$.

After subtracting an appropriate constant, we can formally write our
Hamiltonian as follows,

$H_{N,\beta,\kappa}(\phi,\sigma)=\sum_{e=(v,w)\in
E_{N}}\kappa[f(\sigma_{e},\phi_{v}\phi_{w}^{-1})-f(1,1)]+\beta\sum_{p\in
P_{N}}[\rho((\text{d}\sigma)_{p})-\rho(1)].$ (15)

where $f(1,1)=E_{1},f(1,-1)=E_{2},f(-1,1)=E_{3},f(-1,-1)=E_{4}$ and our
measure on the lattice is

$\mu_{N,\beta,\kappa}(\sigma,\phi)=\frac{1}{Z_{\Lambda,\beta}}\exp[H_{N,\beta,\kappa}(\sigma,\phi)].$
(16)

Here $Z_{\Lambda,\beta}$ is the partition function,

$Z_{\Lambda,\beta}=\prod_{v\in V_{N}}\sum_{\phi_{v}\in\\{-1,1\\}}\prod_{e\in
E_{N}}\sum_{\sigma_{e}\in Z_{2}}\exp[H_{N,\beta,\kappa}(\sigma_{,}\phi)].$
(17)

The action considered above is rather similar to the action in (6) if the
gauge group $G$ were chosen to be $Z_{2}$ and the Higgs boson group $H$ is
$Z_{4}$. In this case, one would only be able to gauge out the Higgs boson
field to take only two values ( $+$ or $-$ ) here; + corresponds to
$\phi_{x}=1$ or $-1$ while $-$ corresponds to $\phi_{x}=i$ or $\phi_{x}=-i$.
In this case, if there is an edge $e=(v,w)$ such that $\phi_{v}$ and
$\phi_{w}$ are assigned the same value, then the lowest energy configuration
would assign $\\{+1\\}$ to $\sigma_{e}$. By contrast, if there is an edge
$e=(v,w)$ such that $\phi_{v}$ and $\phi_{w}$ are assigned different values,
then assigning $\sigma_{e}$ to be $\\{-1\\}$ and $\\{+\\}$ would give the same
energy to the edge; this energy $E$ would be less than the maximum energy
assigned when $\phi_{v}$ and $\phi_{w}$ are the same sign and $\sigma_{e}$ is
assigned the value $\\{+1\\}$.

With the Hamiltonian in (15) and ignoring the effect of the Wilson action for
now, we expect the following behavior. Each of the sites would have the same
sign (either + or -), while we would expect that each of the gauge fields
would be assigned the value $\\{+1\\}$. The fact that we would expect nearly
all vertices to be assigned the same charge is due to the constraints
$E_{3},E_{4}<E_{1}$ and a Peierl’s argument.

Consider an assignment of Higgs boson charges in which the $\phi$’s are not
constant. WLOG, we can consider the case that there is a connected
neighborhood $N$ of vertices assigned a negative charge, but this neighborhood
$N$ is surrounded by an ocean of positive charge. We can let $E(N)$ be the
edges that connect $N$ to its complement $N^{c}$. We now argue that we can
obtain a less excited configuration by flipping the charges of the vertices in
$N$ and changing the assignment of all the edges in $E(N)$ to 1. Clearly, the
change in the energy of this configuration is given by at least
$\kappa|E(N)|(E_{1}-\max(E_{3},E_{4}))$, where $|E(N)|$ is the size of the set
$E(N)$. Thus, we expect that deviations of the Higgs field from the constant
will be suppressed exponentially for large enough $\kappa$.

The previous discussion shows one important concept. When dealing with a pure
gauge field, one only needs to considering fluctuations of the edges
$\sigma_{e}$ from the identity. This leads to an understanding of Wilson loops
expectations via the associated plaquette computations. Here, the fluctuations
comes in two ways; as before, we must still consider fluctuations of edges
from the low energy configurations, but we must further consider the
fluctuations of the Higgs field configurations from the identity.

####  2.1.2 Wilson Loop Action

As is standard, we want to compute the expected value of the Wilson loop
action as the size of the lattice goes to infinity. We consider a closed non-
intersecting loop $\gamma$ in $\Lambda_{N}$ consisting of edges
$e_{1},e_{2},\ldots,e_{m}$.

We can define the Wilson loop action as on a configuration
$\mathcal{C}=(\phi_{v},\sigma_{e})$ as

$W_{\gamma}(\mathcal{C})=\sum_{i=1}^{m}\rho(\sigma_{e})=\rho(\sum_{i=1}^{m}\sigma_{e}).$
(18)

We use the notation
$\langle\gamma,\sigma_{e}\rangle=\sum_{i=1}^{m}\sigma_{e}$.

Recall that on abelian groups we can consider $\sigma_{e}$ to be a one-form
supported on the edges, $E_{N}$; for reference, see the discussion in Section
3 of [4]. Thus, we can interpret $\langle\gamma,\sigma\rangle$ as an integral
on the set of $1$-forms. More importantly, we can apply Stokes’ theorem and
can find a surface $q$ such that $\delta q=\gamma$ and
$(q)_{p}\in\\{-1,0,1\\}$ for all plaquettes $p\in P_{N}$ such that

$\langle\gamma,\sigma\rangle=\sum_{p\in P_{N}}q_{p}(\text{d}\sigma)_{p}.$ (19)

When we write the Wilson loop action in terms of the integration of a two-form
over a surface, we see that we are able directly see the effect of nontrivial
plaquettes those with $(\text{d}\sigma)_{p}\neq 1$; later, we will see that
this decomposition will allow us to determine whether some excitation of the
Higgs and gauge field $\mathcal{C}$ would be independent of the Wilson action
on the loop.

In the next section, we will start rigorously describing some notions we can
use to characterize fluctuations of the Higgs field and the gauge field.

###  2.2 Rigorous Definitions

There is a trivial symmetry that preserves the Hamiltonian; we flip the signs
of all the Higgs boson configurations ( ’+’ to ’-’ and ’-’ to ’+’), while the
gauge field configurations $\sigma_{e}$ are unchanged. For some simplicity in
the proof, we may assume that $N$ is odd. When $N$ is odd, we can always apply
a global flip of the Higgs boson charges to get a unique configuration
satisfying the condition that $\sum_{v\in V_{N}}\phi_{v}>0$. (All this
condition means is that the majority of charges are $+$ instead of $-$.) At
this point, we can start giving definitions of the types of excitations we
will consider so that we can consider a cluster expansion.

######  Definition 1.

[Support of Configurations]

We define $\mathcal{C}_{e}$, our excited edges, as

$\mathcal{C}_{e}=\\{e=(v,w)\in E_{N}:\phi_{v}\neq\phi_{w}\\}\cup\\{e\in
E_{N}:\sigma_{e}=-1\\}.$

The support of our configuration is

$\text{supp}(\mathcal{C})=\\{p\in P_{N}:\exists e\in\mathcal{C}_{e}\text{ s.t.
}e\in\delta p\\}.$ (20)

######  Remark 1.

Without the Higgs field action, an edge with $\sigma_{e}\neq 1$ is not
necessarily excited. For example, one could find a plaquette whose boundary
edges are all $-1$. This plaquette would not be of lower energy in the
Hamiltonian and we would not consider this plaquette to be excited. By
contrast, with the Higgs action, a single edge with $\sigma_{e}=-1$ drives it
to lower energy.

######  Lemma 1.

Consider a configuration $\mathcal{C}=(\sigma_{e},\phi_{v})$ If $p$ is a
plaquette such that $(\text{d}\sigma)_{p}\neq 1$, then
$p\in\text{supp}(\mathcal{C})$.

###### Proof.

Clearly, if $(\text{d}\sigma)_{p}\neq 1$, then there clearly is some edge $e$
in the boundary of $p$ such that $\sigma_{e}\neq 1$. Thus, the only non-
trivial computations to the probability value from our clusters come from what
we define to be the support of our distribution. ∎

We will now define the function $\Phi(P)$ where $P$ is a set of plaquettes in
$P_{N}$; this is a crucial part of our cluster expansion.

######  Definition 2.

We let $E(P)$ denote the set of edges $e$ such that there exists a plaquette
$p\in P$ with $e\in\delta p$. Similarly, we let $V(P)$ denote the set of
vertices such that there is a plaquette $p\in P$ such that $v$ is a boundary
vertex of $p$.

$\displaystyle\Phi(P)=\sum_{\begin{subarray}{c}\mathcal{C}=(\sigma_{e},\phi_{v})\\\
\text{supp}(\mathcal{C})=P\\\ \sum_{v}\phi_{v}>0\end{subarray}}$
$\displaystyle\prod_{p\in\text{supp}(\mathcal{C})}\exp[\beta(\rho((\text{d}\sigma)_{p})-\rho(1))]$
(21) $\displaystyle\prod_{\begin{subarray}{c}e\in E(P)\\\
e=(v,w)\end{subarray}}\exp[\kappa(f(\sigma_{e},\phi_{v}\phi_{w}^{-1})-f(1,1))].$

We also have a similar cluster expansion formula incorporating the effects of
the Wilson action.

$\displaystyle\Phi_{W}(P)=$
$\displaystyle\sum_{\begin{subarray}{c}\mathcal{C}=(\sigma_{e},\phi_{v})\\\
\text{supp}(\mathcal{C})=P\\\
\sum_{v}\phi_{v}>0\end{subarray}}\prod_{p\in\text{supp}(\mathcal{C})}\exp[\beta(\rho((\text{d}\sigma)_{p})-\rho(1))]$
(22) $\displaystyle\times\prod_{\begin{subarray}{c}e\in E(P)\\\
e=(v,w)\end{subarray}}\exp[\kappa(f(\sigma_{e},\phi_{v}\phi_{w}^{-1})-f(1,1))]\prod_{e\in
E(P)\cap\gamma}\rho(\sigma_{e}).$

######  Remark 2.

Though after this section, we will not use this specific definition of
$\Phi(P)$, we will use $V(P)$ and $E(P)$ frequently throughout the course of
this paper.

Our second definition gives us criterion to determine whether our clusters
would interact with each other, or could be considered to split from each
other.

######  Definition 3.

We define the following graph $G_{2}$ with vertex set $P_{N}$ as follows. We
say that there is an edge between two plaquettes $p_{1}$ and $p_{2}$ if there
exists a $3$-cell such that both $p_{1}$ and $p_{2}$ are on the boundary of
said $3$-cell.

We call a set of plaquettes $V$ a vortex if the plaquettes of $V$ form a
connected set in the graph $G_{2}$.

Furthermore, we say that two sets of plaquettes $P_{1}$ and $P_{2}$ are
compatible if the set $P_{1}$ and $P_{2}$ are disconnected in $G_{2}$.
Otherwise, we say that $P_{1}$ and $P_{2}$ are incompatible.

The decomposition of some plaquette set $P$ into its maximal connected
subcomponents $P=V_{1}\cup V_{2}\ldots\cup V_{N}$ is called a vortex
decomposition of $V$. It is clear to see that $V_{i}$ and $V_{j}$ are
compatible for any distinct pair $i$ and $j$.

We have the following Lemma that allows us compute the function $\Phi(P)$ in
terms of $\Phi$ evaluated on the elements of its vortex decomposition.

######  Lemma 2.

Let $P_{1}$ and $P_{2}$ be two compatible sets of plaquettes. Then,

$\Phi(P_{1}\cup P_{2})=\Phi(P_{1})\Phi(P_{2}).$ (23)

As a consequence, if $V_{1}\cup V_{2}\ldots\cup V_{N}$ is a vortex
decomposition of $P$, then we have,

$\Phi(V_{1}\cup V_{2}\ldots\cup
V_{N})=\Phi(V_{1})\Phi(V_{2})\ldots\Phi(V_{N}).$ (24)

The function $\Phi_{W}$ would satisfy a similar property; namely,

$\Phi_{W}(V_{1}\cup V_{2}\ldots\cup
V_{N})=\Phi_{W}(V_{1})\Phi_{W}(V_{2})\ldots\Phi_{W}(V_{N}).$ (25)

###### Proof.

Our goal is to find a bijection
$\mathcal{C}=(\sigma_{e},\phi_{v})\to(\mathcal{C}_{1}=(\sigma^{1}_{e},\phi^{1}_{v}),\mathcal{C}_{2}=(\sigma^{2}_{e},\phi^{2}_{v}))$
where $\mathcal{C}$ is a configuration whose support is $P_{1}\cup P_{2}$ and
$\mathcal{C}_{i}$ is a configuration whose support is $P_{i}$ that satisfies
the following equation.

$\displaystyle\prod_{p\in\mathcal{C}}\exp[\beta(\rho((\text{d}\sigma)_{p}-\rho(1))]\prod_{\begin{subarray}{c}e\in
E(P_{1}\cup P_{2})\\\
e=(v,w)\end{subarray}}\exp[\kappa(f(\sigma_{e},\phi_{v}\phi_{w}^{-1})-f(1,1))]=$
(26)
$\displaystyle\prod_{p_{1}\in\mathcal{C}_{1}}\exp[\beta(\rho((\text{d}\sigma)_{p_{1}}-\rho(1))]\prod_{\begin{subarray}{c}e\in
E(P_{1})\\\
e_{1}=(v_{1},w_{1})\end{subarray}}\exp[\kappa(f(\sigma_{e_{1}},\phi_{v_{1}}\phi_{w_{1}}^{-1})-f(1,1))]$
$\displaystyle\times\prod_{p_{2}\in\mathcal{C}_{2}}\exp[\beta(\rho((\text{d}\sigma)_{p_{2}})-\rho(1))]\prod_{\begin{subarray}{c}e_{2}\in
E(P_{2})\\\
e_{2}=(v_{2},w_{2})\end{subarray}}\exp[\kappa(f(\sigma_{e_{2}},\phi_{v_{2}}\phi_{w_{2}}^{-1})-f(1,1))].$

We will assume that $P_{2}$ is a vortex for simplicity.

Recall that $V(P_{1})$ and $V(P_{2})$(resp. $E(P_{1})$ and $E(P_{2})$) denote
the set of vertices(resp. edges) that form a boundary vertex(resp. edge) of
some plaquette in $P_{1}$ and $P_{2}$.

