# Relaxation time and topology in 1D $O(N)$ models

Pietro Caputo  Università Roma Tre<EMAIL_ADDRESS>,  Sébastien Ott
EPFL<EMAIL_ADDRESS>and  Assaf Shapira  MAP5, Université Paris Cité
<EMAIL_ADDRESS>
###### Abstract.

We discuss the relaxation time (inverse spectral gap) of the one dimensional
$O(N)$ model, for all $N$ and with two types of boundary conditions. We see
how its low temperature asymptotic behavior is affected by the topology. The
combination of the space dimension, which here is always $1$, the boundary
condition (free or periodic), and the spin state $\mathbb{S}^{N-1}$,
determines the existence or absence of non-trivial homotopy classes in some
discrete version. Such non-trivial topology reflects in bottlenecks of the
dynamics, creating metastable states that the system exits at exponential
times; while when only one homotopy class exists the relaxation time depends
polynomially on the temperature. We prove in the one dimensional case that,
indeed, the relaxation time is a proxy to the model’s topological properties
via the exponential/polynomial dependence on the temperature.

##  1\. Introduction

The investigation of the low-temperature behavior of classical spin systems
with continuous symmetry, such as the $O(N)$ model on a lattice, is a source
of many fascinating questions in equilibrium statistical mechanics [7, 17].
For example, in the two dimensional $XY$ model, a deep understanding of the
interplay between the spin wave approximation and topological aspects such as
vortex formation poses significant mathematical challenges, see [5, 13, 15, 9,
8] for some classical works, and see e.g. [16, 10] for more recent studies. To
delve deeper into these questions, it is natural to study the Langevin
dynamics associated to the $O(N)$ model, that is the reversible diffusion
process with stationary distribution given by the $O(N)$ Gibbs measure. In the
mean-field case, a comprehensive analysis of the relaxation time, or inverse
spectral gap, of Langevin dynamics for the $O(N)$ model, has been achieved
recently in [2]. In particular, these results show that, when $N\geq 2$, in
sharp contrast with the case of the Glauber dynamics for the Ising model
($N=1$), the relaxation time of the mean field $O(N)$ model is at most
linearly growing with the size of the system, at any fixed temperature. It is
widely believed that such bounds should continue to hold for short range
models as well, see e.g. [3]. In particular, it can be conjectured that for
all lattice dimension $d\geq 2$, the $O(N)$ model on a lattice box with side
$L$, for all $N\geq 2$, has relaxation time at most of order $L^{d}$ at any
fixed temperature, regardless of the boundary conditions. However, even
establishing that relaxation times grow at most at a polynomial rate with $L$
is a notoriously difficult open problem.

As a much more modest objective, in this note, we explore the presence of
significant topological effects in the simpler one-dimensional setting. In a
one-dimensional system, it is well known that the relaxation time is of order
$1$ at any fixed temperature. However, it was recently observed in [4] that in
the one-dimensional XY model ($N=2$), when periodic boundary conditions are
imposed, as the inverse temperature $\beta$ grows logarithmically with the
size $L$ of the chain, topologically induced metastable phases emerge, which
correspond to distinct global winding numbers of the spin chain. When $\beta$
increases as $\log L$, the free energy barrier between these states also
increases linearly with $\beta$, leading to hitting times that are
exponentially large in $\beta$. As we will see, this system exhibits
relaxation times that grow exponentially with $\beta$. This phenomenon is
specific of the periodic chain, and cannot occur for e.g. free boundary
conditions. Indeed, it is a consequence of the fact that the global winding
number of a periodic XY chain is a topological invariant, and two distinct
phases cannot be connected by a homotopy, giving rise to a topological
bottleneck. The main goal of this paper is to show that these topological
effects on the dynamics are not present when $N\geq 3$. The point is that when
$N\geq 3$, one can connect any two configurations of the spin chain by a
continuous deformation. In particular, by using a continuous version of the
canonical path method, we will show that, as a function of $\beta$, the
relaxation time of the one-dimensional $O(N)$ model with $N\geq 3$ can grow at
most polynomially in a periodic chain. We will also note that the same holds
true for $O(N)$ models, this time for any $N\geq 2$, if one takes free
boundary conditions instead.

###  1.1. Model and results

Given integers $N\geq 2$, and $L\geq 2$, the one-dimensional $O(N)$ model of
size $L$, with free or periodic boundary conditions is defined, respectively,
by the Hamiltonians

$\displaystyle H^{\mathrm{f}}_{L}(S)=-\sum_{i=1}^{L-1}S_{i}\cdot
S_{i+1},\qquad H^{\mathrm{per}}_{L}(S)=-\sum_{i=1}^{L}S_{i}\cdot S_{i+1}.$ (1)

Here $S_{i}\in\mathbb{S}^{N-1}$ denotes the $i$-th spin, $S_{i}\cdot S_{i+1}$
is the usual scalar product for vectors in $\mathbb{R}^{N}$, and we set
$S_{L+1}\equiv S_{1}$ to obtain periodic boundary conditions in
$H^{\mathrm{per}}_{L}(S)$. We write $\nu$ for Lebesgue’s measure on
$\mathbb{S}^{N-1}$ and let $\nu_{L}$ denote the corresponding product measure
on the space $\Omega_{L}=(\mathbb{S}^{N-1})^{L}$ of the spin chain
configurations. Thus, the free and periodic boundary condition $O(N)$ Gibbs
measure at inverse temperature $\beta>0$ is given, respectively, by the
probability measures on $\Omega_{L}$ defined as

$\displaystyle\mu_{L,\beta}^{\mathrm{f}}(dS)=\frac{\exp\left(-\beta
H^{\mathrm{f}}_{L}(S)\right)}{Z^{\mathrm{f}}_{L,\beta}}\,\nu_{L}(dS),\qquad\mu_{L,\beta}^{\mathrm{per}}(dS)=\frac{\exp\left(-\beta
H^{\mathrm{per}}_{L}(S)\right)}{Z^{\mathrm{per}}_{L,\beta}}\,\nu_{L}(dS),$ (2)

where the partition function is defined, respectively, by

$Z^{\mathrm{f},\mathrm{per}}_{L,\beta}=\int_{\Omega_{L}}\exp\left(-\beta
H^{\mathrm{f},\mathrm{per}}_{L}(S)\right)\nu_{L}(dS).$

The Langevin dynamics is defined as the reversible diffusion process on
$\Omega_{L}$ with infinitesimal generator

$\displaystyle\mathcal{L}^{\mathrm{f},\mathrm{per}}_{L,\beta}$
$\displaystyle=\sum_{i=1}^{L}\left(\frac{1}{\beta}\,D_{i}^{2}-(D_{i}H_{L}^{\mathrm{f},\mathrm{per}})\cdot
D_{i}\right),$ (3)

where $D_{i}$ denotes the gradient $\nabla_{\mathbb{S}^{N-1}}$ on the unit
sphere acting on the $i$-th spin $S_{i}$. The generator (3) defines a self-
adjoint operator on $L^{2}(\Omega_{L},\mu_{L,\beta}^{\mathrm{f}})$ and
$L^{2}(\Omega_{L},\mu_{L,\beta}^{\mathrm{per}})$, respectively. For any smooth
function $f:\Omega_{L}\to\mathbb{R}$, the associated Dirichlet form is given
by

$\displaystyle\mathcal{D}^{\mathrm{f},\mathrm{per}}_{L,\beta}(f,f)=\frac{1}{\beta}\sum_{i=1}^{L}\int_{\Omega_{L}}\|D_{i}f(S)\|^{2}\mu_{L,\beta}^{\mathrm{f},\mathrm{per}}(dS),$
(4)

and $\|\cdot\|$ denotes the vector norm. The spectral gap is defined by the
variational principle

$\displaystyle\mathop{\rm
gap}\nolimits^{\mathrm{f},\mathrm{per}}_{L,\beta}=\inf_{f}\frac{\mathcal{D}^{\mathrm{f},\mathrm{per}}_{L,\beta}(f,f)}{{\rm
Var}^{\mathrm{f},\mathrm{per}}_{L,\beta}(f)}\,,$ (5)

where ${\rm
Var}^{\mathrm{f},\mathrm{per}}_{L,\beta}(f)=\mu^{\mathrm{f},\mathrm{per}}_{L,\beta}(f^{2})-\mu^{\mathrm{f},\mathrm{per}}_{L,\beta}(f)^{2}$
denotes the variance functional with respect to
$\mu^{\mathrm{f},\mathrm{per}}_{L,\beta}$ and $f$ ranges over all non-constant
smooth functions on $\Omega_{L}$. The relaxation time is defined as the
inverse of the spectral gap

$\displaystyle T_{\rm
rel}^{\mathrm{f},\mathrm{per}}(L,\beta)=\frac{1}{\mathop{\rm
gap}\nolimits^{\mathrm{f},\mathrm{per}}_{L,\beta}}.$ (6)

At any fixed $\beta$ the one-dimensional nearest neighbor spin system
satisfies exponential decay of covariances uniformly in the boundary
conditions [7]. Then it is not difficult to prove, see e.g. [14], that the
system has a uniformly positive spectral gap, that is there exists a constant
$C(N,\beta)$ independent of $L$, such that

$\displaystyle T_{\rm rel}^{\mathrm{f},\mathrm{per}}(L,\beta)\leq C(N,\beta).$
(7)

Here we are interested in detecting topologically induced slowdown effects on
the relaxation to equilibrium which could appear in the case where $\beta$
grows with $L$. In particular, as shown in [4], these phenomena do occur in
the $XY$ model ($N=2$) in the case of periodic boundary conditions, when
$\beta$ is at least of order $\log L$.

We start by showing that in the case of free boundary conditions there is no
topologically induced slowdown, in the sense that the relaxation time is upper
bounded as follows.

######  Theorem 1.1 ($O(N)$ model with free boundary).

For any $N\geq 2$, $L\in\mathbb{N}$, and $\beta\geq 1$,

$\displaystyle T_{\rm rel}^{\mathrm{f}}(L,\beta)\leq
C(N)\,L^{2}\beta^{(N+1)/2},$ (8)

where $C(N)$ depends only $N$. In particular, for each fixed $N$, it grows at
most polynomially in $\beta$.

We note that, in light of the bound (7), the above estimate becomes relevant
only when $\beta$ grows with $L$. In this case, if $\beta$ is at least $C\log
L$ for some large $C$, as we discuss in Section 1.2 below, a diffusive scaling
of the relaxation time should be expected, and thus at least qualitatively,
the bound in Theorem 1.1 should be tight. In the case of periodic boundary
conditions, in agreement with [4], we show the following bounds quantifying
the topologically induced slowdown for $N=2$.

######  Theorem 1.2 ($XY$ model on the cycle).

Let $N=2$. For $L\in\mathbb{N}$, and $\beta\geq 1$,

$\displaystyle T_{\rm rel}^{\mathrm{per}}(L,\beta)\leq
C\beta^{3/2}e^{2\beta}L^{2},$ (9)

for some absolute constant $C$. Moreover, for any $L\in\mathbb{N}$, there
exists a constant $c(L)$ depending on $L$ such that for all $\beta\geq 1$,

$\displaystyle T_{\rm rel}^{\mathrm{per}}(L,\beta)\geq
c(L)\beta\,e^{(2-C_{0}L^{-1})\beta},$ (10)

where $C_{0}$ is an absolute constant.

The lower bound (10) is based on a rather crude argument and provides no
meaningful $L$-dependance. However, as discussed in Section 3.2, the leading
exponential term $e^{2\beta}$ in (9)-(10) captures the correct metastable
behavior associated to the energy barrier of size $2\beta$ between states with
winding number zero and states with non-zero winding number. We refer to
Remark 3.1 for the sketch of a finer energy-entropy argument providing
quantitative $L$ dependance in the metastable regime $\beta\geq C\log L$.

Finally, we prove that there is no topologically induced slowdown for $N\geq
3$.

######  Theorem 1.3 ($O(N)$ model on the cycle, $N\geq 3$).

For $N\geq 3$, $L\in\mathbb{N}$, and $\beta\geq 1$,

$\displaystyle T_{\rm rel}^{\mathrm{per}}(L,\beta)\leq C(L)\beta.$ (11)

Moreover, in the case of the Heisenberg model ($N=3$), one can take
$C(L)=e^{CL\log L}$ for some constant $C$ not depending on $L$ and $\beta$.

###  1.2. Discussion, conjectures and open problems

We emphasize that these estimates are far from optimal and do not capture all
features of relaxation to equilibrium of the spin chain. However, they are
sufficient to rule out the presence of topological bottlenecks in the
relaxation process for all $N\geq 2$ in the case of free boundary and for
$N\geq 3$ in the case of periodic boundary. Let us give some comments on our
proofs. Roughly speaking, for Theorem 1.1 we use the fact that the spin chain
has a product structure in the case of free boundary conditions, when one
considers the “increment” variables $S_{i+1}-S_{i}$. This is achieved by a
suitable change of variables that allows a convenient representation for the
Hamiltonian. In this setup, a simple tensorization argument applies and one
obtains the estimate (8) by changing back to the original spin variables. The
explicit dependance on the variable $\beta$ in (8) is obtained by a
quantitative bound on the spectral gap for a single increment variable. The
upper bound in (9) is obtained by reducing the problem to the free boundary
case via a perturbation argument. On the other hand, for the lower bound in
(9) we use an upper bound on the Cheeger constant. This is based on the choice
of a suitable bottleneck event that was already analysed in [4].

