# Dynamical Loop Equation

Vadim<EMAIL_ADDRESS>[ Jiaoyang
<EMAIL_ADDRESS>[ University of California,
Berkeleypresep=, ]e1  University of Pennsylvania presep=, ]e2

###### Abstract

We introduce dynamical versions of loop (or Dyson-Schwinger) equations for
large families of two–dimensional interacting particle systems, including
Dyson Brownian motion, Nonintersecting Bernoulli/Poisson random walks,
$\beta$–corners processes, uniform and Jack-deformed measures on Gelfand-
Tsetlin patterns, Macdonald processes, and $(q,\kappa)$-distributions on
lozenge tilings. Under technical assumptions, we show that the dynamical loop
equations lead to Gaussian field type fluctuations.

As an application, we compute the limit shape for $(q,\kappa)$–distributions
on lozenge tilings and prove that their height fluctuations converge to the
Gaussian Free Field in an appropriate complex structure.

60F05,

60K35,

82B44,

Interacting Particle Systems,

Random Tilings,

Dyson-Schwinger Equations,

Gaussian Fluctuations,

###### keywords:

[class=MSC]

###### keywords:

,

###### Contents

  1. 1 Introduction
    1. 1.1 Overview
    2. 1.2 Dynamical loop equations
    3. 1.3 Height fluctuations of $(q,\kappa)$-distributions on lozenge tilings
  2. 2 Dynamical Loop Equation
    1. 2.1 Proof of Theorem 1.1
    2. 2.2 Particle-Hole Duality
  3. 3 Examples of Dynamical Loop Equations
    1. 3.1 Nonintersecting Bernoulli random walks
    2. 3.2 Nonintersecting Poisson random walks
    3. 3.3 Dyson Brownian motion
    4. 3.4 Measures on Gelfand-Tsetlin Patterns
    5. 3.5 $\beta$–Corners processes
    6. 3.6 Macdonald processes
    7. 3.7 $(q,\kappa)$-distributions on tilings and Koornwinder polynomials
  4. 4 Analysis of Dynamical Loop Equation
    1. 4.1 Statement of the asymptotic theorem
    2. 4.2 Plan of the proof for Theorem 4.5
    3. 4.3 Properties of ${\mathcal{A}}(z)$ and ${\mathcal{C}}(z)$
    4. 4.4 Asymptotic expansion of ${\mathcal{A}}(z)$
    5. 4.5 Mean Estimate
    6. 4.6 Covariance Estimate
  5. 5 Law of Large Numbers for random tilings
    1. 5.1 Setup
    2. 5.2 Assumptions in Theorem 4.5
    3. 5.3 Application of Theorem 4.5
    4. 5.4 Equivalent descriptions of the limit shape
    5. 5.5 Proof of Theorem 5.1
    6. 5.6 Proof of Theorems 1.3 and 1.6
  6. 6 Gaussian Free Field
    1. 6.1 Stochastic evolution of fluctuations
    2. 6.2 Fluctuations along the characteristic flow
    3. 6.3 Matching the covariance structure with the Gaussian Free Field
  7. A Proofs of claims in Section 3

##  1 Introduction

###  1.1 Overview

Interacting particle systems in one time and one space dimension with non-
local interactions appear in various probability models, such as random
matrices, random tilings, and random growth models.

In the continuous setting, the most famous example is [48], where Dyson
observed that the eigenvalues of a matrix-valued Hermitian Brownian motion
form an interacting particle system with a logarithmic Coulomb interaction and
quadratic potential. In many important cases one dimensional marginal
densities of those $(1+1)$-dimensional interacting particle systems can be
computed explicitly, and identified with log-gases. For such one dimensional
log-gas type systems, loop (or Dyson-Schwinger) equation is an important tool
to study their macroscopic fluctuations. They were first used in the
theoretical physics literature (e.g., in the work of Migdal [84]) and were
later introduced to the mathematical community by Johansson [73] to derive
macroscopic central limit theorems for general $\beta$-ensembles of
eigenvalues of random matrices, see also [23, 24, 79]. One advantage of the
asymptotic arguments based on the loop equations is that they tend to be very
universal and extend to wide families of stochastic systems.

On the discrete side, marginals given by appropriate versions of log–gases
were used in the analysis of random tilings starting from [38, 75, 19] and in
the study of random interface growth models starting from [74]. Generalizing
these distibutions, Borodin, Guionnet and Gorin [18] introduced discrete
$\beta$-ensembles, and studied their Gaussian macroscopic fluctuations using
discrete loop equations, which originated in the work of Nekrasov and his
collaborators [86, 90, 89]; see also [39].