Consider a configuration $\mathcal{C}$ whose support is $P_{1}\cup P_{2}$. Let
$e$ be an edge in the complement of $E(P_{1}\cup P_{2})$. Then, $\sigma_{e}=1$
for this configuration; otherwise, if $\sigma_{e}\neq 1$, then a plaquette
that contains $e$ as a boundary edge would be in the support of $\mathcal{C}$
by our definition of support. This would imply $e\in E(P_{1}\cup P_{2})$,
which is a contradiction.

Now consider an edge $e\in E(P_{1})\cap E(P_{2})$. If $\sigma_{e}\neq 1$, then
all the plaquettes $p$ that contain $e$ as a boundary edge are in
$\text{supp}(\mathcal{C})$ and are connected to each other in $G_{2}$.
However, since $e$ is in $E(P_{1})$, at least one of these plaquettes must be
in $P_{1}$. For the same reason, one of these plaquettes must also be in
$P_{2}$. However, this implies that $P_{1}$ and $P_{2}$ are connected to each
other; this is a contradiction.

Thus, there is a simple way to describe the gauge field configurations
$\sigma_{e}$ for the configurations $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$.
For edges $e$ in $E(P_{1})$, we will set $\sigma^{1}_{e}=\sigma_{e}$,
$\sigma^{2}_{e}=1$. for edges $e$ in $E(P_{2})$, we will set
$\sigma^{1}_{e}=1$ and $\sigma^{2}_{e}=\sigma_{e}$. For edges $e$ not in
either $E(P_{1})$ or $E(P_{2})$, we merely set $\sigma_{e}=1$. Note that this
is well defined since for edges $e\in E(P_{1})\cap E(P_{2})$, we know that
$\sigma_{e}=1$.

It is more complicated to describe the map on the Higgs field. Just as in the
Ising model, one can expect to see islands of charges that include each other.
Our procedure for mapping the Higgs field configurations from $\mathcal{C}$ to
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ does not merely involve applying a
restriction map as we have done for the edges. Instead, one must apply
appropriate charge flips in order to ensure the gauge constraints
$\sum\phi^{1}_{v}\geq 0,\sum\phi^{2}_{v}\geq 0$ are satisfied.

We will describe the problem formally as follows. Divide $V(P_{1}\cup
P_{2})^{c}$ into connected components as $B_{1}\cup B_{2}\cup\ldots\cup B_{N}$
and, similarly, $V(P_{i})^{c}$ as $B_{1}^{i}\cup B_{2}^{i}\cup\ldots\cup
B_{N_{1}}^{i}$. We see from Lemma 19 in the Appendix that each set $B_{i}$ is
monocharged. If we propose an charge assignment $\mathcal{C}_{1}$ with support
$P_{1}$, then we too must make sure that each of the sets $B_{i}^{1}$ are
monocharged.

Note that each set $B_{i}^{1}$ can be decomposed as follows $B_{i}^{1}\subset
V(P_{2})\cup B_{i_{1}}\cup\ldots B_{i_{m_{i}}}$, where the union is minimal in
$|m_{i}|$. If it were the case that all the sets $B_{i_{k}}$ were assigned the
same charge, e.g. $c_{i_{l}}=c_{i_{1}}$ for all
$l\in\\{1,\ldots,m_{i_{1}}\\}$, then we would be able to safely assign the
charge $c_{i_{1}}$ to all vertices in $B_{i}^{1}$ in the configuration
$\mathcal{C}_{1}$.

The problem occurs when there are two components $B_{i_{k_{1}}}$ and
$B_{i_{k_{2}}}$ in the decomposition of $B^{1}_{i}$ that are assigned two
different charges $c_{i_{k_{1}}}\neq c_{i_{k_{2}}}$. For simplicity of
notation, WLOG, we are considering the set $B^{1}_{1}$ with two subcomponents
$B_{1}$ and $B_{2}$ of different charges. For simplicity of later description,
let us also assume that $B_{1}$ and $B_{2}$ do not contain a boundary vertex
of $V_{N}$.

In the remainder of this proof, we will frequently use notation from Lemma 20
in the Appendix. Let $\mathcal{V}(B_{1})$ and $\mathcal{V}(B_{2})$ be as in
Lemma 20, the set of vertices connected to $B_{1}$ and $B_{2}$ respectively,
having the same charge. From the same Lemma, the exterior boundaries
$EB(\mathcal{V}(B_{1}))$ and $EB(\mathcal{V}(B_{2}))$ are connected sets.
Thus, $EB(\mathcal{V}(B_{i}))$ must be entirely contained in either $P_{1}$ or
$P_{2}$. If both of these external boundaries were in $P_{1}$, then this would
imply that $B_{1}$ and $B_{2}$ would be in different connected components in
the splitting of $B_{i}^{1}$. Therefore, it must be the case that one of the
external boundaries, say $EB(\mathcal{V}(B_{2}))$ was in $P_{2}$.

In this case, what one can do is the following. We can embed our lattice
inside $\mathbb{Z}^{d}$. In the full lattice $\mathbb{Z}^{d}$, find the set
$S$ containing infinity whose unique(internal) boundary is $EB(V_{2})$. Now
consider the complement of $S$ in $\mathbb{Z}^{d}$. We see that
$\mathcal{V}(B_{2})$ is a subset of $S^{c}$. Now, what one can do is to
completely flip the signs of all vertices in $S^{c}$. Under this
transformation, the only edges whose Hamiltonian action value changes are
those edges in $\mathcal{E}(S^{c})$. However, all of these edges must belong
to $E(P_{2})$ rather than $E(P_{1})$. Thus, this flip applied to $\mathcal{C}$
will not change the Hamiltonian action when restricted to edges of
$E(P_{2})^{c}\cup E(P_{1})$. As a consequence of this, if we consider an edge
connecting a vertex of $V(P_{1})$ to its complement, then this edge will still
have its adjacent vertices assigned the same charge even after this flip.
Furthermore, this flip changes the charges so that the charges of $B_{2}$ will
match those of its closest neighbors in $S^{c}$.

Through a careful ordering, one can come up with a series of charge flips that
do not affect the Hamiltonian action’s value on $E(P_{2})^{c}\cup E(P_{1})$
and will ensure that for each component $B^{1}_{i}\subset V(P)\cup
B_{i_{1}}\cup B_{i_{2}}\cup\ldots\cup B_{i_{i_{m}}}$ the charges associated to
the $B_{i_{l}}$’s are the same. Intuitively, this procedure involves flipping
the outermost island, and then further applying flips to correct the internal
islands. At this point, one can assign this charge to all the vertices in
$B^{1}_{i}$ in $\mathcal{C}_{1}$. When restricted to edges in
$E(P_{2})^{c}\cup E(P_{1})$, the Hamiltonian actions of $\mathcal{C}_{1}$ and
$\mathcal{C}$ are the same. If after this procedure $\sum\phi_{v}<0$, then one
can perform a global flip of the Higgs boson charge.

Through this procedure, we get a configuration $\mathcal{C}_{1}$ whose
Hamiltonian action on the set of edges $E(P_{2})^{c}\cup E(P_{1})$ are the
same as those of $\mathcal{C}$. We can similarly construct $\mathcal{C}_{2}$.
Furthermore, if we consider an edge in $(E(P_{1})^{c}\cup
E(P_{2}))\cap(E(P_{2})^{c}\cup E(P_{1}))$, then the action of this edge is
trivial; the vertices attached to it are assigned the same color and
$\sigma_{e}=1$.

One can see for this map $\mathcal{C}\to(\mathcal{C}_{1},\mathcal{C}_{2})$,
the desired relationship (26) holds. In addition, given two different
configurations $(\mathcal{C}_{1},\mathcal{C}_{2})$ supported on $P_{1}$ and
$P_{2}$ respectively, one can reverse the construction here to get a
configuration $\mathcal{C}$ whose support is $P_{1}\cup P_{2}$. This procedure
is fairly tedious, but is still based on finding the appropriate sets to flip
iteratively until one gets the desired configuration. The construction of this
bijection completes the proof of Lemma 2.

∎

In summary, the previous lemma has shown that the cluster expansion splits
based on a vortex decomposition. We will now present the link between our
cluster expansion function $\Phi$ and probabilities under our Hamiltonian.

######  Lemma 3.

Let $Z_{\Lambda,\beta}$ be the partition function associated to the
Hamiltonian (15) on the lattice $\Lambda$. We have the following relations,

$Z_{\Lambda,\beta,\kappa}=2\sum_{P\subset P_{N}}\Phi(P),$ (27)
$P(\text{supp}(\mathcal{C})=P)=\frac{\Phi(P)}{\sum_{P\subset P_{N}}\Phi(P)}.$
(28)

###### Proof.

In the sum that appears in the definition of $Z_{\Lambda,\beta,\kappa}$, one
can first sum over all subsets $P$ that could possibly be a support of the
configuration, and then over all configurations $\mathcal{C}$ that have $P$ as
its support. After possibly applying a trivial global flip to ensure that
$\sum_{v\in V_{N}}\phi_{v}>0$, the first equation is merely the definition of
$\Phi(P)$. The global flip is the reason for the existence of the factor of
$2$ on the right hand side of equation (27). The second equation comes from
the definition of the probability distribution and the previous equation (27)
on $Z_{\Lambda,\beta,\kappa}$. Note that we have implicitly canceled a factor
of $2$ that comes from the global flip symmetry. ∎

###  2.3 Analysis of the Wilson Loop Action

Our ultimate goal in this section is to argue that the main contribution to
the Wilson loop expectations come from minimal vortices, $P(e)$.

######  Definition 4 (Minimal Vortices).

Minimal vortices $P(e)$ have the following structure: they consist of the edge
$e$ and the $2(d-1)$ plaquettes that use the edge $e$ as a boundary edge. This
should be thought of as the smallest excitations, as the easiest way to create
a minimal vortex is to excite the single edge $\sigma_{e}\neq 1$ and to set
all other surrounding edges, $e^{\prime}$, to $\sigma_{e^{\prime}}=1$.

######  Remark 3.

Minimal vortices will be a common structure appearing in the previous works
[4] and [9] and will appear in later sections as well. We remark that in this
section, that a configuration $\mathcal{C}$ that has a minimal vortex $P(e)$
in its support must assign all the vertices of $V(P(e))$ the same Higgs boson
charge. Otherwise, one would be able to find at least $2$ edges in $V(P(e))$
connecting opposite charges. One of these edges will be $e^{\prime}\neq e$.
Under our definition of support, all of the plaquettes that use $e^{\prime}$
as a boundary vertex will be in the support of $\mathcal{C}$. This means that
$P(e)$ cannot be a vortex in the support of $\mathcal{C}$. Furthermore, all
edges $e^{\prime}$ in $E(P(e))$ aside from $e$ itself must be assigned
$\sigma_{e^{\prime}}=1$.

We let $W_{\gamma}$ represent our Wilson loop action on the loop $\gamma$. In
addition, we let $S_{\gamma}$ be a surface whose boundary is $\gamma$.

The analysis in this section depends on describing which plaquette
configurations form the main contribution to the Wilson loop equation and
bounding away the contributions of rare configurations by the function $\Phi$
constructed earlier.

####  2.3.1  Trivial Configurations

One difference from the case of a pure gauge field is that there are excited
configuration $\mathcal{C}$ whose support would intersect the loop $\gamma$,
but would not have any contribution to the Wilson loop action. In this
subsection, we find some we give some conditions on the configuration which
will ensure that the configuration will not have any contribution to the
Wilson loop action. By ‘removing’ the contribution of these trivial
configurations, we can improve our error analysis later.

Our first definition describes a geometric condition for a vortex $V$ to have
no contribution to the Wilson loop action.

######  Definition 5.

We call a vortex $V$ non-contributing to the Wilson loop action $\gamma$ if it
satisfies one of the following two properties:

  * •

There is no edge $e\in\gamma$ that is simultaneously a boundary edge of some
plaquette $p\in V$.

  * •

It is a minimal vortex not centered around an edge $e\in\gamma$.

We will show later that if a vortex is non-contributing then, as its name
states, it will not have a contribution to the Wilson loop action.

To appropriately account for the Wilson loop contribution in orders of
$e^{-\beta}$, we need to describe other configurations $\mathcal{C}$ that are
not immediately removed due to the geometric condition above. In order to do
this, we start by first describing a modification of $\phi$ that only takes
into account the ’nontrivial’ plaquette excitations with respect to the Wilson
loop functional.

######  Definition 6.

We call a configuration $\mathcal{C}=(\phi_{v},\sigma_{e})$ Wilson loop non-
trivial if there is a plaquette $p$ in $\text{supp}\text{ }\mathcal{C}$ such
that there is a boundary edge in $p$ that intersects the loop $\gamma$ and
there is another plaquette $p^{\prime}$ in $\text{supp}\text{ }\mathcal{C}$
with $(\text{d}\sigma)_{p^{\prime}}\neq 1$. A configuration that is not Wilson
loop non-trivial is Wilson loop trivial.

We define the functional $\phi_{NT}(V)$ which takes values on sets of
plaquettes as follows,

$\displaystyle\Phi_{NT}(P)=$
$\displaystyle\sum_{\begin{subarray}{c}\mathcal{C}=(\sigma_{e},\phi_{v})\\\
\text{supp}(\mathcal{C})=P\\\ \sum_{v}\phi_{v}>0\\\ \mathcal{C}\text{ is
Wilson Loop
Nontrivial}\end{subarray}}\prod_{p\in\text{supp}(\mathcal{C})}\exp[\beta(\rho((\text{d}\sigma)_{p})-\rho(1))]$
(29) $\displaystyle\times\prod_{\begin{subarray}{c}e\in E(P)\\\
e=(v,w)\end{subarray}}\exp[\kappa(f(\sigma_{e},\phi_{v}\phi_{w}^{-1})-f(1,1))].$

The purpose of introducing the above Definition 6 is to separate out those
configurations which will have a non-trivial effect on the Wilson loop action
$W_{\gamma}$. Namely, there needs to be some plaquette with
$(\text{d}\sigma)_{p}\neq 1$ in order to ensure that $W_{\gamma}\neq 1$.