The proof of Theorem 1.3 requires more work. We use a continuous version of
the so-called canonical path method, see e.g. [18] for a classical formulation
in the discrete setting. The main idea is to construct a path consisting of a
continuous, energy decreasing transformation, which allows one to move any
given configuration of $L$ points on a sphere $\mathbb{S}^{N-1}$ to a
configuration where all points lie in a small neighbourhood of a pole of the
sphere. Once the system is confined to such a neighbourhood, convexity
considerations allow us to conclude the desired statement. Checking that such
a construction is possible, and controlling the entropy associated to the
contracting path requires some non-trivial analysis. Note that this can only
work in the case $N\geq 3$, since for $N=2$ it is prevented by the topological
obstruction discussed above.

Concerning dimension higher than one, in light of the above, it is natural to
conjecture that if one considers the $O(N)$ model on the $d$-dimensional torus
$(\mathbb{Z}/L\mathbb{Z})^{d}$, then topological bottlenecks are related to
the homotopy structure. More precisely, one expects the relaxation time to
grow exponentially with $\beta$ when there are some non-trivial homotopy
classes of maps from $\mathbb{T}^{d}\to\mathbb{S}^{N-1}$.

Indeed, when $\beta$ is large the angle between two neighboring spins is
small, and the discrete configuration of the spin system looks like a
continuous field, i.e., a map from $\mathbb{T}^{d}$ to $\mathbb{S}^{N-1}$.
With this in mind, the dynamics of the spin system corresponds to homotopy of
the continuous field, and if there are several homotopy classes, then moving
from one to the other requires the creation of a discontinuity. The energetic
cost of this discontinuity creates the topological bottleneck.

While this description provides a good intuition to explain the results in
this paper, in a more general setting we expect such metastability to depend
also on the Riemannian structure of $\mathbb{S}^{N-1}$ and not solely its
topology. Consider for example an hourglass spin state, which has the same
topology as $\mathbb{S}^{2}$ but non-constant curvature. As in the $O(3)$
model, the ground state is when all spins point in the same direction.
However, this system contains a metastable state, where spins are placed on
the narrow part of the hourglass, equally spaced, so that the configuration
has non-zero winding. In this example, the energetic cost is due to the
continuous transformation and not the creation of discontinuities. We
therefore see that endowing the same topological sphere with a Riemannian
structure other than the standard one (i.e., the homogeneous metric induced by
the scalar product in $\mathbb{R}^{3}$) could change the metastability
properties.

This example suggests that homotopy classes of maps are not the correct
candidate to be a marker for metastability. To remedy this, introduce the
following energy functional. For $M,M^{\prime}$ two compact Riemannian
manifolds, and $f:M\to M^{\prime}$, let $\mathcal{E}(f)$ be the energy of $f$:

$\mathcal{E}(f)=\int_{M}\|Df(x)\|^{2}_{f(x)}\differential x,$ (12)

where $\|\ \|_{p}$ is the norm on the tangent space of $M^{\prime}$ at $p$.

Extrema of the functional $\mathcal{E}$, called _harmonic maps_ , are well
studied mathematical objects, see e.g. [6]. Loosely stated, our general
conjecture is that if one considers a model on some graph $G$ with spin taking
value in the manifold $M^{\prime}$, and if

  * •

the graph $G$ “approximates well” the manifold $M$ (for example: the graph
Laplacian on $G$ is close in some sense to the Laplacian on $M$),

  * •

the Hamiltonian of the model is a “approximate discrete version” of the energy
functional $\mathcal{E}$. This is the generic behavior of nearest-neighbor
attractive interaction near low energy states.

Then, the existence of metastable states is equivalent to the existence of
non-trivial local minimizers of $\mathcal{E}$. Specializing to the spin $O(N)$
model, one obtains the next (more precise) conjecture.

######  Conjecture 1.4.

Let $M$ be a compact Riemannian manifold, and define the energy functional
operating on $f:M\to\mathbb{S}^{N-1}$ as

$\mathcal{E}(f)=\int_{M}\|Df(x)\|^{2}\differential x,$ (13)

where the norm is taken with respect to the standard Riemannian metric on the
sphere. Consider the $O(N)$ model on a graph $G$ discretizing $M$. Then the
relaxation time of the corresponding Langevin dynamics grows exponentially
fast with $\beta$ if and only if $\mathcal{E}$ has non-trivial (i.e., non-
constant) local minima.

In particular, for the case of $\mathbb{Z}^{d}$ with free boundary conditions,
there should be no such bottlenecks, that is relaxation times growing
polynomially as a function of $\beta$, for all $N\geq 2$, since here the
homotopy group is always trivial and the harmonic mappings are all constant.
However, we leave it as an open problem to obtain quantitative bounds in
dimension $d>1$.

An interesting example to study would be the case $N=4$, $M=\mathbb{S}^{3}$:
there are several homotopy classes of continuous maps from $\mathbb{S}^{3}$ to
$\mathbb{S}^{3}$ (even countably many, by the Hurewicz theorem), but the
energy $\mathcal{E}$ has no non-trivial local minima [6], so our conjecture is
that there is no metastable behaviour in this case.

###  1.3. Scaling limit in low temperature heuristics

We conclude this introduction with a brief informal discussion of the behavior
of the system for extremely low temperature, that is when $\beta$ is large as
a function of $L$, that could serve as a heuristic guide for a more precise
analysis in this regime. For the sake of simplicity, we discuss the problem
only for the XY model ($N=2$) and give only a brief comment on the case $N\geq
3$ afterwards. When $N=2$ we may parametrize $S_{i}$ by a single angle
$X_{i}\in[-\pi,\pi]$ and if $\beta=\beta(L)$ is very large we may assume there
is a well defined lift to $\mathbb{R}$, so that our variables are now
$X_{i}\in\mathbb{R}$, and the center of mass
$\overline{X}_{L}=\frac{1}{L}\sum_{i=1}^{L}X_{i}$ satisfies the relation

$\displaystyle\differential\overline{X}(t)$
$\displaystyle=\frac{\sqrt{2}}{\beta\sqrt{L}}\,dB(t)\,,$ (14)

where $B(t)$ denotes the standard brownian motion. To see this, observe that
by definition (3), the dynamics is given by the SDEs

$\displaystyle\differential X_{i}(t)=-\partial_{X_{i}}H(X)\,\differential
t+\frac{\sqrt{2}}{\beta}dB_{i}(t)\,,\qquad i=1,\dots,L$ (15)

where the $B_{i}$’s are independent standard Brownian motions, and the
interaction has the form

$H(X)=\sum_{i\sim j}h_{ij}\left(X_{i}-X_{j}\right)=\sum_{i\sim
j}h_{ij}\left(X_{j}-X_{i}\right),$

where the sum ranges over the edges of some finite graph, and
$h_{ij}=h_{ji}=\cos(\cdot)$. The graph is the segment $\\{1,\dots,L\\}$ in the
case of free boundary conditions, and it is the $L$-cycle in case of periodic
boundary. Therefore,

$\sum_{i}\partial_{X_{i}}H(X)=\sum_{i}\sum_{j:\,i\sim
j}h_{ij}^{\prime}(X_{i}-X_{j})=-\sum_{i}\sum_{j:\,i\sim
j}h_{ij}^{\prime}(X_{j}-X_{i})=0.$

In particular, (14) shows that the center of mass relaxes on a time scale
proportional to $L$. Clearly, the above holds in any dimension $d\geq 1$, for
the $d$-dimensional cube with side $L$ with free or periodic boundaries,
provided $L$ is replaced by $L^{d}$. This can be seen as the starting point to
establish volume order relaxation times estimates for the low temperature XY
model. However, one has to keep in mind that this center of mass motion can be
interpreted as a meaningful mode of the system, namely the global phase, only
when all spins point in approximately the same direction. As noted in [4], for
$d=1$, the condition $\beta\gg L$ suffices to ensure that with large
probability all spins are closely aligned, that is the winding number is zero,
and there is a well defined lift $X$ as above. When $\beta\not\gg L$, one
needs to consider the sum of all spins, and of the corresponding Brownian
motions in $\mathbb{R}^{2}$, as actual vectors, which makes the analysis
considerably more involved; see [2] for a treatment of the mean field case.

Beyond the center of mass discussed above, one can also consider a stochastic
PDE describing the continuum limit of the full configuration of the system.
Consider the field $\phi:S^{1}\times\mathbb{R}\to S^{1}$ defined by

$\displaystyle\phi(\xi,\tau)=\frac{1}{\sqrt{L}}X_{\left\lfloor
L\xi/2\pi\right\rfloor}(L^{2}\tau).$ (16)

When $\beta$ is very large, we may approximate $\partial_{X_{i}}H(X)\approx
X_{i}-X_{i-1}-(X_{i+1}-X_{i})\approx 4\pi^{2}L^{-3/2}\Delta\phi(\xi,\tau)$,
so, with $t=L^{2}\tau$ one has $L^{-1/2}\partial_{X_{i}}H(X)\differential
t\approx 4\pi^{2}\Delta\phi(\xi,\tau)\differential\tau$. Moreover, reasoning
as in [11, Section 2],

$L^{-1/2}\frac{\differential B_{\left\lfloor
L\xi/2\pi\right\rfloor}(t)}{\differential t}\approx W(\xi,\tau)\,,$

where $W$ is space-time white noise on $S^{1}\times\mathbb{R}$. In conclusion,
from (15), the field $\phi$ satisfies the Edwards-Wilkinson equation, or
stochastic heat equation

$\displaystyle\partial_{\tau}\phi(\xi,\tau)$ $\displaystyle\approx
4\pi^{2}\Delta\phi(\xi,\tau)+\sqrt{2}\beta^{-1}W(\xi,\tau).$ (17)

We note that the continuum approximation discussed above should give a valid
description of the system, on suitable time scales, provided $\beta$ grows at
least logarithmically with $L$. In particular, this suggests a diffusive time
scale of order $L^{2}$ for relaxation, up to polylog corrections, when the
system has free boundary conditions, as pointed out after Theorem 1.1. After
this diffusive time scale, the system will relax by pure diffusion of the
global phase. If we consider periodic boundary conditions, then the situation
is different, since one has to impose the condition that
$\phi(2\pi,\tau)-\phi(0,\tau)=\text{winding number}$, and thus the equation
(17) is valid only within a given homotopy class, i.e. it describes the system
for times much smaller than the metastable times $T_{\text{MS}}\approx
e^{2\beta-\log(L)}$ detected in [4], at which the XY chain changes winding
number. On the metastable time scale $T_{\text{MS}}$ the dynamics will involve
a random walk between the adjacent homotopy classes corresponding to $\pm 1$
jumps of the global winding number.

The above heuristic analysis can be in principle repeated for any $N\geq 3$,
by working with the local coordinates chosen to parametrize the sphere
$\mathbb{S}^{N-1}$. One gets, independently for each coordinate, a SDE for the
center of mass and a stochastic heat equation as above. However, because of
the dependence on the choice of local coordinates on the manifold, the
interpretation of these equations is no longer obvious in this case.

##  2\. Free boundary conditions

In this section we prove Theorem 1.1. We start by choosing appropriate
coordinates.

###  2.1. Coordinates

For $N=2$, we parametrize the system using angles
$\theta_{1},\dots,\theta_{L}\in[0,2\pi]$ and by setting

$S_{i}=\big{(}\cos(\theta_{1}+\dots+\theta_{i}),\sin(\theta_{1}+\dots+\theta_{i})\big{)}.$

The uniform measure on $(\mathbb{S}^{1})^{L}$ is then the image of the uniform
measure on $[0,2\pi]^{L}$ by the above mapping.

For $N\geq 3$, a vector $s\in\mathbb{S}^{N-1}$ can be parametrized as

$s=s(\theta,v)=\cos(\theta){\rm e}_{1}+\sin(\theta)v\,,$

$({\rm e}_{1},\dots,{\rm e}_{N})$ is the canonical orthonormal basis of
$\mathbb{R}^{N}$, $\theta\in[0,\pi]$, and $v$ is a unit vector in the
orthogonal complement of ${\rm e}_{1}$. Then, sampling $s$ uniformly on
$\mathbb{S}^{N-1}$ is equivalent to sample $v$ uniformly on $\\{x:\ x\cdot{\rm
e}_{1}=0,\|x\|=1\\}$, and $\theta$ proportionally to $\sin(\theta)^{N-2}$. For
$\theta,v$ as before, denote $R_{v,\theta}$ the rotation matrix (in the
standard basis) rotating the plane spanned by $v$ and ${\rm e}_{1}$ by an
angle $\theta$ so that $R_{v,\theta}{\rm e}_{1}=\cos(\theta){\rm
e}_{1}+\sin(\theta)v$. In other words, $R_{v,\theta}$ is given by

$R_{v,\theta}x=x+\big{(}(\cos(\theta)-1)x\cdot{\rm e}_{1}-\sin(\theta)x\cdot
v\big{)}{\rm e}_{1}+\big{(}\sin(\theta)x\cdot{\rm
e}_{1}+(\cos(\theta)-1)x\cdot v\big{)}v.$

Let then $v_{i},i=1,\dots,L$ be a sequence of uniform random variables on
$\\{x:\ x\cdot{\rm e}_{1}=0,\|x\|=1\\}$, let $\theta_{i},i=1,\dots,L$ be an
sequence of random variables on $[0,\pi]$ with density proportional to
$\sin(\theta_{i})^{N-2}$. Suppose $v_{i},\theta_{i},i=1,\dots,L$ forms an
independent family. Set

$R_{i}=R_{v_{i},\theta_{i}},\quad S_{i}=R_{1}\dots R_{i}{\rm e}_{1}.$

The first ingredient we need is the next simple Lemma.