Central limit theorems for macroscopic fluctuations of log-gases (in
investigations of random matrices, random tilings, and random partitions)
reveal that the limiting fields in these theorems can be viewed as
one–dimensional slices of a unique two–dimensional limiting object, the
_Gaussian Free Field_ , see [16, 12, 39, 43, 30] for the results of such type.
This fits into a general belief that the Gaussian Free Field should appear as
a universal limit of the macroscopic fluctuations for random two-dimensional
surfaces of various origins.

The aim of this paper is to lift the asymptotic analysis from one-dimensional
sections to full two-dimensional fields by developing very general dynamical
versions of loop equations, the _dynamical loop equations_. They provide a new
toolbox to study macroscopic fluctuations of two dimensional interacting
particle systems, directly applicable to many well-known stochastic dynamics
both in discrete and continuous settings, such as: Dyson Brownian motion [48],
nonintersecting Bernoulli/Poisson random walks [59, 78, 67], $\beta$–corner
processes [60, 16], measures on Gelfand-Tsetlin patterns [31, 97], Macdonald
processes [14], and $(q,\kappa)$-distributions on lozenge tilings [19, 39]. We
prove that under mild technical assumptions, the dynamical loop equations lead
to fluctuations described by Gaussian fields with accessible covariance.

There were hints in the literature, suggesting that the framework of the loop
equations might be adaptable to multi-time two-dimensional setting, as in [4,
67, 40, 68, 41]. However, these results were mostly isolated and relied on
specifics of the stochastic systems they dealt with. In contrast, we work
directly with transition probabilities of the stochastic evolutions of
interest and allow for them a very general form, parameterized by three
analytic functions and an arbitrary real parameter, playing the same role as
$\beta$ in the random matrix theory. Our approach covers many of the old
examples and simultaneously allows asymptotic analysis of new stochastic
systems, which were not accessible by previous tools.

As our main application, we study the height fluctuations of
$(q,\kappa)$-distributions on lozenge tilings introduced in [19]. In the
special case $\kappa=0$, the $(q,\kappa)$-distributions on lozenge tilings
become the $q^{\text{volume}}$–weighted lozenge tilings (or plane partitions),
whose particular cases were investigated in [104, 94, 34, 2]. We compute the
limit shapes for the $(q,\kappa)$–random tilings of a special class of
polygonal domains (“trapezoids”) with arbitrary many sides and discover that
the shapes can be parameterized by algebraic curves in an appropriate
coordinate system, thus giving the positive answer to the question raised in
[19, end of Section 2]. We further show that the centered height fluctuations
of $(q,\kappa)$-random tilings of these domains converge to the Gaussian free
field. In particular, this verifies the conjecture in [39, Conjecture 8.4.1]
for $(q,\kappa)$-distributions on lozenge tilings of hexagonal domains111For
another proof of this conjecture see the forthcoming paper [44]., and the
prediction in [76] for $q^{\text{volume}}$–weighted lozenge tilings.

In the next two subsections we, first, state the dynamical loop equation in
its most general form, and then present in details our results on
$(q,\kappa)$–random lozenge tilings.

###  1.2 Dynamical loop equations

In the most general discrete setting of our interest, we deal with discrete-
time Markov chains with deterministic initial conditions. The state space of
the chains, denoted $\mathbb{GT}_{n}$, consists of $n$–tuples of integers
${\bm{\lambda}}:\lambda_{1}\geqslant\dots\geqslant\lambda_{n}$ which we call
signatures of rank $n$ or (in the case $\lambda_{n}\geqslant 0$) Young
diagrams with at most $n$ rows. 222The notation $\mathbb{GT}$ comes from the
names Gelfand and Tsetlin, who used combinatorics of signatures in the study
of representation theory of classical Lie groups. In this text we allow
transitions of two types for the Markov chains:

  1. 1.

_Ascending transition_ : $n$ is fixed and at each step coordinates
$\lambda_{i}$ either stay the same or grow by one. I.e., non-zero transition
probabilities are from $\bm{\lambda}\in\mathbb{GT}_{n}$ to
$\bm{\mu}\in\mathbb{GT}_{n}$ under restriction

$\mu_{i}-\lambda_{i}\in\\{0,1\\}\qquad\text{ for }i=1,2,\dots,n.$

  2. 2.

_Descending transition_ : $n$ decreases by $1$ at each step and we transition
to _interlacing_ signatures. I.e., non-zero transition probabilities are from
$\bm{\lambda}\in\mathbb{GT}_{n+1}$ to $\bm{\mu}\in\mathbb{GT}_{n}$ under the
restriction

$\lambda_{1}\geqslant\mu_{1}\geqslant\lambda_{2}\geqslant\dots\geqslant\mu_{n}\geqslant\lambda_{n+1}.$
(1)

Although requiring all transitions to be of one of these two types might seem
restrictive, we present in Section 3 a large number of natural examples
fitting into our formalism. In fact, the two types of transitions are closely
related to each other: as we show in Section 2.2 one can be obtained from
another by the particle-hole duality. Hence, in this section it suffices to
present our results only in the ascending setting.