######  Remark 4.

We remark that if there is a configuration $\mathcal{C}$ with support
$P_{1}\cup P_{2}$ with $P_{1}$, $P_{2}$ compatible such that $\mathcal{C}$ is
Wilson loop non-trivial when restricted to $P_{2}$, then after the splitting
construction in Lemma 2, then the configuration $\mathcal{C}_{2}$ is Wilson
loop non-trivial on $P_{2}$. This is due to the fact that our construction is
merely a restriction map applied to the gauge field configuration $\sigma$.
This allows us to establish multiplicative analogues of our result in 2 with
$\Phi_{NT}$. For example, the sum of $\exp[H_{N,\beta,\kappa}(\sigma,\phi)]$
over configurations with support $P_{1}\cup P_{2}$ and are Wilson loop non-
trivial when restricted to $P_{2}$ is $\Phi(P_{1})\Phi_{NT}(P_{2})$.

The following lemma takes into account both the sources of trivialities to the
Wilson loop action.

######  Lemma 4.

Consider a vortex $V$ whose associated edge set $E(V)$ has no edge in common
with $\gamma$. Then,

$\mathbb{E}[W_{\gamma}|\text{supp}\text{ }\mathcal{C}=V]=1.$ (30)

For similar reasons, we also have the following statement.

$\mathbb{E}[W_{\gamma}|\mathcal{C}\text{ is Wilson Loop Trivial}]=1.$ (31)

###### Proof.

Due to our definition of support, it is clear that if $V$ is a vortex whose
edge set $E(V)$ shares no edge with $\gamma$, then $\sigma_{e}=1$ for all
edges $e\in\gamma$. Otherwise, if $\sigma_{e}\neq 1$ for an edge in $\gamma$,
then the plaquettes surrounding that edge would be in the support of our
configuration. This would imply that $E(V)$ contains an edge of $\gamma$.
Finally, it is clear that if $\sigma_{e}=1$ for all edges of $\gamma$, then
the Wilson loop action $W_{\gamma}$ is clearly 1. This deals with the first
case of non-contributing vortex from Definition 5.

The second case is that the non-contributing vortex is minimal $P(e)$, but the
central edge $e$ is not in $\gamma$. From our discussion in 3, we know that
all edges $e^{\prime}\in E(P(e))$ that are not $e$ itself are set to $1$.
Again, this would imply that $\sigma_{e}$ is $1$ along all edges of $\gamma$
and, furthermore, the Wilson loop action is $1$. This deals with the case of
(30).

Now, we deal with (31). Note that if a configuration $\mathcal{C}$ is Wilson
loop trivial then either $E(\text{supp}\text{ }\mathcal{C})$ will not contain
an edge of $\gamma$ or $(\text{d}\sigma)_{p}=1$ for all plaquettes $p\in
P_{N}$. We have already dealt with the former possibility in the last
paragraph. We now deal with the second possibility.

A benefit of the abelian case is that we can rewrite our Wilson loop
expectation as the integral of $\gamma$ with respect to a $1$-from $\sigma$.
As we have mentioned earlier, Stokes’ theorem allows us to express this Wilson
loop integral as the integral of a surface $S_{\gamma}$ whose boundary is
$\gamma$ with respect to the $2$-from $(\text{d}\sigma)$. Now, if
$(\text{d}\sigma)_{p}=1$ for all $p\in P_{N}$, then clearly the integral
$\langle S_{\gamma},\text{d}\sigma\rangle=1$. Thus, the Wilson loop action is
still $1$. This completes the proof of equation (31).

∎

####  2.3.2 The main term contribution

The following lemma describes the contribution to the Wilson loop action from
the minimal vortices. These should be the simplest excitations.

######  Lemma 5.

Consider an excitation $\mathcal{C}$ and let $\text{supp}\text{
}\mathcal{C}=V_{1}\cup V_{2}\cup\ldots V_{N}$ be the vortex decomposition of
$\text{supp}\text{ }\mathcal{C}$.

We say that $\mathcal{C}$ belongs to the event $E$ if satisfies certain
properties regarding the vortices in the support. To be in the set $E$, at
least one of the following holds for each $i$:

  * •

The vortex $V_{i}$ is minimal and is centered around an edge $e$ in $\gamma$ .

  * •

$V_{i}$ is non-contributing to the Wilson loop action.

  * •

$V_{i}$ is not non-contributing to the Wilson loop action, but $\mathcal{C}$
restricted to $V_{i}$ is Wilson loop trivial.

Restricted to the event $E$, we have

$\mathbb{E}[W_{\gamma}|E]=\rho(-1)^{M_{\gamma}},$ (32)

where $M_{\gamma}$ is the number of minimal vortices centered along an edge of
$\gamma$.

###### Proof.

By definition,

$\mathbb{E}[W_{\gamma}|\text{supp}(\mathcal{C})=P]=\frac{\Phi_{W}(P)}{\Phi(P)},$

.

Consider a decomposition into compatible sets of plaquettes of the form,
$\mathcal{V}_{1}\cup\mathcal{V}_{2}\cup\mathcal{V}_{3}$, where each
$\mathcal{V}_{i}$ has the vortex decomposition
$\mathcal{V}_{1}=V^{1}_{1}\cup\ldots\cup V^{1}_{a}$,
$\mathcal{V}_{2}=V^{2}_{1}\cup\ldots\cup V^{2}_{b}$,
$\mathcal{V}_{3}=V^{3}_{1}\cup\ldots\cup V^{3}_{c}$.

Define $E_{\mathcal{V}_{1},\mathcal{V}_{2},\mathcal{V}_{3}}$ to be the set of
configurations $\mathcal{C}$ on which $\mathcal{V}_{1}$ is non-contributing
$\mathcal{V}_{2}$ are minimal and centered on some edge of $\gamma$, and
$\mathcal{V}_{3}$ are not non-contributing, but $\mathcal{C}$ is trivial on
$\mathcal{V}_{3}$. Assume that
$E_{\mathcal{V}_{1},\mathcal{V}_{2},\mathcal{V}_{3}}$ is non-empty.

We explicitly see that,

$\displaystyle\mathbb{E}[W_{\gamma}|\mathcal{C}\in
E_{\mathcal{V}_{1},\mathcal{V}_{2},\mathcal{V}_{3}}]$
$\displaystyle=\frac{\prod_{i=1}^{a}\Phi(V^{1}_{i})\prod_{j=1}^{b}\Phi_{W}(V^{2}_{j})\prod_{k=1}^{c}[\Phi(V^{3}_{k})-\Phi_{NT}(V^{3}_{k})]}{\prod_{i=1}^{a}\Phi(V^{1}_{i})\prod_{j=1}^{b}\Phi(V^{2}_{j})\prod_{k=1}^{c}[\Phi(V^{3}_{k})-\Phi_{NT}(V^{3}_{k})]}$
(33) $\displaystyle=\rho(-1)^{|\mathcal{V}_{2}|}.$

Noting that
$E=\cup_{\begin{subarray}{c}\mathcal{V}_{1},\mathcal{V}_{2},\mathcal{V}_{3}\\\
\text{all
compatible}\end{subarray}}E_{\mathcal{V}_{1},\mathcal{V}_{2},\mathcal{V}_{3}}$,
a disjoint union, this gives the result (32) after a removal of the
conditioning.

∎

####  2.3.3 Bounding the probability of large vortices

The following lemma bounds the probability of observing a vortex $V$ in the
support by the functions $\Phi$ we have defined earlier.

######  Lemma 6.

Let $V$ be a vortex of cardinality $|V|=m$, where the cardinality counts the
number of plaquettes in $V$. Then, we have the following equation,

$\mathbb{P}(V\text{is a vortex }\in\text{supp}\text{
}\mathcal{C})\leq\Phi(V).$ (34)

Similarly,

$\mathbb{P}(V\text{ is a vortex }\in\text{supp}\text{
}\mathcal{C},\mathcal{C}\text{ is non-trivial on }V)\leq\Phi_{NT}(V).$ (35)

###### Proof.

Let $\mathcal{P}_{V}$ be the set of possible supports of configurations
$\mathcal{C}$ that include $V$ in the vortex decomposition. If $P$ is a vortex
decomposition $\mathcal{P}_{V}$, we use $P\setminus V$ to denote the remaining
vortices. We remark that $P\setminus V$ is still a legitimate vortex
decomposition corresponding to some configuration $\mathcal{C}$. By applying
Lemma 2, we see that,

$\sum_{P\in\mathcal{P}_{V}}\Phi(P)=\sum_{P\in\mathcal{P}_{V}}\Phi(V)\Phi(P\setminus
V).$ (36)

Since each vortex decomposition $p\setminus V$ can be constructed to be the
support of some configuration $Z$, we see that we immediately have the
inequality,

$Z\geq 2\sum_{P\in\mathcal{P}_{V}}\Phi(P\setminus V).$ (37)

Thus,

$\displaystyle\mathbb{P}(V\text{ is a vortex of }\text{supp}\text{
}\mathcal{C})$ $\displaystyle=\frac{2\sum_{P\in\mathcal{P}_{V}}\Phi(P)}{Z}$
(38) $\displaystyle\leq\Phi(V)\frac{\sum_{P\in\mathcal{P}_{V}}\Phi(P\setminus
V)}{\sum_{P\in\mathcal{P}_{V}}\Phi(P\setminus V)}=\Phi(V),$

as desired.

For the second part of the theorem, one can see that we can count the
contribution of excitations whose support contains $V$ but is non-trivial on
$V$ as

$2\Phi_{NT}(V)\sum_{P\in P_{V}}\Phi(P\setminus V).$ (39)

This is due to the fact that the construction in Lemma 2 merely projects the
value of $\sigma_{e}$ to its appropriate subset. Thus, if
$(\text{d}\sigma)_{p}\neq 1$ for some plaquette, it will hold true for the
appropriate projection. Since we assumed $\mathcal{C}$ was nontrivial on $V$,
the same must hold true for the projection to $V$.

∎

Our next lemma uses our previous lemma to prove that large vortices are
exponentially suppressed in probability.

######  Lemma 7.

Define the constant $C_{1}$ as

$\displaystyle C_{1}:=2^{8}\max\bigg{(}$
$\displaystyle\exp[-2\beta\text{Re}(\rho(1)-\rho(-1))],$ (40)
$\displaystyle\exp[-\frac{\kappa}{(d-1)}(\text{Re}f(1,1)-\max_{(a,b)\neq(1,1)}\text{Re}f(a,b))]]\bigg{)}.$

Let $V$ be a vortex with $2k$ oriented plaquettes (alternatively, $k$ pairs of
plaquettes $p,-p$), then we have the following bound.

$\phi(V)\leq(C_{1})^{k},$ (41)

As a consequence of this bound and the previous Lemma 6, the probability that
there is a configuration $\mathcal{C}$ whose support contains a vortex of size
$k$ that is not non-contributing to the Wilson loop action is bounded above by

$2(d-1)|\gamma|(C(d)C_{1})^{k},$ (42)

where $C(d)$ is a constant that only depends on the dimension $d$.

In the case that,

$2\beta(\rho(1)-\rho(-1))>\frac{\kappa}{(d-1)}[\text{Re}f(1,1)-\max_{(a,b)\neq(1,1)}\text{Re}f(a,b)],$

then we have a better estimate when we consider Wilson loop non-trivial
configurations.

$\Phi_{NT}(V)\leq\exp[-4(d-1)\beta\text{Re}[\rho(1)-\rho(-1)]](C_{1})^{k-2(d-1)}.$
(43)

Similarly, the probability that there is a configuration $\mathcal{C}$ such
that there is a vortex $V$ of size $2k$ that is not non-contributing to the
Wilson loop and such that $\mathcal{C}$ is nontrivial when restricted to $V$
is bounded above by

$2(d-1)|\gamma|(C(d)C_{1})^{k-2(d-1)}(2^{8}C(d)\exp[-2\beta\text{Re}(\rho(1)-\rho(-1))])^{2(d-1)}.$
(44)

###### Proof.

Let $\mathcal{C}$ be a configuration with support
$V=\\{P_{1},P_{2},\ldots,P_{k}\\}$.

If $2k_{1}$ of these plaquettes $p$ satisfy the property that
$(\text{d}\sigma)_{p}\neq 1$ (here, we implicitly include a plaquette and its
negative), then the probability of this configuration is immediately reduced
by $\exp[-k_{1}2\beta\text{Re}(\rho(1)-\rho(-1))]$.

Now, if the plaquette $p$ does not satisfy $(\text{d}\sigma)_{p}=1$, then
necessarily, either one of the edges on its boundary satisfies $\sigma_{e}=-1$
or there is an edge $e=(v,w)$ on its boundary with $\phi_{v}\neq\phi_{w}$. In
any case, the presence of such an edge (and thus also its negative) reduces
the probability of the configuration by a factor of,

$\exp[-2\kappa(\text{Re}f(1,1)-\max_{(a,b)\neq(1,1)}\text{Re}f(a,b))].$ (45)

Now a single edge can be the cause of the excitation of at most $4(d-1)$
oriented plaquettes. Thus, there must be at least $\frac{2(k-k_{1})}{4(d-1)}$
excited oriented edges pairs $e$ and $-e$.

Thus, the probability of seeing this configuration is at most

$\exp[-k_{1}2\beta\text{Re}(\rho(1)-\rho(-1))-\frac{k-k_{1}}{(d-1)}\kappa(\text{Re}f(1,1)-\max_{(a,b)\neq(1,1)}\text{Re}f(a,b))].$
(46)

At this point, we now need to calculate the number of configurations with
support $V$. Note that there are at most $4k$ vertices that are boundary
vertices of plaquettes in $V$. These are the only vertices that could be
assigned the Higgs boson charge $-1$. There will be at most $2^{4k}$ ways to
assign the Higgs boson charge. Similarly, there are $2^{4k}$ ways to assign
the value of $\sigma_{e}$ at the edges that bound some plaquette in the
support.

Finally, we must find a way to bound the number of ways to get a connected set
of $k$ plaquettes that would intersect one of the edges of $\gamma$.

Lemma 3.4.2 of [4] gives a useful structure theorem. In dimension $4$, there
are at most $(20e)^{k}$ vortices of size $2k$ that could contain any given
plaquette $p$. A modification to general dimension will show that there is
some constant $C(d)$ such that the number of vortices of size $2k$ that
contain any given plaquette is $(C(d))^{k}$. Here, $C(d)e^{-1}$ is the number
of plaquettes that are adjacent to any given plaquette in the adjacency graph
of plaquette $G_{2}$ from Definition 3.