######  Lemma 2.1.

The sequence $S_{i},i=1,\dots,L$ is an i.i.d. sequence of uniform random
variables on $\mathbb{S}^{N-1}$.

###### Proof.

It is sufficient to check that for every realization of $R_{1},\dots,R_{i-1}$,
$S_{i}$ is uniformly distributed on $\mathbb{S}^{N-1}$. This follows directly
from rotation invariance of the spherical measure and the fact that $R_{i}{\rm
e}_{1}$ is uniform on $\mathbb{S}^{N-1}$. ∎

The final ingredient we will need is a control over partial derivatives of
$S_{i}$ with respect to the angles $\theta_{j}$s and the vectors $v_{j}$s. We
note here that by $\frac{\partial}{\partial v_{j}}S_{i}$ we mean the
differential with respect to $v_{j}$ when fixing all other variables; in local
coordinates it is given by the $(N-1)\times(N-1)$ Jacobian matrix, and we
write $\Big{\|}\cdot\Big{\|}$ for the associated operator norm.

######  Lemma 2.2.

Let $N\geq 3$. For every $i,j\in\\{1,\dots,L\\}$,

$\Big{\|}\frac{\partial}{\partial\theta_{j}}S_{i}\Big{\|}\leq
1,\quad\Big{\|}\frac{\partial}{\partial v_{j}}S_{i}\Big{\|}\leq 4.$ (18)

###### Proof.

If $j>i$, both quantities are $0$ and there is nothing to prove. Otherwise,

$\Big{\|}\frac{\partial}{\partial\theta_{j}}S_{i}\Big{\|}^{2}=\Big{\|}\frac{\partial}{\partial\theta_{j}}R_{j}x\Big{\|}^{2}=(x\cdot{\rm
e}_{1})^{2}+(x\cdot v)^{2}\leq 1.$

where $x=R_{j+1}\dots R_{i}{\rm e}_{1}$, and we used the fact that ${\rm
e}_{1},v$ are orthogonal and of norm $1$. Then, for any $x\in\mathbb{R}^{N}$
and $h$ of norm $1$,

$\lim_{\epsilon\searrow 0}\frac{1}{\epsilon}(R_{v+\epsilon
h,\theta}x-R_{v,\theta}x)=\\\ =-\sin(\theta)x\cdot h{\rm
e}_{1}+(\cos(\theta)-1)x\cdot hv+\big{(}\sin(\theta)x\cdot{\rm
e}_{1}+(\cos(\theta)-1)x\cdot v\big{)}h,$

so that for $h$ of norm $1$ in the tangent space of $\\{y:\ y\cdot{\rm
e}_{1}=0,\|y\|=1\\}$ at $v_{j}$, one has (as ${\rm e}_{1},v_{j},h$ are
orthogonal)

$\Big{\|}\frac{\partial}{\partial
v_{j}}S_{i}(\theta,v)\Big{\|}^{2}=\Big{\|}\frac{\partial}{\partial
v_{j}}R_{v_{j},\theta_{j}}x\Big{\|}^{2}\leq 14$

where again $x=R_{j+1}\dots R_{i}{\rm e}_{1}$. It follows that $\|\partial
S_{i}/\partial v_{j}\|\leq\sqrt{14}\leq 4$. ∎

The interest of those parametrizations lies in the following identity: the
Hamiltonian in (1) becomes

$-H^{\mathrm{f}}_{L}(S(v,\theta))=\sum_{i=1}^{L-1}(R_{1}\dots R_{i+1}{\rm
e}_{1})\cdot(R_{1}\dots R_{i}{\rm e}_{1})=\sum_{i=1}^{L-1}(R_{i+1}{\rm
e}_{1})\cdot{\rm e}_{1}=\sum_{i=2}^{L}\cos(\theta_{i}),$ (19)

which gives a nice factorisation of the Boltzmann weight. The same identity
holds for $N=2$.

Therefore, we have that

$\int_{\Omega_{L}}f(S)e^{-\beta H^{\mathrm{f}}_{L}(S)}\nu_{L}(dS)\propto\int
d\theta
dv\,f(S(\theta,v))\sin(\theta_{1})^{N-2}\prod_{i=2}^{L}\sin(\theta_{i})^{N-2}e^{\beta\cos(\theta_{i})},$

where

  * •

in the case $N\geq 3$, the right-hand-side integral is over
$[0,\pi]^{L}\times(\mathbb{S}^{N-2})^{L}$, and we identified $\\{x:\
\|x\|=1,x\cdot{\rm e}_{1}=1\\}$ with $\mathbb{S}^{N-2}$,

  * •

in the case $N=2$, the the right-hand-side integral is over $[0,2\pi]^{L}$ and
$v$ is off the picture.

Next, for $N\geq 3$, we introduce the probability measures
$\rho_{1},\dots,\rho_{L}$ on $[0,\pi]\times\mathbb{S}^{N-2}$ given by

$\displaystyle
d\rho_{1}(\theta_{1},v_{1})\propto\nu(dv_{1})\sin(\theta_{1})^{N-2}d\theta_{1},$
$\displaystyle
d\rho_{i}(\theta_{i},v_{i})\propto\nu(dv_{i})\sin(\theta_{i})^{N-2}e^{\beta\cos(\theta_{i})}d\theta_{i},\
i=2,\dots,L.$

For $N=2$, we use instead measures on $[0,2\pi]$ given by

$\displaystyle d\rho_{1}(\theta_{1})\propto d\theta_{1},$ $\displaystyle
d\rho_{i}(\theta_{i})\propto e^{\beta\cos(\theta_{i})}d\theta_{i},\
i=2,\dots,L.$

With this notation, we rewrite the expected value of some
$f:\Omega_{L}\mapsto\mathbb{R}$ with respect to the Gibbs measure
$\mu_{L}^{\mathrm{f}}$ as

$\mu_{L}^{\mathrm{f}}(f)=\otimes_{i=1}^{L}\rho_{i}\big{(}f(S(\theta,v))\big{)}.$
(20)

###  2.2. Poincaré inequalities for increments measures

For $f:[0,\pi]\times\\{x:\ \|x\|=1,x\cdot{\rm e}_{1}=0\\}\to\mathbb{R}$, which
maps $(\theta,v)$ to $f(\theta,v)$, denote as before
$\frac{\partial}{\partial\theta}$ the partial derivative with respect to
$\theta$ and $\frac{\partial}{\partial v}$ the partial derivative with respect
to $v$ (so that $\frac{\partial}{\partial v}f(\theta,v)$ is a linear function
from the tangent space of $\\{x:\ \|x\|=1,x\cdot{\rm e}_{1}=0\\}$ at $v$ to
$\mathbb{R}$). We also write ${\rm Var}_{P}(f)$ for the variance of $f$ with
respect to the probability measure $P$.

When $N\geq 3$, the measures $\rho_{i}$ are product measures: the product of
the uniform measure on $\\{x:\ \|x\|=1,x\cdot{\rm
e}_{1}=0\\}\equiv\mathbb{S}^{N-2}$ and of a measure on $[0,\pi]$. We start by
proving Poincaré inequalities for these “elementary constituents” which in
turn imply Poincaré inequalities for the $\rho_{i}$s. We stress that we do not
try to obtain the optimal constants, but we need a reasonable control over
their dependency on the parameters.

We first prove the bounds used for $N\geq 3$.

######  Lemma 2.3.

Let $a,b\geq 0$. Let $P_{a,b}$ be the probability measure on $[0,\pi]$ with
density proportional to $\sin(\theta)^{a}e^{b\cos(\theta)}$. Then, for any
$f:[0,\pi]\to\mathbb{R}$ smooth,

${\rm Var}_{P_{a,b}}(f)\leq c_{1}(a,b)E_{a,b}\big{(}|f^{\prime}|^{2}\big{)},$

where $E_{a,b}$ is the expectation with respect to $P_{a,b}$, and

$c_{1}(a,b)=\frac{\pi^{3}2^{a}b^{(a+1)/2}}{\int_{0}^{\sqrt{b}\pi/2}dxx^{a}e^{-x^{2}/2}}\text{
if }b>0,\quad c_{1}(a,0)=\frac{\pi^{2}2(a+1)4^{a}}{\pi^{a}}.$

Moreover, $c_{1}(a,b)\leq 8\pi^{3}2^{a}b^{(a+1)/2}$ for $b\geq 1$.

###### Proof.

Start with $b>0$. Let $P=P_{a,b}$. Set
$C=\int_{0}^{\pi}dx\sin^{a}(x)e^{b\cos(x)}$. Then,

$\displaystyle{\rm Var}_{P}(f)$
$\displaystyle=\frac{1}{2C^{2}}\int_{0}^{\pi}dx\int_{0}^{\pi}dy(f(x)-f(y))^{2}\sin^{a}(x)e^{b\cos(x)}\sin^{a}(y)e^{b\cos(y)}$
$\displaystyle\leq\frac{\pi^{2}}{2C^{2}}\int_{0}^{\pi}dx\int_{0}^{\pi}dy\int_{0}^{1}dt|f^{\prime}(tx+(1-t)y)|^{2}\sin^{a}(x)e^{b\cos(x)}\sin^{a}(y)e^{b\cos(y)}.$

We can then use that on $[0,\pi]$, $g(x)=-a\ln(\sin(x))$ is convex and non-
negative, therefore

$g(x)+g(y)\geq g(tx+(1-t)y)+g((1-t)x+ty)\geq g(tx+(1-t)y),$

and so $\sin^{a}(x)\sin^{a}(y)\leq\sin^{a}(tx+(1-t)y)$. Also,

$\cos(x)+\cos(y)-\cos(tx+(1-t)y)\leq 1,$

as $\cos$ is non-increasing on $[0,\pi]$ and less or equal to $1$. Using these
and changing variable to $z=tx+(1-t)y$, we obtain

$\displaystyle\int_{0}^{\pi}dx\int_{0}^{\pi}dy\int_{0}^{1}dt|f^{\prime}(tx+(1-t)y)|^{2}\sin^{a}(x)e^{b\cos(x)}\sin^{a}(y)e^{b\cos(y)}$
$\displaystyle\leq
e^{b}\int_{0}^{1}dt\int_{0}^{\pi}\frac{dz}{t}\int_{0}^{\pi}dy|f^{\prime}(z)|^{2}\sin^{a}(z)e^{b\cos(z)}\mathbf{1}_{z-(1-t)y\in[0,t\pi]}$
$\displaystyle\leq 2\pi
e^{b}\int_{0}^{\pi}dz|f^{\prime}(z)|^{2}\sin^{a}(z)e^{b\cos(z)}$

as
$\int_{0}^{1}dt\frac{1}{t}\int_{0}^{\pi}dy\mathbf{1}_{z-(1-t)y\in[0,t\pi]}\leq
2\pi$. Now,

$C\geq\int_{0}^{\pi}dx\sin^{a}(x)e^{b-bx^{2}/2}\geq
e^{b}\int_{0}^{\pi/2}dx\frac{x^{a}}{2^{a}}e^{-bx^{2}/2}=\frac{e^{b}}{2^{a}b^{(a+1)/2}}\int_{0}^{\sqrt{b}\pi/2}dxx^{a}e^{-x^{2}/2},$

as $\cos(\gamma)\geq 1-\gamma^{2}/2$ for $\gamma\in[0,\pi]$, and
$\sin(\gamma)\geq\gamma/2$ for $\gamma\in[0,\pi/2]$. Combining all the
estimates gives the main claim. The last point follows from (for $b\geq 1$)

$\int_{0}^{\sqrt{b}\pi/2}dxx^{a}e^{-x^{2}/2}\geq\int_{1}^{\pi/2}dxe^{-x^{2}/2}\geq
1/8.$

The case $b=0$ follows the exact same path with the lower bound
$C=\int_{0}^{\pi}\sin^{a}(x)\geq\frac{\pi^{a+1}}{2^{2a+1}(a+1)}$. ∎

The complete spectrum and eigenfunctions of the spherical Laplacian are known,
and in particular its spectral gap is equal $N-1$ (see, e.g., [1, Section
2.2.3]).

######  Lemma 2.4.

Let $N\geq 2$. Let $\nu$ be the uniform probability measure on
$\mathbb{S}^{N-1}$. Then, for any $f:\mathbb{S}^{N-1}\to\mathbb{R}$ smooth,

${\rm Var}_{\nu}(f)\leq c_{2}(N)\nu\big{(}\|Df\|^{2}\big{)},$

with $c_{2}(N)=\frac{1}{N-1}$.

The last bound is for the XY model case ($N=2$).

######  Lemma 2.5.

Let $b\geq 0$. Let $P_{b}$ be the probability measure on $[0,2\pi]$ with
density proportional to $e^{b\cos(\theta)}$. Then, for any smooth
$2\pi$-periodic $f:\mathbb{R}\to\mathbb{R}$,

${\rm Var}_{P_{b}}(f)\leq c_{3}(b)E_{b}\big{(}|f^{\prime}|^{2}\big{)},$

where $E_{b}$ is the expectation with respect to $P_{b}$, and

$c_{3}(b)=\frac{\pi^{3}\sqrt{b}}{\int_{-\sqrt{b}\pi}^{\sqrt{b}\pi}e^{-x^{2}/2}}\text{
if }b>0,\quad c_{3}(0)=\frac{\pi^{2}}{2}.$

Moreover, $c_{3}(b)\leq 2\pi^{3}\sqrt{b}$ for $b\geq 1$.

###### Proof.

Let $b>0$, $C=\int_{0}^{2\pi}dxe^{b\cos(x)}$. One has

$\displaystyle{\rm Var}_{P_{b}}(f)$
$\displaystyle=\frac{1}{2C^{2}}\int_{0}^{2\pi}dx\int_{-\pi}^{\pi}dy(f(x)-f(x+y))^{2}e^{b(\cos(x)+\cos(x+y))}$
$\displaystyle\leq\frac{\pi^{2}}{2C^{2}}\int_{0}^{1}dt\int_{0}^{2\pi}dx\int_{-\pi}^{\pi}dy|f^{\prime}(x+ty)|^{2}e^{b(\cos(x)+\cos(x+y))}$

(we simply shifted the integration domain of $y$ to the full period
$[x-\pi,x+\pi]$ which preserves the integral by periodicity). We can then use

$\cos(x)+\cos(x+y)-\cos(x+ty)\leq 1$

as $y\in[-\pi,\pi]$, and proceed as in the proof of Lemma 2.3 to obtain

$\displaystyle{\rm Var}_{P_{b}}(f)$
$\displaystyle\leq\frac{\pi^{2}e^{b}}{2C^{2}}\int_{0}^{1}dt\int_{0}^{2\pi}dx\int_{-\pi}^{\pi}dy|f^{\prime}(x+ty)|^{2}e^{b\cos(x+ty)}$
$\displaystyle=\frac{2\pi^{3}e^{b}}{2C^{2}}\int_{0}^{2\pi}dx|f^{\prime}(x)|^{2}e^{b\cos(x)}=\frac{\pi^{3}e^{b}}{C}E_{b}(|f^{\prime}|^{2})$

where we used periodicity in the second line. Now, as in the proof of Lemma
2.3,
$C\geq\int_{-\pi}^{\pi}e^{b-bx^{2}/2}=\frac{e^{b}}{\sqrt{b}}\int_{-\sqrt{b}\pi}^{\sqrt{b}\pi}e^{-x^{2}/2}$,
which gives the claim. A simplified version of the above treats the case $b=0$
(there, $C=2\pi$). ∎

We end this section by noticing that we have everything we need to control
$\mu_{L}^{\mathrm{f}}=\otimes_{i=1}^{L}\rho_{i}$, as the $\rho_{i}$s are
themselves product of one or two of the above cases. More precisely, for
$f:[0,\pi]\times\mathbb{S}^{N-2}\to\mathbb{R}$ smooth ($N\geq 3$), $i\geq 2$,
and $(\theta,v)\sim\rho_{i}$, by the tensorization property of variance,

$\displaystyle{\rm Var}_{\rho_{i}}(f)$ $\displaystyle\leq
P_{N-2,\beta}\otimes\nu\big{(}{\rm Var}_{\nu}(f(\theta,v))+{\rm
Var}_{P_{N-2,\beta}}(f(\theta,v))\big{)}$ $\displaystyle\leq
P_{N-2,\beta}\otimes\nu\big{(}c_{2}(N-2)\nu(\|\partial_{v}f(\theta,v)\|^{2})+c_{1}(N-2,\beta)E_{N-2,\beta}(|\partial_{\theta}f(\theta,v)|^{2})\big{)}$
$\displaystyle=c_{2}(N-2)\rho_{i}\big{(}\|\partial_{v}f(\theta,v)\|^{2}\big{)}+c_{1}(N-2,\beta)\rho_{i}\big{(}|\partial_{\theta}f(\theta,v)|^{2}\big{)},$
(21)

where $P_{a,b}$ is the measure of Lemma 2.3. A similar bound holds for $i=1$.

###  2.3. Proof of Theorem 1.1

Introduce

$c(\beta,N)=\begin{cases}\max(c_{3}(\beta),c_{3}(0))&\text{ if }N=2,\\\
\max(c_{1}(N-2,\beta),c_{1}(N-2,0),c_{2}(N-2))&\text{ if }N\geq 3.\end{cases}$
(22)

By Lemmas 2.3, 2.4, and 2.5, one has that for $\beta$ larger than $1$,

$c(\beta,N)\leq\pi^{3}2^{N+1}\beta^{(N-1)/2}.$ (23)

The upper bound in Theorem 1.1 is a consequence of the following Lemma and of
(23).

######  Lemma 2.6.

Let $f:\Omega_{L}\to\mathbb{R}$ be smooth. Then,

${\rm
Var}_{\mu_{L}^{\mathrm{f}}}(f)\leq\beta\,17c(\beta,N)L^{2}\,\mathcal{D}^{\mathrm{f}}_{L,\beta}(f,f)\,,$

where $c(\beta,N)$ is given by (22).

###### Proof.

Since $\mu_{L}^{\mathrm{f}}$ is the product measure (20), we have that by the
tensorization of variance,

${\rm
Var}_{\mu_{L}^{\mathrm{f}}}(f)\leq\otimes_{i=1}^{L}\rho_{i}\Big{(}\sum_{i=1}^{L}{\rm
Var}_{\rho_{i}}\big{(}f(\theta_{1},v_{1},\dots,\theta_{L},v_{L})\big{)}\Big{)}.$

Applying (2.2), one obtains

${\rm Var}_{\mu_{L}^{\mathrm{f}}}(f)\leq
c(\beta,N)\otimes_{i=1}^{L}\rho_{i}\Big{(}\sum_{i=1}^{L}\big{(}\|\partial_{v_{i}}f\|^{2}+|\partial_{\theta_{i}}f|^{2}\big{)}\Big{)}.$
(24)

Moreover, by the chain rule, for $f:(\mathbb{S}^{N-1})^{L}\to\mathbb{R}$
smooth, one has

$\partial_{v_{i}}f\big{(}S(\theta,v)\big{)}=\sum_{j=i}^{L}D_{j}f(S(\theta))\cdot\partial_{v_{i}}S_{j}(\theta_{1},v_{1},\dots,\theta_{j},v_{j}).$

and

$\partial_{\theta_{i}}f\big{(}S(\theta,v)\big{)}=\sum_{j=i}^{L}D_{j}f(S(\theta))\cdot\partial_{\theta_{i}}S_{j}(\theta_{1},v_{1},\dots,\theta_{j},v_{j}).$

Observe that by our choice of coordinates (Lemma 2.2), for any $L\geq j\geq
i\geq 1$,

$\|\partial_{\theta_{i}}S_{j}(\theta_{1},v_{1},\dots,\theta_{L},v_{L})\|\leq
1,\quad\|\partial_{v_{i}}S_{j}(\theta_{1},\dots,\theta_{L},v_{L})\|\leq 4.$

Therefore, by Cauchy-Schwarz,

$\displaystyle\sum_{i=1}^{L}\big{(}\|\partial_{v_{i}}f\|^{2}+|\partial_{\theta_{i}}f|^{2}\big{)}$
$\displaystyle\leq\sum_{i=1}^{L}\Big{(}16L\sum_{j=i}^{L}\|D_{j}f\|^{2}+L\sum_{j=i}^{L}\|D_{j}f\|^{2}\Big{)}$
$\displaystyle\leq 17L^{2}\sum_{i=1}^{L}\|D_{i}f\|^{2}.$

Plugging this in (24), and recalling (4) concludes the proof. ∎

##  3\. $XY$ model on the cycle

In this section we prove Theorem 1.2. We start with the proof of the upper
bound.

###  3.1. Upper bound

The proof is based on the upper bound for free boundary conditions in Theorem
1.1 and simple comparison between the free and the periodic boundary condition
system. The first observation is that from the definitions (1)-(2) and the
fact that $|S_{i}\cdot S_{i+1}|\leq 1$ it follows that the relative densities
$\differential\mu_{L}^{\mathrm{f}}/\differential\mu_{L}^{\mathrm{per}},\differential\mu_{L}^{\mathrm{per}}/\differential\mu_{L}^{\mathrm{f}}$
satisfy

$\left\|\frac{\differential\mu_{L}^{\mathrm{f}}}{\differential\mu_{L}^{\mathrm{per}}}\right\|_{\infty}\leq\frac{Z^{\mathrm{per}}_{L,\beta}}{Z^{\mathrm{f}}_{L,\beta}}\,\exp\left(\beta\right),\qquad\left\|\frac{\differential\mu_{L}^{\mathrm{per}}}{\differential\mu_{L}^{\mathrm{f}}}\right\|_{\infty}\leq\frac{Z^{\mathrm{f}}_{L,\beta}}{Z^{\mathrm{per}}_{L,\beta}}\,\exp\left(\beta\right).$
(25)

At this point, for any smooth function $f:\Omega_{L}\to\mathbb{R}$ one has

$\displaystyle{\rm Var}_{\mu_{L}^{\mathrm{per}}}(f)$
$\displaystyle=\inf_{c\in\mathbb{R}}\mu_{L}^{\mathrm{per}}((f-c)^{2})\leq\mu_{L}^{\mathrm{per}}((f-\mu_{L}^{\mathrm{f}}(f))^{2})$
$\displaystyle\leq\frac{Z^{\mathrm{f}}_{L,\beta}}{Z^{\mathrm{per}}_{L,\beta}}\,\exp\left(\beta\right)\mu_{L}^{\mathrm{f}}((f-\mu_{L}^{\mathrm{f}}(f))^{2})=\frac{Z^{\mathrm{f}}_{L,\beta}}{Z^{\mathrm{per}}_{L,\beta}}\,\exp\left(\beta\right){\rm
Var}_{\mu_{L}^{\mathrm{f}}}(f).$

Similarly,

$\displaystyle\mathcal{D}^{\mathrm{f}}_{L,\beta}(f,f)\leq\frac{Z^{\mathrm{per}}_{L,\beta}}{Z^{\mathrm{f}}_{L,\beta}}\,\exp\left(\beta\right)\mathcal{D}^{\mathrm{per}}_{L,\beta}(f,f).$

Therefore, from Theorem 1.1 it follows that

$\displaystyle{\rm Var}_{\mu_{L}^{\mathrm{per}}}(f)\leq
e^{2\beta}C(2)L^{2}\beta^{3/2}\mathcal{D}^{\mathrm{per}}_{L,\beta}(f,f).$

This implies the upper bound (9).

###  3.2. Lower bound

We use the variational principle (5). In order to construct an appropriate
test function, we follow [4], where a proxy for the winding number of the spin
chain was defined as follows. Let $S_{i}\in\mathbb{S}^{1}$ denote the $i$-th
spin, with $S_{L+1}=S_{1}$, and write $[\theta]$ for the representative in the
interval $(-\pi,\pi]$ of any $\theta\in\mathbb{S}^{1}$. If
$S\in(\mathbb{S}^{1})^{L}$ is such that
$S_{i+1}-S_{i}\in\mathbb{S}^{1}\setminus\\{\pi\\}$ for all $i=1,\dots,L$,
define the function

$\displaystyle\mathcal{W}(S)=\frac{1}{2\pi}\sum_{i=1}^{L}[S_{i+1}-S_{i}].$
(26)

Because of the periodic boundary condition, $\mathcal{W}$ is an integer, which
can be interpreted as the winding number of the spin configuration. Moreover,
the function $\mathcal{W}$ is continuous in its domain of definition $D$ given
by

$\displaystyle
D=\left\\{S\in(\mathbb{S}^{1})^{L}:\;S_{i+1}-S_{i}\in\mathbb{S}^{1}\setminus\\{\pi\\},\;i=1,\dots,L\right\\}\,.$
(27)

Next define the events

$\displaystyle B=\\{S\in D:\;\mathcal{W}(S)=0\\}\,,$ (28) $\displaystyle
A_{\delta}=\\{S\in D:\;\mathcal{W}(S)=0\;\,\text{and
}S_{i}-S_{i+1}\in[\pi-\delta,\pi+\delta]\text{ for some }i\\}\,.$ (29)

We let $f:(\mathbb{S}^{1})^{L}\mapsto\mathbb{R}$ denote a
$\mathcal{C}^{\infty}$ function such that $f=0$ on $B^{c}$, $f=1$ on
$B\setminus A_{\delta}$ and such that $\|D_{i}f\|_{\infty}\leq C_{\delta}$,
where $C_{\delta}=O(1/\delta)$ is a constant independent of $\beta,L$. Since
$D_{i}f(S)\neq 0$ implies that $S\in A_{\delta}$, using this function $f$ in
the variational principle (5) one obtains

$\displaystyle\mathop{\rm
gap}\nolimits^{\mathrm{per}}_{L,\beta}\leq\frac{1}{\beta}\,C_{\delta}^{2}L\,\frac{\mu^{\mathrm{per}}_{L,\beta}(A_{\delta})}{{\rm
Var}^{\mathrm{per}}_{L,\beta}f}.$ (30)

Moreover, by definition of $f$ one has

$\displaystyle{\rm
Var}^{\mathrm{per}}_{L,\beta}f\geq\mu^{\mathrm{per}}_{L,\beta}(B\setminus
A_{\delta})\mu^{\mathrm{per}}_{L,\beta}(B^{c}).$ (31)

It remains to estimate $\mu^{\mathrm{per}}_{L,\beta}(A_{\delta})$ and the
probabilities in (31).