The logic applied is similar. Each vortex is a connected subgraph of $G_{2}$
and thus has a connected spanning tree. The problem of counting the number of
vortices of size $2k$ can now be divided into two parts. The first part is to
count the number of non-isomorphic rooted spanning trees of size $k$. Now
given a spanning tree $T$, the second part is to count the number of ways to
embed this spanning tree in $G_{2}$ with the root of the spanning three the
same as our special plaquette $p$. Since no two vortices of size $k$ can have
the same spanning tree when embedded in $G_{2}$, our counting procedure will
clearly be an upper bound on the number of vortices containing $p$.

The number of non-isomorphic spanning trees of size $k$ can be bounded by
$k(k!)^{-1}k^{k-2}$. To get this, we divide the number of labeled spanning
trees from Cayley’s Theorem by the number of permutations of the labels. We
also have $k$ choices for the root. This is less than $e^{k}$ and is good
enough for our purposes.

Now to embed the graph in $G_{2}$, we use the following graph. Assume that we
have already embedded the vertex $v$ in $G_{2}$. Let $w$ be a neighbor of $v$
that we have not yet fixed. There are $C(d)e^{-1}$ ways to choose to embed the
vertex $w$; it has to be one of the $C(d)e^{-1}$ neighbors of $v$ in $G_{2}$.
If we inductively perform this procedure starting from the root, we see that
for every tree $T$, there are at most $(C(d)e^{-1})^{k}$ ways to embed the
spanning tree in $G_{2}$. Combining our two estimates show that there are at
most $(C(d))^{k}$ different vortices of size $k$ that could contain any given
plaquette $p$.

Now, if we consider the $2(d-1)|\gamma|$ possible plaquettes that are adjacent
to edges of $\gamma$, then we see that there are at most
$2(d-1)|\gamma|(C(d))^{k}$ vortices of size $k$ that are not-noncontributing
to the Wilson loop action.

By a union bound, the total probability of seeing one of these excitations is
bounded from above by,

$\displaystyle 2(d-1)|\gamma|(2^{8}C(d))^{k}\max_{k_{1}}$
$\displaystyle\exp\bigg{[}-k_{1}2\beta\text{Re}(\rho(1)-\rho(-1))$ (47)
$\displaystyle-\frac{k-k_{1}}{(d-1)}\kappa(\text{Re}f(1,1)-\max_{(a,b)\neq(1,1)}\text{Re}f(a,b))\bigg{]}.$

If we consider the case that

$\beta\text{Re}(\rho(1)-\rho(-1))<\frac{\kappa(\text{Re}f(1,1)-\max_{(a,b)\neq(1,1)}\text{Re}f(a,b))}{2(d-1)},$
(48)

then the maximum is located at $k_{1}=k$. Otherwise, the maximum is located at
$k_{1}=0$.

By considering only configurations that are Wilson loop non-trivial, we can
get further cancellations in the case detailed in (48).

A non-trivial configurations necessarily has $k_{1}>0$. In fact, $k_{1}\geq
2(d-1)$ for any configuration $\sigma$ such that there exists a plaquette $p$
with $(\text{d}\sigma)_{p}\neq 0$. This is Lemma 3.4.6 of [4] generalized to
higher dimensions.

Thus, we could say that the probability of seeing a configuration whose
support contains a vortex $V$ of size $k$ that is not non-contributing and is
a vortex of size $k$ and, simultaneously, is non-trivial when restricted to
this $V$ is at most,

$\displaystyle 2(d-1)|\gamma|(2^{8}C(d))^{k}$
$\displaystyle\exp\bigg{[}-4(d-1)\beta\text{Re}(\rho(1)-\rho(-1))$ (49)
$\displaystyle-\frac{k-2(d-1)}{(d-1)}\kappa(\text{Re}f(1,1)-\max_{(a,b)\neq(1,1)}\text{Re}f(a,b))\bigg{]}.$

∎

As a consequence, we can now state the main result of our section. This
theorem shows that the main contribution to the Wilson loop expectation comes
from the minimal vortices we considers.

######  Theorem 1.

Recall the constants $C(d)$ and $C_{1}$ from our previous Lemma 7. Assume that
$C(d)C_{1}<1$ and define the constant $D$ as

$\displaystyle D:=\max\bigg{[}$
$\displaystyle\frac{e^{-2\beta\text{Re}(\rho(1)-\rho(-1))}}{1-2^{8}C(d)e^{-2\beta\text{Re}(\rho(1)-\rho(-1))}},$
(50)
$\displaystyle\frac{e^{-\frac{\kappa}{(d-1)}(\text{Re}f(1,1)-\max_{(a,b)\neq(1,1)}\text{Re}f(a,b))}}{1-2^{8}C(d)e^{-\frac{\kappa}{(d-1)}(\text{Re}f(1,1)-\max_{(a,b)\neq(1,1)}\text{Re}f(a,b))}}\bigg{]}.$

We have the following bounds on the expectation of the Wilson loop action,

$\displaystyle|\mathbb{E}[W_{\gamma}]-\mathbb{E}[\rho(-1)^{M_{\gamma}}]|\leq$
(51) $\displaystyle\hskip
28.45274pt2(d-1)|\gamma|(2^{8}C(d))^{2(d-1)+1}\exp[-4(d-1)\beta\text{Re}(\rho(1)-\rho(-1)]D,$

where $M_{\gamma}$ is the number of minimal vortices that are centered around
an edge $e$ of $\gamma$.

###### Proof.

Recall the event $E$ from Lemma 5.

The result of Lemma 7 shows that the probability of the event $E^{c}$ is
bounded by the union of the events described by probability (44) for $k$
varying between 7 and $\infty$. Adding all the terms together and performing
the union bound gives the desired result. ∎

###  2.4 Approximating $M_{\gamma}$ as a Poisson Random Variable

We can write $M_{\gamma}$ as the sum of random events,

$M_{\gamma}:=\sum_{e\in\gamma}\mathbbm{1}[F_{P(e)}],$ (52)

where $F_{P(e)}$ is the event that $e$ is the center of a minimal vortex
$P(e)$ in the support of our configuration.

To show $M_{\gamma}$ behaves approximately as a Poisson random variable, we
would want to show that for $e\neq e^{\prime}$, the events $F_{P(e)}$ and
$F_{P(e^{\prime})}$ are approximately independent from each other. For each
edge $e$, we let $\mathcal{B}_{e}$ to be the set of edges $e^{\prime}$ in
$\gamma$ such that a minimal vortex centered around an edge $e^{\prime}$
connect to the minimal vortex centered around $e$ based on adjacency in the
graph $G_{2}$ of plaquettes from Definition 3.

The following Theorem from [6] details exactly the degree of approximation of
some some of random variables to the Poisson distribution.

######  Theorem 2.

Consider the following constants.

$\displaystyle b_{1}:=\sum_{e\in\gamma}\sum_{e^{\prime}\in
B_{e}}\mathbb{P}(F_{P(e)})\mathbb{P}(F_{P(e^{\prime})}),$ (53) $\displaystyle
b_{2}:=\sum_{e\in\gamma}\sum_{e^{\prime}\in B_{e}\setminus
e}\mathbb{P}(\mathbbm{1}(F_{P(e)})\mathbbm{1}(F_{P(e^{\prime})})),$
$\displaystyle
b_{3}:=\sum_{\gamma}\mathbb{E}\left[|\mathbb{E}[\mathbbm{1}(F_{P(e)})|\mathbbm{1}(F_{P(e^{\prime})})e^{\prime}\not\in
B_{e}]-\mathbb{P}(F_{P(e)})|\right].$

Let $\mathcal{L}(M_{\gamma})$ denote the law of $M_{\gamma}$ and let
$\lambda=\mathbb{E}[M_{\gamma}]$. Then

$d_{TV}(\mathcal{L}(M_{\gamma}),\text{Poisson}(\lambda))\leq
min(1,\lambda^{-1})(b_{1}+b_{2})+\min(1,1.4\lambda^{-1/2})b_{3}.$ (54)

We start with the following corollary of the proof of Lemma 6.

######  Corollary 1 (of Lemma 6).

Let $E_{1}$ and $E_{2}$ be two set of edges. Let $M(E_{1},E_{2})$ be the event
that our configuration has a minimal vortex centered on each edge of $E_{1}$
and no minimal vortices centered on any edge of $E_{2}$. Assume that
$M(E_{1},E_{2})>0$ (this amounts to assuming that the minimal vortices
centered around $E_{1}$ do not intersect each other). Now consider some other
union of vortices $U$. The probability that some union of vortices $U$ appears
in the support of some configuration, conditionally on being in the event
$M(E_{1},E_{2})$ is less than $\Phi(U)$.

###### Proof.

Let $(\sigma,\phi)$ be a configuration with support $U\bigcup_{e_{i}\in
E_{1}}P(e_{i})\cup R$ where $P(e_{i})$ is the minimal vortex centered around
$e_{i}$. In addition, $R$ is some union of vortices that do not intersect $U$
or $P(e_{i})$ and does not contain a minimal vortex centered at an edge of
$E_{2}$.

The sum of $\exp[H_{N,\beta,\kappa}(\sigma,\phi)]$ of all configurations with
this support is given by $2\Phi(U)\prod_{e_{i}\in E_{1}}\Phi(e_{i})\Phi(R)$.

Thus, the sum of $\exp[H_{N,\beta,\kappa}(\sigma,\phi)]$ over all
configurations such that the support contains $U$ will be

$2\Phi(U)\prod_{e_{i}\in E_{1}}\Phi(e_{i})\sum_{R}\Phi(R),$ (55)

where the sum in $R$ goes over all unions of vortices that do not intersect
$U$, the $P(e_{i})$’s and do not contain a minimal vortex centered around an
edge of $e_{2}$.

All configurations with support $\bigcup_{e_{i}\in E_{1}}P(e_{i})\cup R$ will
be in $M(E_{1},E_{2})$.

Thus, the sum of $\exp[H_{N,\beta,\kappa}]$ for all configurations found in
$M(E_{1},E_{2})$ (we will call this $Z(M(E_{1},E_{2}))$ the partition function
restricted to the event $M(E_{1},E_{2})$) will be satisfy the relation

$Z(M(E_{1},E_{2}))\geq 2\prod_{e_{i}\in E_{1}}\Phi(P(e_{1}))\sum_{R}\Phi(R).$

The probability that restricted to $M(E_{1},E_{2})$ that $U$ will be in the
support is the ratio of the quantity in (55) with the partition function
$Z(M(E_{1},E_{2}))$. This ratio is clearly less than $\Phi(U)$. ∎

This is one of our major tools in computing
$\mathbb{E}[\mathbbm{1}[F_{P(e)}]|\mathbbm{1}[F_{P(e^{\prime}}],e^{\prime}\not\in
B_{e}]$.

######  Lemma 8.

Let $\gamma$ be a loop that has no self-intersection. In addition, recall the
dimension dependent constant $C(D)$ and the constant $D$ from Theorem 1. If
$2(d-1)C(d)D<1$, we have the following bounds on the constants $b_{1},b_{2}$
and $b_{3}$ and $\lambda$.

$\displaystyle b_{1}\leq 8(d-1)|\gamma|\Phi(P(e))^{2},$ (56) $\displaystyle
b_{2}=0,$ $\displaystyle
b_{3}\leq\frac{2|\gamma|\Phi(P(e))(d-1)C(d)D}{1-2(d-1)C(d)D},$
$\displaystyle|\lambda-|\gamma|\Phi(P(e))|\leq\frac{2|\gamma|\Phi(P(e))(d-1)C(d)D}{1-2(d-1)C(d)D}.$

As a consequence of Theorem 2, we see that

$\displaystyle
d_{TV}(\mathcal{L}(M_{\gamma}),\text{Poisson}(|\gamma|\Phi(P(e))))$ (57)
$\displaystyle\hskip 28.45274pt\leq
8(d-1)|\gamma|(\Phi(P(e)))^{2}+\frac{4|\gamma|\Phi(P(e))(d-1)C(d)D}{1-2(d-1)C(d)D}.$

######  Remark 5.

The function $\Phi$ evaluated on a minimal plaquette $P(e)$ does not depend on
the minimal plaquette chosen. Thus, in a minor abuse of notation, we treat it
like a constant. Explicitly, we may always substitute,

$\Phi(P(e))=\exp[-4(d-1)\text{Re}(\rho(1)-\rho(-1))]\exp[-2\kappa\text{Re}(f(1,1)-f(-1,1))]$
(58)

###### Proof.

We start with the most difficult part: computing a bound on $b_{3}$. Fix an
edge $e$. We now attempt to compute

$\mathbb{E}\left[|\mathbb{E}[\mathbbm{1}(F_{P(e)})|\mathbbm{1}(F_{P(e^{\prime})})e^{\prime}\not\in
B_{e}]-\mathbb{P}(F_{P(e)})|\right],$

for our fixed edge $e$. Recall the notation $M(E_{1},E_{2})$ and
$Z(M(E_{1},E_{2}))$ from the proof of Corollary 1. Let $E_{1}$ be some set of
edges in $\gamma\setminus B_{e}$ and $E_{2}$ be the remaining edges in
$\gamma\setminus B_{e}$ that are not contained in $E_{1}$.

We know compute $\mathbb{E}[\mathbbm{1}[F_{P(e)}]|M(E_{1},E_{2})]$. Events in
$M(E_{1},E_{2})$ can be divided into two parts,

  1. 1.

$G(E_{1},E_{2})$: These are configuration $(\sigma,I)$ in $M(E_{1},E_{2})$
whose support has a vortex decomposition $V_{1}\cup\ldots\cup V_{m}$ such that
$P(e)$ does not intersect any $V_{i}$. Thus, $P(e)\bigcup_{i=1}^{m}V_{i}$ is a
valid vortex decomposition.

  2. 2.

$B(E_{1},E_{2})$: These are configurations $(\sigma,\phi)$ in $M(E_{1},E_{2})$
whose vortex decomposition $V_{1}\ldots V_{m}$ does contain a vortex $V_{j}$
such that $V_{j}$ contains a plaquette that is in $P(e)$. Thus,
$P(e)\bigcup_{i=1}^{m}V_{i}$ is not a valid vortex decomposition.

If we let $Z(G(E_{1},E_{2}))$ and $Z(B(E_{1},E_{2}))$ be the sum of
$\exp[H_{N,\beta,\kappa}]$ for configurations in them ( the partition
functions), then we see that the probability of seeing $P(e_{i})$ as a
plaquette in the support conditional on the event $M(E_{1},E_{2})$ is given
by,

$\frac{\Phi(V)Z(G(E_{1},E_{2}))}{Z(G(E_{1},E_{2}))+Z(B(E_{1},E_{2}))}.$ (59)

If we show that $Z(B(E_{1},E_{2}))\leq cZ(M(E_{1},E_{2}))$ for some constant
$c<1$, then we see that the probability of seeing $P(e_{i})$ in the support
conditional on $M(E_{1},E_{2})$ is greater than $(1-c)\Phi(P(e))$ and less
than $\Phi(P(e))$.

By applying Corollary 1, we can bound
$\frac{Z(B(E_{1},E_{2}))}{Z(M(E_{1},E_{2}))}$ by the sum of $\Phi(V)$ over all
configurations $V$ that contain a plaquette of $P(e_{i})$ in its support. We
have performed a version of his sum when trying to compute the probability of
the event $E^{c}$ from 5. Recall the bound (41) on the probability of
observing a vortex excitation of size $2k$. Notice that a minimal vortex has
$2(d-1)$ plaquettes attached to it. We can provide a union bound over all of
these $2(d-1)$ vortices and all vortices of size $2k$ that intersect these
plaquettes. Recall that there are at most $(C(d))^{k}$ vortices of size $2k$
that contain any given plaquette.

We see that the probability of the event $B(E_{1},E_{2})$ conditioned on
$G(E_{1},E_{2})$ is less than,

$2(d-1)\sum_{k=1}C_{1}^{k}(C(d))^{k}\leq 2(d-1)C(d)D,$ (60)

provided $C(d)C_{1}<1$.

We thus see that the expectation
$\mathbb{E}[\mathbbm{1}[F_{P(e)}]|M(E_{1},E_{2})]$ is greater than
$\frac{\Phi(P(e))}{1-2(d-1)C(d)D}$, provided $2(d-1)C(d)D<1$.

Through the same logic, we can show that
$\mathbb{P}(F_{P(e)})\geq\frac{\Phi(P(e))}{1-2(d-1)C(d)D}$. Recalling our
previous upper bound of $\Phi(P(e))$ on both of these quantities, we see that,

$|\mathbb{E}[\mathbbm{1}[F_{P(e)}]|M(E_{1},E_{2})]-\mathbb{E}[F_{P(e)}]|\leq\frac{2\Phi(P(e))(d-1)C(d)D}{1-2(d-1)C(d)D}.$
(61)

We can now integrate over our conditioning on $M(E_{1},E_{2})$ to show

$\mathbb{E}\left[|\mathbb{E}[\mathbbm{1}(F_{P(e)})|\mathbbm{1}(F_{P(e^{\prime})})e^{\prime}\not\in
B_{e}]-\mathbb{P}(F_{P(e)})|\right]<\frac{2\Phi(P(e))(d-1)C(d)D}{1-2(d-1)C(d)D}.$

Since this bound did not depend on the specific choice of edge $e$, we see
that our bound on $b_{3}$ is,

$b_{3}\leq\frac{2|\gamma|\Phi(P(e))(d-1)C(d)D}{1-2(d-1)C(d)D}.$ (62)

$b_{2}$ is trivially $0$ since by definition, a support of a configuration
cannot have support on both $P(e)$ and $P(e^{\prime})$ when $e^{\prime}$ is in
$B_{e}$.

Now, to estimate $b_{1}$, we use the bound
$\mathbb{P}(F_{P(e)})\leq\Phi(P(e))$ for all edges. Now for any edge $e$,
there are at most $8(d-1)$ other edges $e^{\prime}$ in $B_{e}$ that lie in
$\gamma$. Here, we use the assumption that $\gamma$ has no self-intersection.
We can now apply 2 to assert that,

$\text{d}_{TV}(\mathcal{L}(M_{\gamma}),\text{Poisson}(\lambda))\leq
8(d-1)|\gamma|(\Phi(P(e)))^{2}+\frac{2|\gamma|\Phi(P(e))(d-1)C(d)D}{1-2(d-1)C(d)D}.$
(63)

Finally, an immediate consequence of the fact that

$\frac{\Phi(P(e))}{1-2(d-1)C(d)D}\leq\mathbb{P}(F_{P(e)})\leq\Phi(P(e))$

shows that the
$|\lambda-|\gamma|\Phi(P(e))|\leq\frac{2|\gamma|\Phi(P(e))(d-1)C(d)D}{1-2(d-1)C(d)D}$.

Corollary 3.1 of [1] shows that
$\text{d}_{TV}(\text{Poisson}(\lambda),\text{Poisson}(|\gamma|\Phi(P(e)))\leq|\lambda-|\gamma|\Phi(P(e))|$.
We can combine this with our earlier bound on the total variation to show the
desired equation (57). ∎

Combining the results of Lemma 8 and Theorem 1 will give us our following main
result.

######  Theorem 3.

Assume that the conditions of Theorem 1 and 8 hold. Let $X$ be a Poisson
random variable with expectation parameter
$|\gamma|\Phi(P(e))=\exp[-4(d-1)\beta\text{Re}(\rho(1)-\rho(-1))]\exp[-2\kappa\text{Re}(f(1,1)-f(-1,1))]$.
(Recall that the only way to excite a minimal vortex is to excite the center
of the edge that is on its center.) Then we have that,

$\displaystyle|\mathbb{E}[W_{\gamma}]-\mathbb{E}[\rho(-1)^{X}]|\leq
8(d-1)|\gamma|(\Phi(P(e)))^{2}+\frac{4|\gamma|\Phi(P(e))(d-1)C(d)D}{1-2(d-1)C(d)D}$
(64) $\displaystyle\hskip
34.14322pt+2(d-1)|\gamma|(2^{8}C(d))^{2(d-1)+1}\exp[-4(d-1)\beta\text{Re}(\rho(1)-\rho(-1)]D.$

###### Proof.

This is a simple triangle inequality, provided we characterize the value of
the difference of $\mathbb{E}[\rho(-1)^{X}]$ and
$\mathbb{E}[\rho(-1)^{M_{\gamma}}]$. We can choose a coupling of $X$ and
$M_{\gamma}$ such that $\mathbb{P}(X\neq
M_{\gamma})=d_{TV}(\mathcal{L}(M_{\gamma}),\text{Poisson}(|\gamma|\phi(P(e)))$.
Now, $|\rho(-1)^{M_{\gamma},X}|\leq 1$ for all values of $X$ or $M_{\gamma}$.
Thus, we see that, under this coupling

$|\mathbb{E}[\rho(-1)^{X}]-\mathbb{E}[\rho(-1)^{M_{\gamma}}]|\leq\mathbb{P}(X\neq
M_{\gamma})\leq
d_{TV}(\mathcal{L}(M_{\gamma}),\text{Poisson}(|\gamma|\phi(P(e))).$

We can apply the triangle inequality with our estimate on
$|\mathbb{E}[W_{\gamma}]-\mathbb{E}[\rho(-1)^{M_{\gamma}}]|$ and complete the
proof. ∎

##  3 The Non-Abelian Case

###  3.1 Model and Some Preliminary Discussion

####  3.1.1 Definitions

To treat the non-abelian case, we need significantly different notation. This
preliminary section will discuss many of our new conventions. The Hamiltonian
we will consider in the case of a non-abelian gauge field $G$ will be as
follows:

$\displaystyle H_{N,\beta,\kappa}(\sigma,\phi)$ $\displaystyle=\sum_{p\in
P_{N}}\beta(\psi_{p}(\sigma)-\psi_{p}(1))+\sum_{e\in
E^{U}_{N}}\kappa[g_{e}(\sigma,\phi)-g^{u}(1,1)].$ (65)

$\sigma_{e}$ is still a map from the set of oriented edges $E_{N}\to G$, a
non-abelian group with $\sigma_{-e}=\sigma_{e}^{-1}$, $\rho(\cdot)$ is a $D$
dimensional unitary representation of $G$. Finally, $\phi_{x}$ will be
represented as a field $V_{N}\to H$ taking values in $H$, a finite
multiplicative subgroup of the unit circle. In some sense, we can interpret
$\phi$ as a scalar field.

There are a few technical differences from the presentation of the abelian
gauge model. First of all, we need to be more careful when defining the
‘current’ around a plaquette $p$. If a plaquette $p$ has boundary vertices
$v,w,x$ and $y$ and has an oriented boundary consisting of edges
$e_{1}=(v,w)$, $e_{2}=(w,x)$, $e_{3}=(x,y)$ and $e_{4}=(y,v)$, then the
‘current’ defined around the oriented plaquette $p$ would be the product
$\sigma_{e_{1}}\sigma_{e_{2}}\sigma_{e_{3}}\sigma_{e_{4}}$ and we set
$\psi_{p}(\sigma)=\text{Tr}[\rho(\sigma_{e_{1}}\sigma_{e_{2}}\sigma_{e_{3}}\sigma_{e_{4}})]$,
where $\rho$ is a $D$ dimensional unitary representation of the group $G$. We
remark that even if one chose a different start edge for the boundary(
$e_{2},e_{3},e_{4}$ and $e_{1}$ in that order, for example), the ‘current’
might change, but the Hamiltonian will not change due to the multiplicative
property $\rho$ and the cyclic property of trace. Thus, $\psi_{p}(\sigma)$
will be well defined regardless of how we choose the starting edge of the
boundary of $p$.

Recall also that $P_{N}$ is a set of oriented plaquettes, so $P_{N}$ would
also contain $-p$ whose boundary is $-e_{4},-e_{3},-e_{2}$ and $-e_{1}$ in
that order. We see that our ‘currents’ satisfy
$(\sigma_{e_{1}}\sigma_{e_{2}}\sigma_{e_{3}}\sigma_{e_{4}})^{-1}=\sigma_{-e_{4}}\sigma_{-e_{3}}\sigma_{-e_{2}}\sigma_{-e_{1}}$.With
respect to these ‘currents’, it is obvious
$\psi_{p}(\sigma)=\overline{\psi_{-p}(\sigma)}$. By our earlier remarks, this
relation does not depend on the specific choice of start point. We thus see
that this part of the Hamiltonian will ultimately take real values.

We introduce the notion of unoriented edges $E_{N}^{U}$. In contrast to
$E_{N}$, for every adjacent pair of vertices $(x,y)$ in $\Lambda_{N}$,
$E_{N}^{U}$ includes only a single undirected edge between $x$ and $y$ rather
than both $e=(x,y)$ and $e_{-}=(y,x)$. The function $g^{u}_{e}(\sigma,\phi)$
will consist of the sum
$\phi_{x}\text{Tr}[\rho(\sigma_{e})]\phi_{y}^{-1}+\phi_{y}\text{Tr}[\rho(\sigma_{-e})]\phi_{x}^{-1}$
where $e$ is the oriented edge $(x,y)$ and $-e$ is the oriented edge $(y,x)$.
Namely, it is the sum of the Higgs boson action
$\phi_{x}\text{Tr}[\rho(\sigma_{e})]\phi_{y}^{-1}$ over the oriented edge pair
$\\{e,-e\\}$ in $E_{N}$. The reason we introduce $g$ instead of using our
previous sum over $E_{N}$ is that we will eventually have to perform an
expansion of the Hamiltonian with respect to these unoriented edges. One can
see that the function $g_{e}$ is manifestly real.

######  Remark 6.

Cao [4] introduces many notions from algebraic topology to describe non-
abelian gauge field configurations. Though such concepts would be useful in
places, it is not essential to understanding the logic of the proof in this
section. Whenever we need to refer to some of these topological notions, I
would refer to the appropriate location in [4].

####  3.1.2 Removing Trivial Gauge Invariances

We discuss some gauge invariances that simplify our analysis. First of all, we
can assume that there is no element $\sigma\in G$ such that $\rho(\sigma)=I$,
the identity matrix. The set of all elements $g^{\prime}\in G$ such that
$\rho(g^{\prime})=I$ forms a normal subgroup $G^{\prime}$ of $G$. To
understand our Hamiltonian, it suffices to interpret $\sigma_{e}$ as a map
from $E_{N}$ to the quotient group $G/G^{\prime}$ instead. From a map from
$E_{N}\to G/G^{\prime}$, one can obtain all maps from $E_{N}\to G$ by freely
multiplying each edge with some member of the group $G^{\prime}$. As
$\rho(g)=I$ for all members in $G^{\prime}$, this will not change the
Hamiltonian, nor Wilson loop values. Essentially, the reduction from $G$ to
$G/G^{\prime}$ is the removal of a trivial gauge invariance.

Furthermore, after this reduction, we know that if
$\Psi_{p}(\sigma)=\Psi_{p}(1)$, then we can say in a well-defined sense that
the ‘current’ around the plaquette $p$ is exactly 1. The condition
$\Psi_{p}(\sigma)=\Psi_{p}(1)$ for $p$ surrounded by boundary edges
$e_{1}$,$e_{2}$,$e_{3}$ and $e_{4}$ in that order asserts that the product
$\sigma_{e_{1}}\sigma_{e_{2}}\sigma_{e_{3}}\sigma_{{}_{4}}$ is an element of
$G$ whose image under $\rho$ is the identity matrix. Our reduction shows the
only such element is $1$, so
$\sigma_{e_{1}}\sigma_{e_{2}}\sigma_{e_{3}}\sigma_{e_{4}}=1$. By multiplying
by $\sigma_{e_{1}}^{-1}$ on the left and $\sigma_{e_{1}}$ on the right, we can
derive that $\sigma_{e_{2}}\sigma_{e_{3}}\sigma_{e_{4}}\sigma_{e_{1}}=1$ and
similar for other choices of start vertex.