For $\delta>0$, define the events

$\displaystyle B_{\delta}^{0}$
$\displaystyle=\left\\{S_{i}-S_{i}^{0}\in[-\delta,\delta]\text{ for all
}i=1,\dots,L\right\\},$ $\displaystyle B_{\delta}^{1}$
$\displaystyle=\left\\{S_{i}-S_{i}^{1}\in[-\delta,\delta]\text{ for all
}i=1,\dots,L\right\\},$

where the configurations $S^{0},S^{1}$, seen as variables in the complex
plain, are defined by

$\displaystyle S_{j}^{0}=1\,,\qquad S_{j}^{1}=e^{2\pi i\frac{j}{L}}\,,\qquad
j\in\\{1,\dots L\\}.$

Observe that, for small enough $\delta=\delta(L)$ depending on $L$,

$B_{\delta}^{0}\in\\{\mathcal{W}=0\\}\text{ and
}B_{\delta}^{1}\in\\{\mathcal{W}=1\\},$

and, writing $H(S):=H^{\mathrm{per}}_{L}(S)$,

$\displaystyle H(S)$ $\displaystyle\geq 2-\delta^{2}-L,\qquad S\in
A_{\delta},$ $\displaystyle H(S)$ $\displaystyle\leq-(1-\delta^{2})L,\qquad
S\in B_{\delta}^{0},$ $\displaystyle H(S)$
$\displaystyle\leq-L+C(\delta+L^{-1})+C\delta^{2}L\,,\qquad S\in
B_{\delta}^{1},$

for some absolute constant $C>0$. With the notation
$\text{vol}(A)=\int_{A}\differential S$ for any $A\subset\Omega$,
$\Omega=\Omega_{L}$, we obtain

$\text{vol}(B_{\delta}^{0})e^{\beta(1-\delta^{2})L}\leq\int_{\Omega}e^{-\beta
H(S)}\differential S\leq\text{vol}(\Omega)e^{\beta L}.$

Therefore, writing $\mu=\mu^{\mathrm{per}}_{L,\beta}$,

$\displaystyle\mu(A_{\delta})$ $\displaystyle=\frac{\int_{A_{\delta}}e^{-\beta
H(S)}\differential S}{\int_{\Omega}e^{-\beta H(S)}\differential
S}\leq\frac{\text{vol}(A_{\delta})e^{-\beta\left(2-\delta^{2}-L\right)}}{\text{vol}(B_{\delta}^{0})e^{\beta(1-\delta^{2})L}}$
$\displaystyle=\frac{\text{vol}(A_{\delta})}{\text{vol}(B_{\delta}^{0})}\,e^{-\beta\left(2-\delta^{2}(L+1)\right)}=C_{\delta,L}\,e^{-\beta\left(2-\delta^{2}(L+1)\right)},$

where $C_{\delta,L}$ is a constant depending on $\delta,L$. We can also
estimate

$\mu(B_{\delta}^{0})=\frac{\int_{B_{\delta}^{0}}e^{-\beta H(S)}\differential
S}{\int_{\Omega}e^{-\beta H(S)}\differential
S}\geq\frac{\text{vol}(B_{\delta}^{0})e^{\beta(1-\delta^{2})L}}{\text{vol}(\Omega)e^{\beta
L}}=c_{\delta,L}e^{-\delta^{2}L\beta},$

where $c_{\delta,L}$ is another constant depending on $\delta,L$. Finally,

$\mu(B_{\delta}^{1})=\frac{\int_{B_{\delta}^{1}}e^{-\beta H(S)}\differential
S}{\int_{\Omega}e^{-\beta H(S)}\differential
S}\geq\frac{\text{vol}(B_{\delta}^{1})e^{\beta\left(L-C(\delta+L^{-1})-C\delta^{2}L\right)}}{\text{vol}(\Omega)e^{\beta
L}},$

which implies

$\mu(B_{\delta}^{1})\geq
c_{\delta,L}e^{-\left(C(\delta+L^{-1})+C\delta^{2}L\right)\beta},$

for some other constant $c_{\delta,L}$ depending on $\delta,L$. Summarizing,
we obtain the following estimate.

######  Proposition 3.1.

For all $L\in\mathbb{N}$, $\delta=\delta(L)\leq L^{-1}$, for all $\beta\geq
1$,

$\frac{\mu(A_{\delta})}{\mu^{\mathrm{per}}_{L,\beta}(B\setminus
A_{\delta})\mu^{\mathrm{per}}_{L,\beta}(B^{c})}\leq\frac{\mu(A_{\delta})}{\mu(B_{\delta}^{0})\mu(B_{\delta}^{1})}\leq
C_{\delta,L}e^{-(2-C_{0}L^{-1})\beta},$ (32)

where $C_{\delta,L}$ is a constant depending on $\delta,L$, and $C_{0}$ is an
absolute constant.

The lower bound in (10) now follows from (30)-(31) and Proposition 3.1.

######  Remark 3.1 (A finer lower bound for $\beta\geq C\log L$).

The estimate in Proposition 3.1 captures the correct energy barrier of size
$2\beta$ up to $O(L^{-1})$ corrections, see the matching upper bound (9).
However, it provides no quantitative estimate in terms of the system size $L$.
To resolve this, we observe that Proposition 3.1 can be considerably refined
by adapting the analysis from [4]. Rather than giving an explicit derivation,
we content ourselves with the following observations. One can use the
arguments in [4, Section 4] to prove that for all $L$ and $\beta\geq 1$,

$\mu^{\mathrm{per}}_{L,\beta}(B)\geq\frac{1}{L}\,,\qquad\mu^{\mathrm{per}}_{L,\beta}(B^{c})\geq\frac{1}{C_{0}\sqrt{\beta}L}e^{-C_{0}\beta/L}\,,$

where $C_{0}$ is an absolute constant, and that for any $\delta\in(0,1)$ one
has

$\mu^{\mathrm{per}}_{L,\beta}(A_{\delta})\leq C_{0}\beta L\delta
e^{-\beta(2-\delta^{2}/2)}.$

From these bounds it is not difficult to check that for an appropriate
absolute constant $C_{0}$, if $\beta\geq C_{0}\log L$, $L\geq C_{0}$, then for
all fixed $\delta\in(0,1)$, the left hand side of (32) is bounded from above
by

$C_{0}\beta^{3/2}L^{3}\delta\,e^{-\beta(2-\delta^{2}/2-C_{0}L^{-1})}.$

Thus, from (30) and (31) one arrives at the following relaxation time lower
bound: for all $\delta\in(0,1)$, there exists a constant $c_{\delta}>0$ such
that for $L\geq 1/c_{\delta}$, and $\beta\geq C_{0}\log L$, for an appropriate
absolute constant $C_{0}$, one has

$T_{\rm rel}^{\mathrm{per}}(L,\beta)\geq
c_{\delta}\beta^{-1/2}L^{-4}\,e^{(2-\delta^{2})\beta}\,.$ (33)

##  4\. $O(N)$ on the cycle

Here we prove Theorem 1.3. In order to simplify notation we omit some
subscripts and superscripts; we only discuss in this section the relaxation
time for fixed $N\geq 3$, and fixed $L$ and $\beta$, with periodic boundary
conditions.

###  4.1. General strategy

In order to bound the relaxation time from above, we will show how to bring a
configuration, using a canonical path method, to a small neighborhood where
the hamiltonian in convex.

######  Definition 4.1.

The _arctic_ is defined as the set of configurations where all spins are in
the ball of radius $\arccos(0.99)\approx 0.02\times 2\pi$ around ${\rm
e}_{1}$:

$A=\\{S\in\mathbb{S}^{N-1}:S\cdot{\rm e}_{1}>0.99\\}^{L}.$ (34)

The following is a simple consequence of our definitions.

######  Fact 1.

$H^{\mathrm{per}}_{L}$ is convex on $A$.

######  Definition 4.2.

Fix an open set $U\subseteq\Omega$. A path from $U$ is a continuous function
$\Phi:U\times[0,1]\to\Omega$. We define the path’s energy and entropy
respectively as

$\displaystyle\Delta H_{\Phi}(S,t)$
$\displaystyle=H^{\mathrm{per}}_{L}(\Phi(S,t))-H^{\mathrm{per}}_{L}(S),$ (35)
$\displaystyle\Delta\operatorname{Ent}_{\Phi}(S,t)$
$\displaystyle=\log\left|\frac{d(\Phi(\cdot,t))_{\\#}\nu_{L}}{d\nu_{L}}\right|,$
(36)

where $\left|\frac{d(\Phi(\cdot,t))_{\\#}\nu_{L}}{d\nu_{L}}\right|$ is the
density of the push-forward measure of $\nu_{L}$ with respect to $\nu_{L}$; in
local coordinates it is given by the determinant of the Jacobian matrix. The
_free energy barrier_ of the path is defined by

$\Delta F[\Phi]=\sup_{t\in[0,1]}\sup_{S\in U}\,\Delta
H_{\Phi}(S,t)-\beta^{-1}\Delta\operatorname{Ent}_{\Phi}(S,t).$ (37)

The speed of the path is defined by

$v[\Phi]=\sup_{t\in[0,1]}\sup_{S\in U}\|\partial_{t}{\Phi}(S,t)\|,$ (38)

where $\|\cdot\|$ denotes the euclidean norm

$\|\xi\|^{2}=\sum_{j=1}^{L}\|\xi_{j}\|^{2}\,,\qquad\xi=(\xi_{1},\dots,\xi_{L})\in\mathbb{R}^{L}.$

The paths to be considered below are almost everywhere differentiable, so the
above is well defined as an essential supremum.

######  Lemma 4.1.

Let $U_{1},\dots,U_{K}$ be some finite collection of open subsets of $\Omega$,
and $\Phi_{1},\dots,\Phi_{K}$ a collection of paths,
$\Phi_{i}:U_{i}\times[0,1]\to\Omega$. Assume:

  1. (1)

$\bigcup_{i=1}^{K}U_{i}=\Omega$.

  2. (2)

$\Phi_{i}(S,0)=S$ and $\Phi_{i}(S,1)\in A$ for all $S\in U_{i}$.

  3. (3)

$\Delta F:=\sup_{i}\Delta F[\Phi_{i}]$ and $v:=\sup_{i}v[\Phi_{i}]$ are both
finite.

Then

$T_{\rm rel}\leq 3K^{2}\beta(v^{2}e^{\beta\Delta F}+e^{2\beta\Delta F}\mu(A))$
(39)

In particular, if the energy barrier $\sup_{i}\sup_{t\in[0,1]}\sup_{S\in
U_{i}}\,\Delta H_{\Phi_{i}}(S,t)$ is non-positive then the $\beta$-dependence
of the relaxation time grows at most linearly.

###### Proof.

Fix some test function $f:\Omega\to\mathbb{R}$, and use the notation
$\Phi_{i}(X)=\Phi_{i}(X,1)$. The variance of $f$ with respect to
$\mu:=\mu_{L,\beta}^{\mathrm{per}}$ satisfies

$2{\rm Var}_{L,\beta}^{\mathrm{per}}(f)=\int d\mu(X)d\mu(Y)(f(X)-f(Y))^{2}\\\
\leq\sum_{i=1}^{K}\sum_{j=1}^{K}\int
d\mu(X)d\mu(Y)\mathds{1}_{U_{i}}(X)\mathds{1}_{U_{j}}(Y)(f(X)-f(Y))^{2}\\\
=\sum_{i=1}^{K}\sum_{j=1}^{K}\int
d\mu(X)d\mu(Y)\mathds{1}_{U_{i}}(X)\mathds{1}_{U_{j}}(Y)\\\
\times(f(X)-f(\Phi_{i}(X))+f(\Phi_{i}(X))-f(\Phi_{j}(Y))+f(\Phi_{j}(Y))-f(Y))^{2}\\\
\leq 3\sum_{i=1}^{K}\sum_{j=1}^{K}\int
d\mu(X)d\mu(Y)\mathds{1}_{U_{i}}(X)\mathds{1}_{U_{j}}(Y)(f(X)-f(\Phi_{i}(X)))^{2}\\\
+3\sum_{i=1}^{K}\sum_{j=1}^{K}\int
d\mu(X)d\mu(Y)\mathds{1}_{U_{i}}(X)\mathds{1}_{U_{j}}(Y)(f(\Phi_{i}(X))-f(\Phi_{j}(Y)))^{2}\\\
+3\sum_{i=1}^{K}\sum_{j=1}^{K}\int
d\mu(X)d\mu(Y)\mathds{1}_{U_{i}}(X)\mathds{1}_{U_{j}}(Y)(f(\Phi_{j}(Y))-f(Y))^{2}\\\
\leq 6K\sum_{i=1}^{K}\int
d\mu(X)\mathds{1}_{U_{i}}(X)(f(X)-f(\Phi_{i}(X)))^{2}\\\
+3\sum_{i=1}^{K}\sum_{j=1}^{K}\int
d\mu(X)d\mu(Y)\mathds{1}_{U_{i}}(X)\mathds{1}_{U_{j}}(Y)(f(\Phi_{i}(X))-f(\Phi_{j}(Y)))^{2}\\\
=6K\sum_{i=1}^{K}{\rm(I)}_{i}+3\sum_{i=1}^{K}\sum_{j=1}^{K}{\rm(II)}_{i,j}$
(40)

We start with the first term. For any $i$ and $X\in\Omega$,

$\displaystyle(f(X)-f(\Phi_{i}(X)))^{2}$
$\displaystyle=\left(\int_{0}^{1}dt\sum_{j=1}^{L}D_{j}f(\Phi_{i}(X,t))\cdot(\partial_{t}\Phi_{i}(X,t))_{j})\right)^{2}$
(41) $\displaystyle\leq
v^{2}\int_{0}^{1}dt\sum_{j=1}^{L}\|D_{j}f(\Phi_{i}(X,t))\|^{2},$ (42)

where $D_{j}$ denotes the gradient on the sphere $\mathbb{S}^{N-1}$ acting on
the $j$-th spin. Plugging this into (I), recalling the definition of $\mu$,
and then changing variable to $X^{\prime}=\Phi_{I}(X,t)$, yield