Now, $|\text{Tr}(\rho(\sigma_{e}))|<D$ unless
$\rho(\sigma_{e})=e^{\frac{2\pi\text{i}j}{n}}I$ for some integers $j$ and $n$.
Observe that since we are considering a unitary representation, all
eigenvalues of $\rho(g)$ have absolute value less than $1$. Thus, we know that
$|\phi_{x}\text{Tr}[\rho(\sigma_{e})]\phi_{y}^{-1}+\overline{\phi_{x}\text{Tr}[\rho(\sigma_{e})]\phi_{y}^{-1}}|\leq
2D$ for all possible choices of $\phi_{x},\sigma_{e}$ and $\phi_{y}$. Equality
can only occur if $\rho(\sigma_{e})=e^{\frac{2\pi\text{i}j}{n}}I$ for some
integer $j$ and if $e^{\frac{2\pi\text{i}j}{n}}\in H$.

Let $X$ be the set of values $e^{\frac{2\pi\text{i}j}{n}}$ such that there
exists an element $g$ of $G$ such that $\rho(g)=e^{\frac{2\pi\text{i}j}{n}}I$.
We see that $X$ is a subgroup of the multiplicative group of the unit circle,
since if $\rho(g_{1})=c_{1}I$ and $\rho(g_{2})=c_{2}I$, where $X_{1}$ and
$X_{2}$ are roots of unity, then $\rho(g_{1}g_{2})=c_{1}c_{2}I$. This implies
that if $c_{1}\in X$ and $c_{2}\in X$, then the product $c_{1}c_{2}$ is also
in $X$. Furthermore, $\rho(g)=\rho(g)^{-1}$, so if $\rho(g)=cI$, then
$\rho(g^{-}1)=c^{-1}I$. This shows that $X$ contains inverses. Manifestly $X$
has the identity, so $X$ must be a subgroup of the unit circle.

As we have done in the abelian case, we can consider the Higgs field to take
values in $H/X$ instead of $H$. Observe that the Wilson loop expecation can be
computed as the ratio,

$\langle W_{\gamma}\rangle=\frac{\sum_{\begin{subarray}{c}\sigma_{e}\in G\\\
e\in E_{N}\end{subarray}}\sum_{\begin{subarray}{c}\phi_{x}\in H\\\ x\in
V_{N}\end{subarray}}W_{\gamma}(\sigma)\exp[H_{N,\beta,\kappa}(\sigma,\phi)]}{\sum_{\begin{subarray}{c}\sigma_{e}\in
G\\\ e\in E_{N}\end{subarray}}\sum_{\begin{subarray}{c}\phi_{x}\in H\\\ x\in
V_{N}\end{subarray}}\exp[H_{N,\beta,\kappa}(\sigma,\phi)]}.$ (66)

Fix some representative $h_{i}\in H$ for each coset class $C_{i}$ in $H/X$.
Now, fix a particular map $\phi_{x}=V_{N}\to H$; we will denote this map by
$M$. For every vertex choose an element $\zeta^{M}_{v}$ and $\eta^{M}_{v}$
(depending on $M$) such that $\phi_{v}(\zeta^{M}_{v})^{-1}=h_{i}$ for some $i$
and $\rho(\eta^{M}_{v})=\zeta^{M}_{v}I$ . Notice that the transformation
$(\\{\phi_{v}\\},\\{\sigma_{e}\\})\to(\\{\phi_{v}(\zeta^{M}_{v})^{-1}\\},\\{\eta^{M}_{v}\sigma_{e}(\eta^{M}_{w})^{-1}\\}$
for $e=(v,w)$ does not change the Wilson loop action nor the value of the
Hamiltonian. Define $\phi^{M}_{v}:=\phi_{v}(\zeta^{M}_{v})^{-1}$ and
$\sigma^{M}_{e}=\eta_{v}^{M}\sigma_{e}(\eta^{M}_{w})^{-1}$.

Another way to represent the Wilson loop expectation is as,

$\langle W_{\gamma}\rangle=\frac{\sum_{M=\phi_{x}:V_{N}\to
H}\sum_{\sigma_{e}:E_{N}\to
G}W_{\gamma}(\sigma^{M})\exp[H_{N,\beta,\kappa}(\sigma^{M},\phi^{M})]}{\sum_{M=\phi_{x}:V_{N}\to
H}\sum_{\sigma_{e}:E_{N}\to G}\exp[H_{N,\beta,\kappa}(\sigma^{M},\phi^{M})]}.$
(67)

In both the numerator and denominator, when considering the internal sum over
$\sigma$, the map $\sigma\to\sigma^{M}$ can be treated as a change of
parameters for the summation. That is, for fixed $M$, if $\sigma_{e}$ is
uniform over all maps from $E_{N}\to G$, then $\sigma^{M}_{e}$ is uniform for
all maps from $E_{N}\to G$.

We can reparameterize the sum as,

$\langle W_{\gamma}\rangle=\frac{\sum_{M=\phi_{x}:V_{N}\to
H}\sum_{\sigma_{e}:E_{N}\to
G}W_{\gamma}(\sigma)\exp[H_{N,\beta,\kappa}(\sigma,\phi^{M})]}{\sum_{M=\phi_{x}:V_{N}\to
H}\sum_{\sigma_{e}:E_{N}\to G}\exp[H_{N,\beta,\kappa}(\sigma,\phi^{M})]}.$
(68)

Now, the map $\phi^{M}$ is a set of maps from $V_{N}\to H/X$. The map
$M\to\phi^{M}$ is onto and the preimage of any $\phi^{M}$ is of size
$|X|^{|V_{N}|}$. This shows that we can reduce our problem to computing Wilson
loop expectations with a Higgs field taking values in $H/X$. After these
reductions, we see that
$|\phi_{x}\text{Tr}[\rho(\sigma_{e})]\phi_{y}^{-1}+\overline{\phi_{x}\text{Tr}[\rho(\sigma_{e})]\phi_{y}^{-1}}|<2d$
unless $\phi_{x}=\phi_{y}$ and $\rho(\sigma_{e})=1$. Thus, we may consider
edges $e=(x,y)$ with $\sigma_{e}\neq 1$ or $\phi_{x}\neq\phi_{y}$ as real
excitations of the Hamiltonian.

######  Remark 7.

If instead of choosing $\phi_{x}$ to be a scalar, we do the following steps

  1. 1.

We let $\phi_{x}$ take values in a general group $H$.

  2. 2.

We assume $H$ has a unitary representation $\rho_{\phi}$ with the same
dimension as the unitary representation on $G$, $\rho$ and the intersection of
the groups $\rho_{\phi}(H)\cap\rho(G)$ is a normal subgroup of the image
$\rho_{\phi}(H)$.

  3. 3.

We change the Higgs boson interaction to,

$\sum_{e=(x,y)\in
E^{u}_{N}}\text{Tr}[\rho_{\phi}(\phi_{x})\rho(\sigma_{e})\rho_{\phi}(\phi_{y})]+\overline{\text{Tr}[\rho_{\phi}(\phi_{x})\rho(\sigma_{e})\rho_{\phi}(\phi_{y}^{-1})]}.$
(69)

Then, after similar transformations as we have detailed above, we can show we
have an excitation with exponential suppression $\exp[-O(\kappa)]$ whenever we
have an edge $e=(x,y)$ that satisfies $\sigma_{e}\neq 1$ or
$\phi_{x}\neq\phi_{y}$.

To show this, not that since
$\rho_{\phi}(\phi_{x})\rho(\sigma_{e})\rho_{\phi}(\phi_{y}^{-1})$ is a unitary
matrix, so the absolute value of its trace is at most $D$, the dimension of
the representation. Furthermore, the trace plus its conjugate is less than
$2D$ unless the unitary matrix considered is the identity.

If $\rho_{\phi}(\phi_{x})\rho(\sigma_{e})\rho_{\phi}(\phi_{y}^{-1})$ is the
identity, then $\rho(\sigma_{e})=\rho_{\phi}(\phi_{y}\phi_{x}^{-1})$, and
there is a common element in the image of $\rho_{\phi}(H)$ and $\rho(G)$. We
can quotient out $\rho_{\phi}(H)$ by its intersection with $\rho(G)$ through
the same gauge transformation procedure we outlined previously if
$\rho_{\phi}(H)\cap\rho(G)$ is a normal subgroup of $\rho_{\phi}(H)$. If we
also quotient out $G$ and $H$ by the elements such that $\rho(g)=I$ and
$\rho_{\phi}(h)=I$, respectively, then we will have removed all possibilities
for $\rho_{\phi}(\phi_{x})\rho(\sigma_{e})\rho_{\phi}(\phi_{y}^{-1})=I$.

###  3.2 The case of Low Disorder in the Higgs Field

Before we analyze Wilson loop expectations under this Hamiltonian, we start by
discussing the general difficulty of introducing non-abelian gauge
interactions.

Our key tool in probabalistic computation was based on cluster expansions;
thus, our first step in any problem was to first find an appropriate
definition of cluster that would satisfy nice properties. In the case of a
pure gauge field, the natural choice of clusters would be to find those
plaquettes such that the current around it is $0$, or, in other words,
$\psi_{p}(\sigma)\neq\psi_{p}(1)$. These are the natural excitations that
suppress probability.

In the case of an abelian gauge field, we are actually able to use the set of
excited plaquettes $p$ with $\psi_{p}(\sigma)\neq\psi_{p}(1)$ as the basis of
a legitimate cluster expansion. This due to the fact that a configuration
$\sigma$ whose support consists two compatible sets $P_{1}\cup P_{2}$ ,
according to Definition 3, can be split into two configurations $\sigma_{1}$
and $\sigma_{2}$ whose supports are $P_{1}$ and $P_{2}$ respectively.

In the case of a non-abelian gauge field, it is no longer possible in general
to split a configuration $\sigma$ whose support consists of two compatible
sets $P_{1}\cup P_{2}$ into two configurations $\sigma_{1}$ and $\sigma_{2}$
with support $P_{1}$ and $P_{2}$. As shown in the papers [4] and [15] , there
are topological restrictions that prevent such a splitting. To get around this
difficulty, one must consider a more sophisticated condition to determine
whether we can split a configuration or not. This ‘sophisticated condition’ is
related the knotting properties of the vortices in the support. The
combinatorial analysis of the knotting properties of vortices was done by Cao
in [4].

Now, let us return to our Higgs field model. As we have seen, a basic
difficulty is that our cluster expansion must consider both the excitations of
the Higgs field and the gauge field. We have observed already in the abelian
case with low disorder (large $\kappa$) that considering the Higgs field makes
defining the splitting substantially more complicated.

However, in the large $\kappa$ case, our construction becomes more robust when
considering non-abelian groups. In fact, the cluster expansion we proposed in
Section 2 works very well when considering non-abelian groups. We try now to
explain intuitively why we are able to more easily perform a splitting in the
non-abelian case with low $\kappa$ than the pure gauge field case.

In the case of a pure gauge field, there is a large gauge symmetry. Namely, if
we let $\eta:V_{N}\to G$ be a map from the vertices $V_{N}$ to the group $G$,
then $\sigma_{e}\to\eta_{x}\sigma_{e}\eta_{y}^{-1}$ does not change the value
of the Hamiltonian. Thus, the basic object in the pure gauge field case is not
a configuration, but a gauge equivalence class of configurations. A gauge
equivalence class containing the configuration $\sigma$ will also contain the
configurations $\eta_{x}\sigma\eta_{y}^{-1}$ for any map $\eta:V_{N}\to G$.

When $G$ is abelian, these gauge equivalence classes of a configuration
$\sigma$ can be understood as a $2$-form $(\text{d}\sigma)$. These $2$-forms
can split on disjoint supports by restriction. However, for non-abelian
groups, the gauge equivalence classes are homomorphisms to the fundamental
group; for details, one can refer to Section 4 of [4]. These homomorphisms
cannot split disjoint supports in general; the support components $P_{1}$ and
$P_{2}$ cannot be knotted with each other if one wants to split the support
between $P_{1}$ and $P_{2}$.

By contrast, the introduction of the Higgs field and the $\kappa$ term ensures
that there is a strong breaking of the gauge equivalence symmetry created by
the maps $\eta:V_{N}\to G$. For large $\kappa$, our basic objects are indeed
configurations $(\sigma,\phi)$ rather than gauge equivalence classes of
configurations. The basic object of study in the large $\kappa$ regime of the
Higgs boson stays the same, whether we are consider abelian or non-abelian
groups. For this reason, the analysis in Section 2 is robust to the
introduction of a non-abelian group $G$.

In the analysis of Section 2, we found that any edge with $\sigma_{e}\neq 1$
automatically suppresses the probability by a factor of $\exp[-O(\kappa)]$.
This allowed us to consider plaquettes that contain an edge with
$\sigma_{e}\neq 1$ as part of our support. Immediately, if the support of our
configuration is $P_{1}\cup P_{2}$ where $P_{1}$ and $P_{2}$ are compatible,
then we can split the gauge field part $\sigma$ of our configuration by
restriction to $P_{1}$ and $P_{2}$.

The assignment of Higgs bosons to vertices is similar to what is done in Lemma
2. Lemma 2 treated the case when $H/(\rho(G)\cap H)$ is isomorphic to $Z_{2}$.
The key point of the argument was to identify the phase boundary and perform
appropriate flips to correct phase boundaries when we separate the supports.
The only key fact that we used about the group is that if we switch
$\phi_{x},\phi_{y}$ to $-\phi_{x},-\phi_{y}$ where vertices $x$ and $y$ bound
an edge $e$, then the Hamiltonian for that edge $e$ does not change. In
particular, if we apply this flip for all vertices inside a phase boundary, it
does not change the Hamiltonian except on the phase boundary. For a more
general group $H$, we still see that the Hamiltonian does not change on an
edge if we apply the transformation $\phi_{x}\to h\phi_{x}$, $\phi_{y}\to
h\phi_{y}$ for an arbitrary group element in $H/(H\cap\rho(G))$. This will
allow the argument to go through when we consider Higgs boson groups larger
than $Z_{2}$. Combining these two facts together, we see that all of the
arguments outlined in Section 2 apply verbatim when $\kappa$ is sufficiently
large.

######  Remark 8.

Even if we consider the case outline in Remark 7, with the Higgs boson taking
non-abelian values, we can still apply the vertex assignment procedure outline
in Lemma 2. This is again due to the fact that the transformation $\phi_{x}\to
h\phi_{x}$ and $\phi_{y}\to h\phi_{y}$ does not change the Hamiltonian.

$\displaystyle\text{Tr}[\rho_{\phi}(h\phi_{x})\rho(\sigma_{e})\rho_{\phi}((h\phi_{x})^{-1})]=\text{Tr}[\rho_{\phi}(h)\rho_{\phi}(\phi_{x})\rho(\sigma_{e})\rho_{\phi}(\phi_{y}^{-1})\rho_{\phi}(h^{-1})]$
(70)
$\displaystyle=\text{Tr}[\rho_{\phi}(h^{-1})\rho_{\phi}(h)\rho_{\phi}(\phi_{x})\rho(\sigma_{e})\rho_{\phi}(\phi_{y}^{-1})]=\text{Tr}[\rho_{\phi}(\phi_{x})\rho(\sigma_{e})\rho_{\phi}(\phi_{x}^{-1})].$

The first equality used the fact that $\rho_{\phi}$ is a representation. The
second inequality used that $\text{Tr}[AB]=\text{Tr}[BA]$ for general matrices
$A$ and $B$. The final inequality again used the fact that $\rho_{\phi}$ is a
representation and $\rho_{\phi}(h)\rho_{\phi}(h^{-1})=\rho_{\phi}(1)=I$.

##  4 The High Disorder Regime: Small $\kappa$

###  4.1 Expansion in Random Currents

In contrast to the low disorder regime, we cannot simply consider edges with
$\sigma_{e}\neq 1$ to be excitations blindly. Unfortunately, this means we
would have to consider a fundamentally new definition of cluster. We will
observe later that in our new definition of cluster, the knotting problem is a
serious difficulty when trying to split the configuration. Interestingly, this
difficulty even appears in the case of an abelian gauge group with low
$\kappa$. This means we have to treat the Higgs field model with non-abelian
and abelian gauge field with small $\kappa$ in the same way. To avoid
presenting some long combinatorial estimates, we will restrict to the
important case $d=4$ where we can cite these combinatorial estimates from
previous results and focus on the new ideas.

Intuitively, one can imagine in the small $\kappa$ case, there will be large
fluctuations in the Higgs field which will decorrelate very quickly along
large distances. To quantify this intuition in a cluster expansion, we
introduce the notion of a random cluster expansion. First, let $c$ be some
positive constant such that for any $\sigma_{e}$ $\phi_{x}$ and $\phi_{y}$, we
have that,

$g_{e}(\sigma,\theta)=\phi_{x}\text{Tr}[\rho(\sigma_{e})]\phi_{y}^{-1}+\overline{\phi_{x}\text{Tr}[\rho(\sigma_{e})]\phi_{y}^{-1}}>-c,c\geq
0.$ (71)

We see that the measure generated by the Hamiltonian would be the same whether
we considered $H_{N,\beta,\kappa}$ or the following Hamiltonian,

$\displaystyle\tilde{H}_{N,\beta,\kappa}:$ $\displaystyle=\sum_{p\in
P_{N}}\beta[\psi_{p}(\sigma)-\psi_{p}(1)]+\sum_{e\in
E^{u}_{N}}\kappa[g_{e}(\sigma,\phi)+c].$ (72)

We can relate the above model to a random current model. This has three sets
of random variables, the $\sigma$’s, the $\phi$’s, and a new set of edge
activations $I(e)$. The marginal distributions of $\sigma$’s and $\phi$’s are
those given by the Hamiltonian $\tilde{H}_{N,\beta,\kappa}$. Provided one has
a configuration $\phi_{x}$ and $\sigma_{e}$, the distribution of $I(e)$ is
given as follows:

$\mathbb{P}(I(e)=k)=\frac{[\kappa(g_{e}(\sigma,\phi)+c)]^{k}}{k!\exp[\kappa(g_{e}(\sigma,\phi)+c)]},$
(73)

where $k$ can be any non-negative integer. The benefit of the random current
expansion is that it is equivalent to finding the measure associated with the
following Hamiltonian.

$\displaystyle\mathcal{H}_{N,\beta,\kappa}(\phi,\sigma,I)$
$\displaystyle:=\sum_{p\in P_{N}}\beta(\psi_{p}(\sigma)-\psi_{p}(1))$ (74)
$\displaystyle+\sum_{e\in E_{N}}I(e)[\kappa(g_{e}(\sigma,\phi)+c)]-\log
I(e)!.$

$I(e)$ is only allowed to take non-negative integer values. Essentially, the
idea is that summing over $I(e)$ returns the exponential.

With this in hand, we can start writing up our definition of cluster.

######  Definition 7.

Given an configuration of the form $\mathcal{C}:=(\phi,\sigma,I)$, we can
define the set of excited vertices as

$\mathcal{V}:=\\{v\in V_{N}:\exists e\text{ s.t. }v\in\delta(e)\text{ and
}I(e)\neq 0\\}.$ (75)

We can define the support of the configuration as,

$\displaystyle\text{supp}\text{ }{\mathcal{C}}:=\\{p\in P_{N}:$
$\displaystyle\psi_{p}(\sigma)\neq\psi_{p}(1)\text{ or }$ (76)
$\displaystyle\exists v\in\mathcal{V}\text{ s.t. }v\text{ is a boundary vertex
of }p\\}.$

In words, a plaquette is excited if either the plaquette has a non-trivial
current running through it or it has a vertex on its boundary that is adjacent
to an edge that is excited with non-zero $I(e)$ value.

We still have the same notion of compatibility of plaquette sets $P_{1}$ and
$P_{2}$ from Definition 3. We also remark that the condition
$\psi_{p}(\sigma)\neq\psi_{p}(1)$ here is the same as the condition
$\sigma_{p}\neq 1$ in Section 4 of [4]. In addition, the support of a
configuration only depends on the values of $\sigma$ and $I$, not on $\phi$.

######  Remark 9.

We can show that, with regards to this definition, the excitations with
smallest support are still minimal vortices $P(e)$ centered around an edge
$e$. We remark that if there is an excited edge $e$ with $I(e)\neq 0$, then
the support will contain all plaquettes that share a boundary vertex with this
plaquette. This is certainly more than 12 oriented plaquettes ( 6 plaquette
pairs $\\{p,-p\\}$). Furthermore, Lemma 4.3.3 of [4] asserts that the smallest
vortex of the form $\text{supp}(\sigma,0)$( i.e. all $I(e)=0$) has 12 excited
oriented plaquettes( 6 plaquette pairs $\\{p,-p\\}$).

Furthermore, in order to avoid dealing with tedious edge cases, we will assume
a boundary condition on which we will not find excitation whose support
contains a plaquette on the boundary of $\Lambda_{N}$. One can show that for
loops sufficiently far away from the boundary, the effect of excitations on
the boundary is of size $e^{-N}$, where $N$ is the size of our lattice.

We now define the analogue of the function $\Phi$ as ,

$\displaystyle\Phi(P)$
$\displaystyle=\frac{1}{|G|^{|V_{N}|-1}|H/X|^{|V_{N}|}}\sum_{\text{supp}(\sigma,I)=P}\prod_{p\in
P_{N}}\exp[\beta(\psi_{p}(\sigma)-\psi_{p}(1))]$ (77)
$\displaystyle\sum_{\phi}\prod_{e=(x,y)\in
E_{N}}\frac{(\kappa(g_{e}(\sigma,\phi)+c))^{I(e)}}{I(e)!}.$

The normalization is introduced in order to derive some multiplicativity
relations in the future. Because we are dealing with non-abelian gauge groups,
we have a more complicated condition that ensures that if the support can be
decomposed into two compatible plaquette sets $P_{1}$ and $P_{2}$ satisfying
this complicated condition, then $\Phi(P_{1}\cup
P_{2})=\Phi(P_{1})\Phi(P_{2})$. A sufficient condition for this splitting is
the well-separated condition of Cao [4][Lemma 4.1.21].

######  Definition 8 (Well-Separated).

This definition is derived from the one used in [4]. We modify some of the
notation used in [4] to follow the conventions of this manuscript.

We say that two plaquette sets $P_{1}$ and $P_{2}$ are well-separated if we
can find some rectangle $R$ in $\Lambda_{N}$ that satisfies the following
properties: $P_{1}$ is contained in $P(R)$, the plaquettes inside $R$, while
$P_{2}$ is contained in $P(R)^{c}$, the plaquettes outside $R$. Furthermore,
$P_{1}$ and $P_{2}$ do not contain a plaquette belonging to the boundary of
$R$.

Wit this definition in hand, we can prove the following lemma.

######  Lemma 9.

Consider two plaquette sets $P_{1}$ and $P_{2}$ that are well-separated
according to Definition 8. Then, we can make the following assertion,

$\Phi(P_{1}\cup P_{2})=\Phi(P_{1})\Phi(P_{2}).$ (78)

###### Proof.

Part 1: Splitting the configuration $\sigma$

Let $P_{1}$ and $P_{2}$ be separated by the rectangle $R$. Let $T$ be a
spanning tree of $\Lambda_{N}$ that is simultaneously a spanning tree of $R$,
the complement $R^{c}$, and the boundary of $R$ with some basepoint $b$.

Notice by how we defined the support of our configuration $P_{1}\cup P_{2}$,
all plaquettes $p$ such that $\psi_{p}(\sigma)\neq\psi_{p}(1)$ are contained
in $P_{1}$ or in $P_{2}$.

We will say that any gauge field configuration $\sigma$ is gauged with respect
to the tree $T$ if $\sigma_{e}=1$ for all edges $e\in T$. We can apply Lemma
4.1.21 of [4] in order to find a unique gauge transformation on the vertices
$\eta^{\sigma}:V_{N}\to G$ such that
$\tilde{\sigma}_{e}:=\eta^{\sigma}_{x}\sigma_{e}(\eta^{\sigma}_{y})^{-1}$ with
$\eta_{b}=1$ at the root and $\tilde{\sigma}$ is gauged with respect to the
tree $T$.

Once defining $\tilde{\sigma}_{e}$, we see that $\tilde{\sigma}$ can be split
into a product $\tilde{\sigma}_{1}$ and $\tilde{\sigma}_{2}$ such that
$\tilde{\sigma}_{e}=(\tilde{\sigma}_{1})_{e}(\tilde{\sigma}_{2})_{e}$ for all
edges $e=(x,y)$ such that $\tilde{\sigma}_{1}$ and $\tilde{\sigma}_{2}$
satisfy the following properties:
$(\tilde{\sigma}_{1})_{e}=\tilde{\sigma}_{e}$ for edges $e$ inside $R$ and
$(\tilde{\sigma}_{1})_{e}=1$ for edges $e$ outside of $R$. In addition,
$(\tilde{\sigma}_{2})_{e}=\tilde{\sigma}_{e}$ for edges $e$ outside of $R$ and
$(\tilde{\sigma}_{2})_{e}=1$ for edges $e$ inside of $R$. This treats how we
would divide $\sigma$ into two disjoint supports. Now, let us see how we would
deal with activated edges.

Using the activated edges of $I(e)$, one can generate a subgraph $\mathcal{I}$
of the lattice $\Lambda_{N}$ whose edges consist of the activated edges with
$I(e)\neq 0$. One can consider the connected clusters
$C_{1},C_{2},\ldots,C_{m}$. We will observe later in the last line of (79)
that our function $\Phi$ will not share variables between the different
clusters; this will allow us to split $\Phi$ appropriately as a product. We
also remark that given a cluster $C_{i}$, all of its vertices are either
boundary vertices of plaquettes of $P_{1}$ or of $P_{2}$ exclusively.
Otherwise, there exists some vertices $v$ and $w$ adjacent in the cluster
$C_{i}$ such that $v$ is the boundary vertex of some plaquette $p_{1}$ in
$P_{1}$ and $w$ is the boundary vertex of some plaquette $p_{2}$ in $P_{2}$.
However, the edge $(v,w)$ is activated; thus all plaquettes that either have
$v$ or $w$ as a boundary vertex are in the support of the configuration. In
the connectivity graph of plaquettes $G_{2}$ from Definition 3, this would
imply that $p_{1}$ and $p_{2}$ are connected. This contradicts our assumption
that $P_{1}$ and $P_{2}$ are disconnected components of the support.

We can let $V_{1}$ be the vertices that bound excited edges and are boundary
vertices of some plaquette in $P_{1}$ and we let $I^{a}_{1}$ be the activated
edges whose boundary vertices are both in $V_{1}$. We define $V_{2}$ and
$I^{a}_{2}$ similarly. From our earlier discussion, $I^{a}_{1}\cup I^{a}_{2}$
is a disjoint union covering all activated edges and $V_{1}$ is disjoint from
$V_{2}$.

For an activation $I$, we define $I_{1}(e)$ to be the activation restricted to
edges of $I_{1}^{a}$ and $I_{2}(e)$ to be the activation restricted to the
edges of $I_{2}^{a}$.

We now claim that the map
$(\tilde{\sigma},I)\to(\tilde{\sigma}_{1},I_{1}),(\tilde{\sigma}_{2},I_{2})$
is a bijection from those configurations $(\tilde{\sigma},I)$ whose support is
$P_{1}\cup P_{2}$ and $\tilde{\sigma}$ is gauged with respect to the tree $T$
and pairs of configuration
$(\tilde{\sigma}_{1},I_{1}),(\tilde{\sigma}_{2},I_{2})$ whose supports are
$P_{1}$ and $P_{2}$ respectively and $\tilde{\sigma}_{1},\tilde{\sigma}_{2}$
are gauged with respect to the tree $T$. To show that this is a map between
the proposed spaces, it suffices to show that the support of
$(\tilde{\sigma}_{1},I_{1})$ is $P_{1}$ exactly. By our construction,
$(\tilde{\sigma}_{1},I_{1})$ has support contained in $P_{1}$. Now, let $p$ be
an arbitrary plaquette in $P_{1}$. Since $p$ was in the support of
$(\tilde{\sigma},P)$, we know that either
$\psi_{p}(\tilde{\sigma})\neq\psi_{p}(1)$ or there is a vertex $v$ on the
boundary of $p$ such that $v$ is adjacent to an excited edge, $e$ with
$I(e)\neq 0$.