$\displaystyle{\rm(I)}_{i}$ $\displaystyle\leq v^{2}\int_{0}^{1}dt\int
d\mu(X)\mathds{1}_{U_{i}}(X)\sum_{j=1}^{L}\|D_{j}f(\Phi_{i}(X,t))\|^{2}$ (43)
$\displaystyle=\frac{v^{2}}{Z}\int_{0}^{1}dt\int d\nu(X)e^{-\beta
H(\Phi_{i}(X,t))+\beta\Delta
H_{\Phi_{i}}(X,t)}\mathds{1}_{U_{i}}(X)\sum_{j=1}^{L}\|D_{j}f(\Phi_{i}(X,t))\|^{2}$
(44) $\displaystyle=\frac{v^{2}}{Z}\int_{0}^{1}dt\int
d\nu(X^{\prime})e^{-\Delta\operatorname{Ent}_{\Phi_{i}}(X,t)}e^{-\beta
H(X^{\prime})}e^{\beta\Delta
H_{\Phi_{i}}(X,t)}\mathds{1}_{U_{i}}(X)\sum_{j=1}^{L}\|D_{j}f(X^{\prime})\|^{2}$
(45) $\displaystyle\leq v^{2}e^{\beta\Delta F}\int_{0}^{1}dt\int
d\mu(X^{\prime})\sum_{j=1}^{L}\|D_{j}f(X^{\prime})\|^{2}=v^{2}e^{\beta\Delta
F}\sum_{j=1}^{L}\mu(\|D_{j}f\|^{2})$ (46) $\displaystyle=\beta
v^{2}e^{\beta\Delta F}\mathcal{D}_{L,\beta}^{\mathrm{per}}(f,f).$ (47)

For the second term, the change of variables
$X^{\prime}=\Phi_{i}(X,t),Y^{\prime}=\Phi_{j}(Y,t)$ leads using the same
calculation to

${\rm(II)}_{i,j}\leq e^{2\beta\Delta F}\int
d\mu(X^{\prime})d\mu(Y^{\prime})\mathds{1}_{A}(X^{\prime})\mathds{1}_{A}(Y^{\prime})(f(X^{\prime})-f(Y^{\prime}))^{2}.$
(48)

This last integral equals $\mu(A)^{2}$ times the variance of $f$ under the
conditional measure $\mu(\cdot|A)$. Since on $A$ the Hamiltonian is convex,
and since $\Omega$ has positive curvature $1$, the Brascamp-Lieb inequality
for $\mu(\cdot|A)$, see e.g. [12, Theorem 1.2], tells us that:

$\frac{1}{2}\int
d\mu(X|A)d\mu(Y|A)(f(X)-f(Y))^{2}\leq\sum_{j=1}^{L}\mu(\|D_{j}f\|^{2}|A).$
(49)

We conclude that

${\rm(II)}_{i,j}\leq 2e^{2\beta\Delta
F}\mu(A)\sum_{j=1}^{L}\mu(\|D_{j}f\|^{2})=2\beta e^{2\beta\Delta
F}\mu(A)\mathcal{D}_{L,\beta}^{\mathrm{per}}(f,f).$ (50)

Together with the bound (47), this ends the proof of the lemma. ∎

###  4.2. Constructing the path: a soft argument for any $N\geq 3$

We will construct paths to be used in Lemma 4.1 in three parts: first, we use
the deterministic flow in order to bring the configuration near a critical
point, where all spins are on a single great circle, defined as the
intersection of the sphere with a $2$ dimensional plane. Then, we pull the
spins to a point perpendicular to that great circle. Finally, we rotate the
configuration to the arctic.

#### Step 1: Approaching a great circle

Define the deterministic flow $\varphi:\Omega\times\mathbb{R}\to\Omega$ as the
solution, for each $S\in\Omega$, of the differential equation

$\displaystyle\frac{\partial\varphi(S,t)}{\partial t}$
$\displaystyle=-DH(\varphi(S,t)),$ (51) $\displaystyle\varphi(S,0)$
$\displaystyle=S,$ (52)

where we use $D=(D_{j})_{j=1,\dots,L}$ for the $L$-vector of sphere gradients
$D_{j}$.

######  Claim 1.

Let $S$ be a stationary point of the flow. Then the spins $S_{1},\dots,S_{L}$
all belong to the same great circle.

###### Proof.

By explicit calculation in $\mathbb{R}^{N}$,

$\displaystyle-D_{i}H(S)$ $\displaystyle=D_{i}\left[S_{i-1}\cdot
S_{i}+S_{i}\cdot S_{i+1}\right]=S_{i-1}-(S_{i-1}\cdot
S_{i})S_{i}+S_{i+1}-(S_{i+1}\cdot S_{i})S_{i}$
$\displaystyle=S_{i-1}+S_{i+1}-(S_{i-1}\cdot S_{i}+S_{i+1}\cdot S_{i})S_{i}.$

Therefore, $D_{i}H(S)=0$ implies that $S_{i+1}$ is in the linear span of
$S_{i-1}$ and $S_{i}$. It follows that if $DH(S)=0$, then this holds for all
$i$, and therefore all the spins belong to the same 2D plane. ∎

#### Step 2: Pulling towards a single point

Once all spins are on a great circle, we move them to a point $s$
perpendicular to that circle. First, we define the flow $\phi$ that does that.
We then need to show that the energy is decreasing along the flow. This is
done in two parts—initially, spins that were on one side of the great circle
move closer to it, and the others move away. After a short period, the spins
on the "wrong" side cross the great circle, and in the second part of the
motion all spins are on a single hemisphere. Claim 2 shows that if we start
$\varepsilon$-close to a great circle, after time $2\varepsilon$ all points
are going to be at the same side of the circle.
We would then like to show that the energy is decreasing up to time
$2\varepsilon$. This unfortunately is not always true—if all spins are in the
hemisphere opposite to $s$, then initially distances grow and the energy
increases. This problem, however, could be easily solved by replacing $s$ with
$-s$, which is now in the right hemisphere. In Claim 4 we show that whenever
we start $\varepsilon$-close to the great circle, pulling the spins to $s$ or
$-s$ results in energy decrease.
Finally, Claim 5 shows that once all spins are in the same hemisphere as $s$,
the energy is decreasing.

######  Definition 4.3.

Fix two spins $s,s^{\prime}\in\mathbb{S}^{N-1}$. We define the function
$\phi_{s}:\mathbb{S}^{N-1}\times\mathbb{R}\to\mathbb{S}^{N-1}$ as the flow on
the sphere given by the following equation in $\mathbb{R}^{N}$:

$\displaystyle\frac{\partial}{\partial t}\phi_{s}(s^{\prime},t)$
$\displaystyle=s-(\phi(s^{\prime},t)\cdot s)\phi(s^{\prime},t),$
$\displaystyle\phi_{s}(s^{\prime},0)$ $\displaystyle=s^{\prime}.$

For a configuration $S=(S_{1},\dots,S_{L})$ we write
$\phi_{s}(S,t)=\left(\phi_{s}(S_{1},t),\dots,\phi_{s}(S_{L},t)\right)$.

######  Observation 1.

If $s^{\prime}$ is perpendicular to $s$, we can solve explicitly

$\phi_{s}(s^{\prime},t)=\tanh(t)\,s+\frac{1}{\cosh(t)}\,s^{\prime},$

which shows that $\lim_{t\to\infty}\phi_{s}(s^{\prime},t)=s$. This last fact
is true in general—we can always choose $\tilde{s}$ and $t_{0}$ such that

$\phi_{s}(s^{\prime},t)=\tanh(t-t_{0})\,s+\frac{1}{\cosh(t-t_{0})}\,\tilde{s},$

by taking $\tilde{s}$ perpendicular to $s$ in the plane spanned by $s$ and
$s^{\prime}$, and determine $t_{0}$ using $\phi_{s}(s^{\prime},0)=s^{\prime}$.

######  Claim 2.

For any $\varepsilon<\frac{1}{2}$, if $\left|s\cdot
s^{\prime}\right|<\varepsilon$ then $\phi_{s}(s^{\prime},2\varepsilon)\cdot
s>0.$

###### Proof.

Note that

$\frac{\differential}{\differential t}\left(\phi_{s}(s^{\prime},t)\cdot
s\right)=1-(\phi(s^{\prime},t)\cdot s)^{2}\geq 0,$

so it is enough to prove that $\phi_{s}(s^{\prime},t)\cdot s$ cannot remain in
the interval $[-\varepsilon,0]$ for all $t\in[0,2\varepsilon]$. But assuming
it does,

$\frac{\differential}{\differential t}\left(\phi_{s}(s^{\prime},t)\cdot
s\right)=1-(\phi(s^{\prime},t)\cdot s)^{2}>\frac{3}{4},$

and therefore $\phi_{s}(s^{\prime},2\varepsilon)\cdot
s>-\varepsilon+\frac{3}{4}\cdot 2\varepsilon=\frac{\varepsilon}{2}>0$. ∎

######  Claim 3.

Fix $s\in\mathbb{S}^{N-1}$, and a configuration $S$ where all spins are on a
great circle perpendicular to $s$. If not all spins are aligned, then

$\left.\frac{\differential^{2}}{\differential
t^{2}}H(\phi_{s}(S,t))\right|_{t=0}<0.$

###### Proof.

Since all spins are perpendicular to $s$, we can write explicitly, setting
$\underline{s}=(s,\dots,s)\in\Omega$,

$\displaystyle\phi_{s}(S,t)$
$\displaystyle=\tanh(t)\,s+\frac{1}{\cosh(t)}\,S,$ $\displaystyle
H(\phi_{s}(S,t))$
$\displaystyle=\frac{1}{2}\sum_{i}\|\phi_{s}(S,t)_{i}-\phi_{s}(S,t)_{i-1}\|^{2}+{\rm
const.}$ $\displaystyle=\frac{1}{2}\frac{1}{\cosh(t)^{2}}H(S)+{\rm const.}$
$\displaystyle\frac{\differential}{\differential t}H(\phi_{s}(S,t))$
$\displaystyle=-\frac{\sinh(t)}{\cosh(t)^{3}}H(S)$
$\displaystyle\frac{\differential^{2}}{\differential t^{2}}H(\phi_{s}(S,t))$
$\displaystyle=\frac{2\sinh(t)^{2}-1}{\cosh(t)^{4}}H(S).$

∎

In the following we use $d(\cdot,\cdot)$ to denote the geodesic distance on
the sphere.

######  Claim 4.

Fix $s\in\mathbb{S}^{N-1}$, and a non-constant configuration $S^{0}$ where all
spins are on a great circle perpendicular to $s$. There exists $\varepsilon$
small enough (depending on $S^{0}$), such that for all $S$ satisfying
$d(S_{i},S_{i}^{0})<\varepsilon$ for all $i$, either $H(\phi_{s}(S,t))$ or
$H(\phi_{-s}(S,t))$ (or both) is decreasing for $t\in[0,2\varepsilon]$.

###### Proof.

We consider two cases. The first, is when $\frac{\differential}{\differential
t}H(\phi_{s}(S,0))\leq 0$. Since $\|\frac{\partial}{\partial
t}\phi_{s}(s^{\prime},t)\|\leq 1$, during the time interval $[0,2\varepsilon]$
all spins of $S$ remain at a distance at most $3\varepsilon$ form $S^{0}$. By
the last claim $\frac{\differential^{2}}{\differential
t^{2}}H(\phi_{s}(S^{0},0))<0$, hence we may choose $\varepsilon$ small enough,
such that $\frac{\differential^{2}}{\differential
t^{2}}H(\phi_{s}(S^{\prime},t))<0$ for all $S^{\prime}$ in a
$3\varepsilon$-neighborhood of $S^{0}$. Then
$\frac{\differential}{\differential t}H(\phi_{s}(S,t))\leq 0$ for all
$t\in[0,2\varepsilon]$.

In the second case $\frac{\differential}{\differential t}H(\phi_{s}(S,0))\geq
0$, but then by symmetry of the flow $\frac{\differential}{\differential
t}H(\phi_{-s}(S,0))\leq 0$ and we are back to the first case. ∎

######  Claim 5.

Fix $s\in\mathbb{S}^{N-1}$ and a configuration $S$, such that $s\cdot S_{i}>0$
for all $i$. Then $H(\phi_{s}(S,t))$ is a decreasing function for
$t\in[0,\infty]$.

###### Proof.

For all $i$, denoting $\underline{s}=(s,\dots,s)\in(\mathbb{S}^{N-1})^{L}$,

$\displaystyle-
D_{i}H(S)\cdot(\underline{s}-(S\cdot\underline{s})S)_{i}=S_{i-1}\cdot(s-(S_{i}\cdot
s)S_{i})+S_{i+1}\cdot(s-(S_{i}\cdot s)S_{i})$
$\displaystyle\qquad-(S_{i-1}\cdot S_{i}+S_{i+1}\cdot
S_{i})\,S_{i}\cdot(s-(S_{i}\cdot s)S_{i})$ $\displaystyle\qquad=S_{i-1}\cdot
s-(S_{i}\cdot s)(S_{i}\cdot S_{i-1})+S_{i+1}\cdot s-(S_{i}\cdot s)(S_{i}\cdot
S_{i+1}).$

Therefore,

$\displaystyle\sum_{i}-D_{i}H(S)\cdot(\underline{s}-(S\cdot\underline{s})S)_{i}$
$\displaystyle\qquad=2\Big{(}\sum_{i}S_{i}\Big{)}\cdot
s-\Big{(}\sum_{i}(S_{i}\cdot S_{i-1})S_{i}\Big{)}\cdot
s-\Big{(}\sum_{i}(S_{i}\cdot S_{i+1})S_{i}\Big{)}\cdot s$
$\displaystyle\qquad=\sum\left((1-(S_{i}\cdot S_{i-1}))S_{i}\right)\cdot
s+\sum\left((1-(S_{i}\cdot S_{i+1}))S_{i}\right)\cdot s>0.$

Note that if $s\cdot S_{i}>0$ then at all positive times $\phi_{s}(S,t)>0$
(since spins always get closer to $s$). Therefore we can plug $\phi_{s}(S,t)$
for $S$ in the above inequality, and obtain:

$\displaystyle\frac{\differential}{\differential t}H(\phi_{s}(S,t))$
$\displaystyle=DH\cdot\frac{\partial\phi_{s}(S,t)}{\partial t}$
$\displaystyle=DH\cdot\left(\underline{s}-(\phi(S,t)\cdot\underline{s})\phi(S,t)\right)<0.$

∎

#### Step 3: Combining all parts of the path

######  Definition 4.4.

Fix $T>0$, $s\in\mathbb{S}^{N-1}$, and a configuration $S^{0}$ where all spins
are on a plane perpendicular to $s$. Let $U_{T,s,S^{0}}$ be the set of
configurations $S$, such that:

  1. (1)

If $S^{0}$ is constant, then under the deterministic flow $\varphi$, all spins
$\varphi(S,T)_{i}$ are at distance at most $\arccos(0.99)$ from $S^{0}$.