Consider the case that $\psi_{p}(\tilde{\sigma})\neq\psi_{p}(1)$. We know that
for $p\in P_{1}$ that $\psi_{p}(\tilde{\sigma})=\psi_{p}(\tilde{\sigma}_{1})$
by construction. Thus, in this case we have that
$p\in\text{supp}(\tilde{\sigma}_{1},I_{1})$. Now consider the case that there
is a vertex $v$ in the boundary of $p_{1}$ such that $v$ is adjacent to an
activated edge $e$ in $I$. Our earlier discussion shows that $v$ must be in
$V_{1}$ and $e$ must be in $I^{a}_{1}$. Thus, $v$ is still adjacent to an
activated edge in $I_{1}$.

Since all of our maps have been defined by restriction, it is easy to show the
proposed map is a bijection. The splitting is clearly unique and one can
combine an arbitrary split by multiplying the $\sigma$’s and combining the
$I$’s. This proves our claim on the properties of the map
$(\tilde{\sigma},I)\to(\tilde{\sigma}_{1},I_{1}),(\tilde{\sigma}_{2},I_{2})$.

Part 2: The Multiplicativity of $\Phi$

We rewrite our sum in $\Phi$ with respect to a base representative
$\tilde{\sigma}_{e}$ gauged with respect to the spanning tree $T$ and a
separate field $\eta:V_{N}\to G$, which re-introduces the gauge invariance.
The sum in $\Phi(P_{1}\cup P_{2})$ can be written as,

$\displaystyle\Phi(P_{1}\cup P_{2})$
$\displaystyle=\frac{1}{|G|^{|V_{N}|}|H/X|^{|V_{N}|}}\sum_{\begin{subarray}{c}\text{supp}(\tilde{\sigma},I)=P_{1}\cup
P_{2}\\\ \tilde{\sigma}\text{ gauged with }T\end{subarray}}\prod_{p\in
P_{1}\cup P_{2}}\exp[\beta(\psi_{p}(\tilde{\sigma})-\psi_{p}(1))]$ (79)
$\displaystyle\sum_{\eta,\phi}\prod_{e=(x,y)\in\\{e:I(e)\neq
0\\}}\frac{(\kappa(g(\eta(\tilde{\sigma}),\phi)+c))^{I(e)}}{I(e)!}.$

The new variables $\eta:V_{N}\to G$ act as follows: it takes a gauge field
configuration $\sigma$ to the configuration $\eta(\sigma)$ that takes values
$\eta(\sigma)_{e}=\eta_{x}\sigma_{e}\eta_{y}^{-1}$ for the edge $e=(x,y)$.
This reintroduces the gauge invariance we removed when defining
$\tilde{\sigma}$. We remark that we get an extra factor of $\frac{1}{|G|}$ due
to the removal of the gauge fixing $\eta_{b}=1$ for the root. This can be done
by a global transformation multiplying each element $\eta$ by the same element
$g\in G$.

Recall our map
$(\tilde{\sigma},I)\to(\tilde{\sigma}_{1},I_{1}),(\tilde{\sigma}_{2},I_{2})$
from earlier. We see that it can split the product as follows.

$\displaystyle\Phi(P_{1}\cup P_{2})$
$\displaystyle=\frac{(|G||H/X|)^{|V_{N}|-|V_{1}|-|V_{2}|}}{(|G|||H/X|)^{|V_{N}|}}$
(80)
$\displaystyle\sum_{\text{supp}(\tilde{\sigma}_{1},I_{1})=P_{1}}\prod_{p\in
P_{1}}\exp[\beta(\psi_{p}(\tilde{\sigma}_{1})-\psi_{p}(1))]$
$\displaystyle\times\sum_{\begin{subarray}{c}\eta_{v},\phi_{v}\\\
vinV_{1}\end{subarray}}\prod_{\begin{subarray}{c}e=(x,y)\\\ I_{1}(e)\neq
0\end{subarray}}\frac{(\kappa(g(\eta(\tilde{\sigma}_{1}),\phi)+c))^{I_{1}(e)}}{I_{1}(e)!}$
$\displaystyle\sum_{\text{supp}(\tilde{\sigma}_{2},I_{2})=P_{2}}\prod_{p\in
P_{2}}\exp[\beta(\psi_{p}(\tilde{\sigma}_{2})-\psi_{p}(1))]$
$\displaystyle\times\sum_{\begin{subarray}{c}\eta_{v},\phi_{v}\\\
vinV_{2}\end{subarray}}\prod_{\begin{subarray}{c}e=(x,y)\\\ I_{2}(e)\neq
0\end{subarray}}\frac{(\kappa(g(\eta(\tilde{\sigma}_{2}),\phi)+c))^{I_{2}(e)}}{I_{2}(e)!}.$

We used the following facts:

  1. 1.

Firstly, the mapping
$(\tilde{\sigma},I)\to(\tilde{\sigma}_{1},I_{1}),(\tilde{\sigma}_{2},I_{2})$
is a bijection.

  2. 2.

Secondly, the product of the $\exp[\beta\psi_{p}]$ factors split between the
disjoint supports $P_{1}$ and $P_{2}$.

  3. 3.

Thirdly, the products $(\kappa(g(\eta(\tilde{\sigma},\phi)+c))^{I_{1}(e)}$
split between the disjoint sets $E_{1}$ and $E_{2}$ where the only variable
values of $\eta$ and $\phi$ affecting the values of the product over $I_{1}$(
$I_{2}$ resp.) are those in $V_{1}$ (resp. $V_{2}$).

  4. 4.

Fourthly, we used the fact that the restriction of $\tilde{\sigma}$ to
$E_{1}$(resp. $E_{2}$) is $\tilde{\sigma}_{1}$(resp. $\tilde{\sigma}_{2}$).

  5. 5.

Finally, the sum over all remaining variables $\phi_{x}$ and $\eta_{x}$
belonging to variables not in $V_{1}$ or $V_{2}$ is
$(|G||H/X|)^{|V|-|V_{1}|-|V_{2}|}$.

Now, in the second line, we can reintroduce a dummy summation over new
variables $\eta^{1}_{v},\phi^{1}_{v}$ for vertices $v\not\in V_{1}$ and a new
dummy summation over variables $\eta^{2}_{v},\phi^{2}_{v}$ for vertices
$v\not\in V_{2}$ in the third line. We will also relabel the variables
$\eta_{v},\phi_{v}$ in the second line as $\eta^{1}_{v},\phi^{1}_{v}$ to
simplify notation; similarly, we relabel $\eta_{v},\phi_{v}$ as
$\eta^{2}_{v},\phi^{2}_{v}$ in the third line.

Compensating for the extra multiplicative factor of
$(|G||H/X|)^{|V_{N}|-|V_{1}|}$ in the second line due to the new variables we
added, as with $(|G||H/X|)^{|V_{N}|-|V_{2}|}$ in the second line, we see that
we can write the above expression as expression

$\displaystyle\Phi(P_{1}\cup P_{2})=$ (81)
$\displaystyle\frac{1}{(|G||H/X|)^{|V_{N}|}}\sum_{\text{supp}(\tilde{\sigma}_{1},I_{1})=P_{1}}\prod_{p\in
P_{1}}\exp[\beta(\psi_{p}(\tilde{\sigma}_{1})-\psi_{p}(1))]$
$\displaystyle\times\sum_{\begin{subarray}{c}\eta^{1}_{v},\phi^{1}_{v}\end{subarray}}\prod_{\begin{subarray}{c}e=(x,y)\\\
I_{1}(e)\neq
0\end{subarray}}\frac{(\kappa(g(\eta^{1}(\tilde{\sigma}_{1}),\phi^{1})+c))^{I_{1}(e)}}{I_{1}(e)!}$
$\displaystyle\frac{1}{(|G||H/X|)^{|V_{N}|}}\sum_{\text{supp}(\tilde{\sigma}_{2},I_{2})=P_{2}}\prod_{p\in
P_{2}}\exp[\beta(\psi_{p}(\tilde{\sigma}_{2})-\psi_{p}(1))]$
$\displaystyle\times\sum_{\begin{subarray}{c}\eta^{2}_{v},\phi^{2}_{v}\end{subarray}}\prod_{\begin{subarray}{c}e=(x,y)\\\
I_{2}(e)\neq
0\end{subarray}}\frac{(\kappa((\eta^{2}(\tilde{\sigma}_{2}),\phi^{2})+c))^{I_{2}(e)}}{I_{2}(e)!}.$

But, this is just the product $\Phi(P_{1})\Phi(P_{2})$.

∎

###  4.2 Knot Expansion

The above Lemma 9 is most useful when paired with the concept of knot
expansions. A knot expansion serves as a more careful way to split the support
of a configuration into disjoint components that would be better for analysis.

######  Definition 9.

Let $V_{1}\cup V_{2}\ldots\cup V_{m}$ be a vortex decomposition. A partition
of these vortices into knots $K_{1}\cup K_{2}\cup\ldots\cup K_{n}$ will be
called a knot decomposition if for each $j$, there is a box $R_{j}$ that well-
separates $K_{j}$ from $K_{j+1}\cup K_{j+2}\cup\ldots\cup K_{n}$. As a
consequence of Lemma 9, we would have that $\Phi(K_{1}\cup K_{2}\cup\ldots\cup
K_{n})=\prod_{i=1}^{n}\Phi(K_{i})$.

A knot decomposition will be maximal if there is no further way to partition
any $K_{j}=K_{j}^{1}\cup K_{j}^{2}$ into non-empty components such that
$K_{j}^{1}$ and $K_{j}^{2}$ are well-separated by a cube in $\Lambda_{N}$.

######  Remark 10.

Due to Lemma 4.1.10 of [4], if one of the vortices in the support of a
configuration is a minimal vortex $P(e)$ centered around some edge $e$, then
one would always be able to separate out this minimal vortex in the knot
decomposition; though this slightly abuses the notion of separating box. We
will usually choose these minimal vortices to be the first knots in the knot
decomposition. This is exactly the convention of [4].

With the notion of knot decomposition and Lemma 9 in hand, we can rather
easily prove the following statement. This is essentially Lemma 4.3.10 of [4].

######  Lemma 10.

The probability that, under the Hamiltonian $\mathcal{H}_{N,\beta,\kappa}$, we
will observe a configuration $(\sigma,\phi,I)$ whose support under the knot
decomposition contains $K$ would be less than $\Phi(K)$.

###### Proof.

Consider a knot decomposition of the form $K_{1}\cup\ldots\cup K_{j}\cup\ldots
K_{m}$ that includes $K=K_{j}$.
$|G|^{|V_{N}|-1}|H/X|^{|V_{N}|}\Phi(K_{1}\cup\ldots\cup K_{j}\cup\ldots\cup
K_{m})$ will be the sum of $\exp[\mathcal{H}_{N,\beta,\kappa}]$ among all
configurations that have knot decomposition $K_{1}\cup\ldots\cup
K_{j}\cup\ldots\cup K_{m}$.

We see that,

$\sum_{K\in\text{supp}(\sigma,\phi,I)}\exp[\mathcal{H}_{N,\beta,\kappa}(\sigma,\phi,I)]=|G|^{|V_{N}|-1}|H/X|^{|V_{N}|}\sum_{K\in
K_{1}\cup K_{2}\ldots K_{m}}\prod_{i=1}^{m}\Phi(K_{i}).$ (82)

Where we abuse notation to say that $K\in K_{1}\cup\ldots\cup K_{m}$ means
that $K$ is one of the knots in the decomposition.

We see that if we remove $K=K_{j}$ from $K_{1}\cup\ldots\cup K_{m}$, then ,
though $K_{1}\cup\ldots\cup K_{j-1}\cup K_{j+1}\ldots\cup K_{m}$ may split
further in a knot decomposition, we would still satisfy a well-separatedness
condition to assert that $\Phi(K_{1}\cup\ldots\cup K_{j-1}\cup K_{j+1}\cup
K_{m})=\prod_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}^{n}\Phi(K_{i})$.

By considering those configurations whose support would be contained in
$K_{1}\cup K_{j-1}\cup K_{j+1}\cup K_{m}$ for all knot decompositions
$K_{1}\cup\ldots\cup K_{m}$ that contain $K=K_{j}$, we see the partition
function can be bounded below by,

$Z_{\mathcal{H}_{N,\beta,\kappa}}\geq|G|^{|V_{N}|-1}|H/X|^{|V_{N}|}\sum_{K\in
K_{1}\cup K_{2}\ldots\cup K_{m}}\prod_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}^{m}\Phi(K_{i}).$ (83)

Taking the ratio of the term in (82) with our lower bound on the partition
function gives us that the probability of seeing a configuration whose support
contains $K$ is bounded by $\Phi(K)$.

∎

We also have the following quantitative bound on $\Phi(K)$.

######  Lemma 11.

Let $K$ be some union of vortices $V_{1}\cup V_{2}\cup\ldots\cup V_{N}$ with
$2k$ oriented plaquettes ( $k$ pairs of plaquettes $\\{p,-p\\}$) in the
support. Define the constant,

$\alpha_{\beta,\kappa}:=2^{4}|G|\max(\exp[2\beta(\max_{a\neq 1\in
G}\text{Re}[\psi_{p}(a)-\psi_{p}(1)])],(\kappa\mathfrak{d}\exp[\kappa\mathfrak{d}])^{\frac{1}{2{8\choose
2}}}),$

where $\mathfrak{d}$ is the maximum value of $g_{e}(\sigma,\phi)+c$ for all
possible values $\sigma$ and $\phi$. Note that this maximum does not depend on
the edge $e$.

Then, we have the following bound on $\Phi(K)$.

$\Phi(K)\leq\alpha_{\beta,\kappa}^{k}.$ (84)

###### Proof.

Choose some spanning tree $\mathcal{T}$. We see that we can express $\Phi(K)$
with respect to configurations gauged with respect to $\mathcal{T}$ as,

$\displaystyle\Phi(K)=$
$\displaystyle\frac{1}{|G|^{|V_{N}|}|H/X|^{|V_{N}|}}\sum_{\text{supp}(\tilde{\sigma},I)=K}\prod_{p\in
P_{N}}\exp[\beta(\psi_{p}(\tilde{\sigma})-\psi_{p}(1)]$ (85)
$\displaystyle\sum_{\eta_{v},\phi_{v}}\prod_{e=(x,y):I(e)\neq
0}\frac{(\kappa(g(\eta(\tilde{\sigma}),\phi)+c))^{I(e)}}{I(e)!}.$

Notice that the support of $\tilde{\sigma},I$ only depends on the set $A_{I}$