  2. (2)

If $S^{0}$ is not constant

    1. (a)

All spins $\varphi(S,T)_{i}$ are at distance at most $\varepsilon$ from
$S_{i}^{0}$; and

    2. (b)

$H(\phi_{s}(S,t))$ is decreasing for $t\in[0,2\varepsilon]$;

for $\varepsilon$ given in Claim 4.

######  Lemma 4.2.

Fix $T,s,S^{0}$ as in definition above. Then there exists a map
$\Psi_{T,s,S^{0}}:U_{T,s,S^{0}}\times[0,1]\to\Omega$ such that:

  1. (1)

$\Psi_{T,s,S^{0}}(S,0)=S$ and $\Psi_{T,s,S^{0}}(S,1)$ is in in the arctic $A$.

  2. (2)

$H(\Psi(S,t))$ is non-increasing for all $t$.

Moreover, $v[\Psi_{T,s,S^{0}}]$ and
$\sup_{S,t}|\Delta\operatorname{Ent}_{\Psi}(S,t)|$ are finite.

###### Proof.

If $S^{0}$ is non-constant, take

$\Psi_{T,s,S^{0}}(S,t)=\begin{cases}\varphi(S,3T\times
t)&t\in\left[0,\frac{1}{3}\right],\\\
\phi_{s}\left(\varphi(S,T),C\times\left(t-\frac{1}{3}\right)\right)&t\in\left[\frac{1}{3},\frac{2}{3}\right],\\\
\psi_{s}\left(\phi_{s}\left(\varphi(S,T),C/3\right),t-\frac{2}{3}\right)&t\in\left[\frac{2}{3},\frac{1}{3}\right],\end{cases}$

where $\psi$ is a rotation from $s$ to ${\rm e}_{1}$ with speed $3d(s,{\rm
e}_{1})$. $C$ should be chosen such that $\tanh(C/3-2\varepsilon)$ is close
enough to $1$, guaranteeing that we end up in the arctic.

If $S^{0}$ is constant, we skip $\phi$ and take $\psi$ to be a rotation of
$S^{0}$ to ${\rm e}_{1}$.

The speed $\partial_{t}\Psi$ can be calculated explicitly on each part of the
path—in the time interval $[0,1/3]$ it is given by $\frac{1}{3T}\|DH\|$ which
is bounded on the entire $\Omega$. During $[1/3,2/3]$ it is bounded by
$\frac{\sqrt{L}}{C}$, since each coordinate of $\partial_{t}\phi(S,t)$ is
bounded by $1$, hence $\|\partial_{t}\phi(S,t)\|\leq\sqrt{L}$. Finally, in the
last interval the speed is bounded by $3\pi\sqrt{L}$, since during time $1/3$
each coordinate crosses distance at most $\pi$.

In order to show that the entropy is bounded we use the fact that for a flow
given by an equation of the type

$\displaystyle\partial_{t}\Phi(S,t)$ $\displaystyle=f(\Phi(S,t)),$
$\displaystyle\Phi(S,0)$ $\displaystyle=S,$

the entropy production is the divergence of $f$:

$\Delta\operatorname{Ent}_{\Phi}(S,t)=\int\limits_{0}^{t}\text{div}f(\Phi(S,u))du,$
(53)

see, e.g., [19, Section 8.2]. In the first part of the path $\text{div}f$ is
bounded by $3T\sup|\Delta H|$, in the second part by $NL$, and in the third
part it is $0$. Since composing the paths results in adding the entropies, the
overall entropy is bounded. ∎

###### Proof of Theorem 1.3.

Since all limit points of the deterministic dynamics are on a great circle,
any configuration $S$ belongs to some set $U_{T,s,S^{0}}$. These are open
sets, and $\Omega$ is compact, therefore there exists a finite cover
$U_{1},\dots,U_{K}$. We set for $i=1,\dots,K$ the function $\Phi_{i}$ to be
equal $\Psi_{T,s,S^{0}}$, for $T,s,S^{0}$ such that $U_{i}=U_{T,s,S^{0}}$. The
hypotheses of Lemma 4.1 are then satisfied, with finite $v$ and
$\Delta\operatorname{Ent}$ (which do not depend on $\beta$), and $\Delta H\leq
0$. ∎

###  4.3. Explicit construction for $N=3$

When $N=3$ we are able to construct the path directly, without using a
compactness argument. This enables us to obtain an explicit (though not
optimal) estimate on the constant $C(L)$ in Theorem 1.3.

The path consists of three parts: first, we align the spins so that they fall
close to a plane. We then contract the spins towards a well chosen pole. This
second step is where the crucial difference between $O(3)$ and $O(2)$ enters.
Finally, we rotate the whole configuration to get it close to ${\rm e}_{1}$
and into the arctic.

The three steps are summarized in Lemmas 4.4, 4.5, and 4.6. We will need a few
preparations. The first is stating that “if all spins are close to _some_
great circle, they are also close to a great circle _in a fixed finite set_ of
great circles”.

######  Lemma 4.3.

There exists $C>0$ such that for any $\epsilon>0$, one can find $K\leq
C\epsilon^{-2}$ integer, and a set
$\\{v_{1},\dots,v_{K}\\}\subset\mathbb{S}^{2}$ such that for any $L$, and any
collection $S_{1},\dots,S_{L}\in\mathbb{S}^{2}$ satisfying $|S_{i}\cdot
s|\leq\epsilon$ for all $i\in\\{1,\dots,L\\}$ and some fixed
$s\in\mathbb{S}^{2}$, there exits $k\in\\{1,\dots,K\\}$ such that $|S_{i}\cdot
v_{k}|<2\epsilon$ for all $i\in\\{1,\dots,L\\}$.

###### Proof.

Let $\epsilon>0$. Let $A\subset\mathbb{S}^{2}$ be such that for every
$s\in\mathbb{S}^{2}$, there is $v\in A$ such that $|s-v|<\epsilon$. One can
find such a set containing at most $C\epsilon^{-2}$ points with $C$ universal
(for example, by projecting on $\mathbb{S}^{2}$ an $\epsilon/2$ mesh-size grid
on the boundary of the cube of side $2$ centred at $0$). Then for such a set
$A$, for any $s\in\mathbb{S}^{2}$, there is $v\in A$ with $|v-s|<\epsilon$.
For this $v$, one has $|S\cdot v|\leq|S\cdot(v-s)|+|S\cdot
s|\leq\epsilon+|S\cdot s|$ for every $S\in\mathbb{S}^{2}$, so

$|S\cdot s|<\epsilon\implies|S\cdot v|<2\epsilon,$

which gives the claim. ∎

Fix $\sin(\frac{\pi}{16})\geq\epsilon>0$, and let
$V_{\epsilon}=\\{v_{1},\dots,v_{K}\\}\subset\mathbb{S}^{2}$ be a set whose
existence is guaranteed by the previous Lemma (with $K\leq C\epsilon^{-2}$).

For $s\in\mathbb{S}^{2}$, define then

$D_{1}(\epsilon,s)=\\{S\in\Omega_{L}:\ |S_{i}\cdot s|<\epsilon,\ \forall
i=1,\dots,L\\}.$

Define also

$\Omega_{L}^{\pm}(s,\epsilon)=\\{S\in\Omega_{L}:\ S_{i}\cdot\pm s\geq
1-2\arcsin(\epsilon)^{2}\,,\;\forall i=1,\dots,L\\}.$

The three steps are then represented by the following three lemmas. For the
first two of them, the precise construction and proof is given in the next
subsections.

######  Lemma 4.4.

The application $\varphi:\Omega_{L}\times[0,\tau]\to\Omega_{L}$ with
$\tau=L-1-\epsilon$, to be constructed in Section 4.4, is such that

  1. (1)

$\varphi(S,0)=S$ and $\varphi(S,\tau)\in D_{1}(\epsilon,v)$ for some $v\in
V_{\epsilon}$;

  2. (2)

$t\mapsto\varphi(S,t)$ is continuous, and piecewise differentiable;

  3. (3)

$t\mapsto H^{\mathrm{per}}_{L}(\varphi(S,t))$ is non-increasing;

  4. (4)

$\|\dot{\varphi}(S,t)\|^{2}\leq 4\pi^{2}L$;

  5. (5)

letting $f_{t}(S)=\varphi(S,t)$, one has
$\Big{|}\frac{d(f_{t})_{\\#}\nu_{L}}{d\nu_{L}}\Big{|}\leq\Big{(}\frac{L}{\epsilon}\Big{)}^{L}$
for every $t\in[0,\tau]$ and $S$ in the image of $f_{t}$.

######  Lemma 4.5.

Let $s\in\mathbb{S}^{2}$. The application
$\phi_{s}:D_{1}(\epsilon,s)\times[0,\pi/2-\arcsin(\epsilon)/2]\to\Omega_{L}$,
to be constructed in Section 4.5, is such that

  1. (1)

$\phi_{s}(S,0)=S$ and
$\phi_{s}(S,\pi/2-2\arcsin(\epsilon))\in\Omega_{L}^{+}(s,\epsilon)\cup\Omega_{L}^{-}(s,\epsilon)$;

  2. (2)

$t\mapsto\phi_{s}(S,t)$ is continuous, and differentiable;

  3. (3)

$t\mapsto H^{\mathrm{per}}_{L}(\phi_{s}(S,t))$ is non-increasing;

  4. (4)

$\|\dot{\phi}_{s}(S,t)\|^{2}\leq 4\pi^{2}L$;

  5. (5)

letting $f_{t}(S)=\phi_{s}(S,t)$, one has
$\Big{|}\frac{d(f_{t})_{\\#}\nu_{L}}{d\nu_{L}}\Big{|}\leq 2^{L}$, for all
$t\in[0,\pi/2-2\arcsin(\epsilon)]$ and $S$ in the image of $f_{t}$.

######  Lemma 4.6.

Let $s\in\mathbb{S}^{2}$. There exist $\tau=\tau(s)\in[0,\pi]$, and
$\psi_{s}:\Omega_{L}^{+}(s,\epsilon)\times[0,\tau]\to\Omega_{L}$ such that

  1. (1)

$\psi_{s}(S,0)=S$ and $\psi_{s}(S,\tau)\in\Omega_{L}^{+}({\rm
e}_{1},\epsilon)$;

  2. (2)

$t\mapsto\psi_{s}(S,t)$ is continuous, and differentiable;

  3. (3)

$t\mapsto H^{\mathrm{per}}_{L}(\psi_{s}(S,t))$ is constant;

  4. (4)

$\|\dot{\psi}_{s}(S,t)\|^{2}\leq 4\pi^{2}L$;

  5. (5)

letting $f_{t}(S)=\psi_{s}(S,t)$, one has
$\Big{|}\frac{d(f_{t})_{\\#}\nu_{L}}{d\nu_{L}}\Big{|}=1$ for every
$t\in[0,\tau]$ and $S$ in the image of $f_{t}$.

###### Proof of Lemma 4.6.

Simply set $\tau$ to be the angle between $s$ and ${\rm e}_{1}$, $R_{t}$ the
rotation of angle $t$ in the plane spanned by $s,{\rm e}_{1}$ (with positive
direction from $s$ to ${\rm e}_{1}$). Setting $\psi(S,t)=R_{t}(S)$ does the
job. ∎

In the next sections, we prove Lemma 4.4 and 4.5. We then use them together
with Lemma 4.1 to bound the spectral gap in Section 4.6.

###  4.4. Local alignment: proof of Lemma 4.4

The path we will use will align $S_{1},S_{2},S_{3}$ by rotating $S_{2}$ around
the axis spanned by $S_{1}$ to end on the geodesic between $S_{1}$ and
$S_{3}$, then align $S_{1},\dots,S_{4}$ by rotating the pair $S_{2},S_{3}$
around the axis spanned by $S_{1}$ so that $S_{3}$ ends on the geodesic
between $S_{1}$ and $S_{4}$, and so on and so forth. We will first consider a
sequence of mappings which is better expressed in a suitable choice of
coordinates. For $1<k\leq L+1$, consider the following coordinate system. Let
$v_{1}=S_{1}$. If $S_{k}\notin\\{v_{1},-v_{1}\\}$, let

$v_{2}=\frac{S_{k}-(S_{k}\cdot v_{1})v_{1}}{\sqrt{1-(S_{k}\cdot v_{1})^{2}}},$

otherwise, let $v_{2}$ be any (fixed) norm one vector orthogonal to $v_{1}$.
Let then $v_{3}$ be any vector such that $(v_{1},v_{2},v_{3})$ is an
orthonormal basis. Express then a point $s$ on the sphere as

$s=(s\cdot v_{1})v_{1}+\sqrt{1-(s\cdot
v_{1})^{2}}\cos(\theta)v_{2}+\sqrt{1-(s\cdot v_{1})^{2}}\sin(\theta)v_{3}$

with $\theta\in[-\pi,\pi)$ the angle between $v_{2}$ and the projection of $s$
in the $(v_{2},v_{3})$-plane (so that $\theta=0$ when $s=S_{k}$). Write
$u_{i}=S_{i}\cdot v_{1}\in[-1,1]$, and $\theta_{i}$ for the angle $\theta$
corresponding to $s=S_{i}$. With this choice of coordinates, one has

$\sum_{i=1}^{k-1}S_{i}\cdot
S_{i+1}=\sum_{i=1}^{k-1}\big{(}u_{i}u_{i+1}+\sqrt{1-u_{i}^{2}}\sqrt{1-u_{i+1}^{2}}\cos(\theta_{i}-\theta_{i+1})\big{)},$

where $\theta_{k}=0$.

Define then the path to be

$\big{(}\varphi_{k}(S,t)\big{)}_{i}=\begin{cases}S_{i}&\text{ if }k\leq i\leq
L+1,\\\
u_{i}v_{1}+\sqrt{1-u_{i}^{2}}\cos(\theta_{i}-t\theta_{k-1})v_{2}+\sqrt{1-u_{i}^{2}}\sin(\theta_{i}-t\theta_{k-1})v_{3}&\text{
if }1<i<k.\end{cases}$

As, in the chosen coordinates,
$d\nu_{L}(S)=d\nu(S_{1})\prod_{i=k}^{L}d\nu(S_{i})\prod_{i=2}^{k-1}du_{i}d\theta_{i}$,
one has

$\Big{|}\frac{d(f_{t})_{\\#}\nu_{L}}{d\nu_{L}}\Big{|}=\frac{1}{1-t},$

where $f_{t}(S)=\varphi_{k}(S,t)$.

Note then that

$\displaystyle-H^{\mathrm{per}}_{L}(\varphi_{k}(S,t))$
$\displaystyle=\sum_{i=1}^{k-1}\big{(}\varphi_{k}(S,t)\big{)}_{i}\cdot\big{(}\varphi_{k}(S,t)\big{)}_{i+1}+\sum_{i=k}^{L}S_{i}\cdot
S_{i+1}$ $\displaystyle=\sum_{i=1}^{k-2}S_{i}\cdot
S_{i+1}+\big{(}u_{k-1}u_{k}+\sqrt{1-u_{k-1}^{2}}\sqrt{1-u_{k}^{2}}\cos((1-t)\theta_{k-1})\big{)}+\sum_{i=k}^{L}S_{i}\cdot
S_{i+1}.$

As the angles are in $[-\pi,\pi]$,
$-H^{\mathrm{per}}_{L}(\varphi_{k}(S,t))\geq-H^{\mathrm{per}}_{L}(S)$.

One has that $\big{(}\varphi_{k}(S,1)\big{)}_{k-1}$ is in the plane spanned by
$S_{1}$ and $S_{k}$, and that the distance to that plane of
$\big{(}\varphi_{k}(S,1-\epsilon)\big{)}_{k-1}$ is at most

$|\big{(}\varphi_{S}^{(k)}(1-\epsilon)\big{)}_{k-1}\cdot
v_{3}|\leq\sin(\epsilon\pi).$

Let $\tau_{0}=0$, $\tau_{k}=\tau_{k-1}+1-\epsilon/L$. Define

$\displaystyle\varphi(S,0)=S,$
$\displaystyle\varphi(S,t)=\varphi_{k}(\varphi(S,\tau_{k-1}),t-\tau_{k-1}),\
t\in[\tau_{k-1},\tau_{k}].$

This defines a continuous, and piecewise differentiable function of
$t\in[0,\tau_{L-1}]$, where $\tau_{L-1}=L-1-\epsilon$. By triangular
inequality, the distance of $(\varphi(S,\tau_{L-1}))_{i}$ to the plane spanned
by $S_{1}$ and $S_{L}$ is at most $\epsilon$. By the previous observations,
the Hamiltonian is decreasing in $t$ and

$\Big{|}\frac{d(f_{t})_{\\#}\nu_{L}}{d\nu_{L}}\Big{|}\leq(L\epsilon^{-1})^{L},$

where $f_{t}(S)=\varphi(S,t)$.

It remains to prove the bound on $\|\dot{\varphi}(S,t)\|^{2}$. On each time
interval $[\tau_{k-1},\tau_{k}]$, the mapping is a rotation of the $k$ first
spins around the axis spanned by $S_{1}$, so $\|\dot{\varphi}(S,t)\|^{2}\leq
4\pi^{4}L$.

###  4.5. Contraction towards a pole: proof of Lemma 4.5

For $s\in\mathbb{S}^{2}$, parametrize $S\in\Omega_{L}$ by
$(\theta,\theta^{\prime})\in([-\frac{\pi}{2},\frac{\pi}{2}]\times[0,2\pi])^{L}$
via
$S_{i}=\cos(\frac{\pi}{2}+\theta_{i})s+\sin(\frac{\pi}{2}+\theta_{i})\cos(\theta_{i}^{\prime})v_{1}+\sin(\frac{\pi}{2}+\theta_{i})\sin(\theta_{i}^{\prime})v_{2}$
where $(s,v_{1},v_{2})$ form an orthonormal basis of $\mathbb{R}^{3}$. Define
the configuration $X_{t}=X_{t}(S)=X_{t}(\theta,\theta^{\prime})$ via

$\big{(}X_{t}(\theta,\theta^{\prime})\big{)}_{i}=\cos(\frac{\pi}{2}+\theta_{i}+t)s+\sin(\frac{\pi}{2}+\theta_{i}+t)\cos(\theta_{i}^{\prime})v_{1}+\sin(\frac{\pi}{2}+\theta_{i}+t)\sin(\theta_{i}^{\prime})v_{2}$

Using $\sin(\frac{\pi}{2}+a)=\cos(a)$ and $\cos(\frac{\pi}{2}+a)=-\sin(a)$,
one obtains

$-H^{\mathrm{per}}_{L}(X_{t}(\theta,\theta^{\prime}))=\sum_{i=1}^{L}\sin(\theta_{i}+t)\sin(\theta_{i+1}+t)+\cos(\theta_{i}+t)\cos(\theta_{i+1}+t)\cos(\theta_{i}^{\prime}-\theta_{i+1}^{\prime}).$

So, as $\sin a\cos b+\sin b\cos a=\sin(a+b)$,

$\frac{d}{dt}-H^{\mathrm{per}}_{L}(X_{t}(\theta,\theta^{\prime}))=\sum_{i=1}^{L}\sin(\theta_{i}+\theta_{i+1}+2t)(1-\cos(\theta_{i}^{\prime}-\theta_{i+1}^{\prime})).$

Moreover,

$\frac{d^{2}}{dt^{2}}-H^{\mathrm{per}}_{L}(X_{t}(\theta,\theta^{\prime}))=2\sum_{i=1}^{L}\cos(\theta_{i}+\theta_{i+1}+2t)(1-\cos(\theta_{i}^{\prime}-\theta_{i+1}^{\prime})).$

First observe three things:

  1. (1)

if $0\leq\theta_{i}+t\leq\pi/2$ for all $i$s, then
$\frac{d}{dt}-H^{\mathrm{per}}_{L}(X_{t}(\theta,\theta^{\prime}))\geq 0$;

  2. (2)

if $-\pi/2\leq\theta_{i}-t\leq 0$ for all $i$s, then
$\frac{d}{dt}-H^{\mathrm{per}}_{L}(X_{-t}(\theta,\theta^{\prime}))\geq 0$;

  3. (3)

if $|\theta_{i}|<\frac{\pi}{8}$ for all $i$s and $t<\frac{\pi}{8}$,
$\cos(\theta_{i}+\theta_{i+1}+2t)\geq 0$ for all $i$s and so
$\frac{d^{2}}{dt^{2}}-H^{\mathrm{per}}_{L}(X_{t}(\theta,\theta^{\prime}))\geq
0$.

When $S\in D_{1}(\epsilon,s)$, we have $|\theta_{i}|\leq\arcsin(\epsilon)$.
Therefore, by our constraint on $\epsilon$, the sign of
$\frac{d}{dt}-H^{\mathrm{per}}_{L}(X_{t}(\theta,\theta^{\prime}))$ at $t=0$
determines the desired monotonicity for either
$t\in[0,\pi/2-2\arcsin(\epsilon)]$ or $-t\in[0,\pi/2-2\arcsin(\epsilon)]$:
indeed, the third observation above gives the monotonicity for all
$|t|\leq\pi/8$, and the first two observations show the monotonicity for
$|t|>\pi/8$ since $|\theta_{i}|\leq\arcsin(\epsilon)$.

We can now construct the path $\phi_{s}$ as follows:

  * •

If $S\in D_{1}(\epsilon,s)$ is such that
$\frac{d}{dt}-H^{\mathrm{per}}_{L}(X_{t}(S))\big{|}_{t=0}\geq 0$, set

$\phi_{s}(S,t)=X_{t}(S),$

for $t\in[0,\pi/2-2\arcsin(\epsilon)]$.

  * •

If $S\in D_{1}(\epsilon,s)$ is such that
$\frac{d}{dt}-H^{\mathrm{per}}_{L}(X_{t}(S))\big{|}_{t=0}<0$, set

$\phi_{s}(S,t)=X_{-t}(S),$

for $t\in[0,\pi/2-2\arcsin(\epsilon)]$.

As $\epsilon<\sin(\frac{\pi}{16})$, the first three points of Lemma 4.5 are
obvious (the first one follows from $(\phi_{s}(S,t))_{i}\cdot
s=\cos(\pi/2+\theta_{i}\pm t)$ depending on the case). To obtain the one-
before-last, observe that the transformation is a rotation of each
coordinates, so $\|\dot{\phi}_{s}(S,t)\|^{2}\leq 4\pi^{2}L$.

The last point of Lemma 4.5 follows from

$d\nu_{L}(S)=\prod_{i=1}^{L}\cos(\theta_{i})d\theta_{i}d\theta_{i}^{\prime},$

and $\frac{\cos(\theta_{i}-t)}{\cos(\theta_{i})}\leq\frac{1}{\cos(\pi/16)}\leq
2$ for $\theta_{i},t$ in the concerned regions.

###  4.6. Spectral Gap

Let $\epsilon>0$. Rather than parametrizing the sets $U_{i}$’s of Lemma 4.1
with numbers, we label them with pairs $(\sigma,v)$ with $\sigma\in\\{-,+\\}$
and $v\in V_{\epsilon}$, where $V_{\epsilon}$ is given as right after Lemma
4.3. Explicitly, we use $U_{\sigma,v}=\\{s:\ \chi_{\sigma,v}(s)=1\\}$, where,
for $v\in V_{\epsilon}$, $\sigma\in\\{-,+\\}$,

$\chi_{\sigma,v}(S)=\mathbf{1}_{\varphi(S,\tau_{1})\in
D_{1}(\epsilon,v)}\mathbf{1}_{\phi_{v}(\varphi(S,\tau_{1}),\tau_{2})\in\Omega_{L}^{\sigma}(v,\epsilon)},$

and $\tau_{1}=L-1-\epsilon$ and $\varphi$ are as in Lemma 4.4, while
$\tau_{2}=\frac{\pi}{2}-2\arcsin(\epsilon)$, and $\phi_{s}$ are as in Lemma
4.5. Finally, let $\tau_{3}(s),\psi_{s}$ be as in Lemma 4.6, and define

$T_{1}=\tau_{1},\quad T_{2}=\tau_{1}+\tau_{2},\quad
T_{3}=\tau_{1}+\tau_{2}+\tau_{3}(\sigma v)\,.$

Next, introduce the path

$\Psi_{\sigma,v}(S,t)=\begin{cases}\varphi(S,t)&\text{ if }t\leq T_{1},\\\
\phi_{v}(\varphi(S,T_{1}),t-T_{1})&\text{ if }t\in[T_{1},T_{2}],\\\
\psi_{v}\big{(}\phi_{v}(\varphi(S,T_{1}),T_{2}-T_{1}),t-T_{2}\big{)}&\text{ if
}t\in[T_{2},T_{3}],\end{cases}$

which is defined on $U_{\sigma,v}$, is continuous, and is piecewise
differentiable in $t$ with the bound

$\|\dot{\Psi}_{\sigma,v}(S,t)\|^{2}\leq 4\pi^{2}L,$ (54)

at every point of differentiability. Moreover, for any $L$ large enough

$\Big{|}\frac{d(f_{t})_{\\#}\nu_{L}}{d\nu_{L}}\Big{|}\leq(L\epsilon^{-1})^{L},$
(55)

where $f_{t}(S)=\Psi_{\sigma,v}(S,t)$. Finally, from the monotonicity of the
Hamiltonian along each piece of the path,

$-H^{\mathrm{per}}_{L}(\Psi_{\sigma,v}(S,t))\geq-H^{\mathrm{per}}_{L}(S).$
(56)

Applying Lemma 4.1 with these estimates on the functions
$(S,t)\mapsto\Psi_{\sigma,v}(S,t/T_{3})$, and choosing $\epsilon$ such that
$\Omega_{L}^{+}(e_{1},\epsilon)\subseteq A$, yields the bound on the
relaxation time stated in Theorem 1.3 in the case $N=3$. ∎

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