Figure 1: We identify
$(\lambda_{1},\lambda_{2},\dots,\lambda_{n})\in\mathbb{GT}_{n}$ with
$n$-particles
$(\lambda_{1},\lambda_{2}-\theta,\lambda_{3}-2\theta,\dots,\lambda_{n}-(n-1)\theta)$.

Fix $\theta>0$ and let $n\in\mathbb{Z}_{>0}$. We identify each signature
${\bm{\lambda}}=(\lambda_{1}\geqslant\lambda_{2}\geqslant\dots\geqslant\lambda_{n})\in\mathbb{GT}_{n}$,
with an $n$-particle system
$(\lambda_{1},\lambda_{2}-\theta,\lambda_{3}-2\theta,\dots,\lambda_{n}-(n-1)\theta)$.
In other words, we use a bijection between $\mathbb{GT}_{n}$ and the
$\theta$-shifted lattice

$\displaystyle\mathbb{W}_{\theta}^{n}\mathrel{\mathop{:}}=\bigl{\\{}(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\mid
x_{1}\in\mathbb{Z},\quad x_{i}-x_{i+1}\in\theta+\mathbb{Z}_{\geqslant 0},\quad
i=1,2,\dots,n-1\bigr{\\}},$ (2)

given by

$\displaystyle\mathbb{GT}_{n}\ni{\bm{\lambda}}\leftrightarrow{\bm{x}}=(x_{1},x_{2},\dots,x_{n})\in\mathbb{W}_{\theta}^{n},\qquad
x_{i}=\lambda_{i}-(i-1)\theta,\quad i=1,2,\dots,n.$ (3)

Graphically this procedure is illustrated in Figure 1. In the special case
$\theta=1$, $\mathbb{W}_{1}^{n}$ is the set of strictly ordered $n$-tuples of
integers and we simply denote it by $\mathbb{W}^{n}$.

Fix $\theta>0$, a particle configuration
${\bm{x}}=(x_{1},x_{2},\dots,x_{n})\in\mathbb{W}_{\theta}^{n}$, an interaction
function $b(z)$, and weight functions $\phi^{+}(z),\phi^{-}(z)$. Our central
object is the following transition probability

$\displaystyle\mathbb{P}({\bm{x}}+{\bm{e}}|{\bm{x}})=\frac{1}{Z({\bm{x}})}\prod_{1\leqslant
i<j\leqslant n}\frac{b(x_{i}+\theta e_{i})-b(x_{j}+\theta
e_{j})}{b(x_{i})-b(x_{j})}\prod_{i=1}^{n}\phi^{+}(x_{i})^{e_{i}}\phi^{-}(x_{i})^{1-e_{i}},$
(4)

where ${\bm{x}},{\bm{x}}+{\bm{e}}\in\mathbb{W}_{\theta}^{n}$,
${\bm{e}}=(e_{1},e_{2},\dots,e_{n})\in\\{0,1\\}^{n}$ and $Z({\bm{x}})$ is a
normalization constant333Here and below we silently assume that such a
constant exists. guaranteeing that

$\sum_{{\bm{e}}\in\\{0,1\\}^{n}}\mathbb{P}({\bm{x}}+{\bm{e}}|{\bm{x}})=1.$ (5)

We remark that, as in [18], $\theta$ can be linked to the parameter $\beta$ of
the random matrix theory (taking values $\beta=1,2,4$ for
real/complex/quaternionic matrices) through $\theta=\beta/2$. Thus, outside
$\theta=1$ we do not expect Markov chains with transitions (4) to be free
fermionic or analysable through the techniques of determinantal point
processes.

Some examples of Markov chains with transitions in the form (4) and their
limits are explained in Section 3. They include nonintersecting
Bernoulli/Poisson random walks, $\beta$–corners processes, Dyson Brownian
motion, uniform and Jack-deformed measures on Gelfand-Tsetlin patterns,
Macdonald processes, and $(q,\kappa)$-distributions on lozenge tilings. The
following dynamical version of the loop (or Schwinger-Dyson) equation provides
a unified approach to study the macroscopic fluctuations of dynamics with
transitions (4).

######  Theorem 1.1 (Dynamical Loop Equation).

Choose an open set $U\subset\mathbb{C}$, a particle configuration
${{\bm{x}}=(x_{1}>x_{2}>\dots>x_{n})\in\mathbb{W}_{\theta}^{n}}$ such that the
interval $[x_{n},x_{1}]\subset U$, a parameter $\theta>0$, two holomorphic
functions $\phi^{+}(z)$, $\phi^{-}(z)$ on $U$ and a conformal (i.e.,
holomorphic and injective) function $b(z)$ on $U$. Assume that the random
$n$–tuple ${\bm{e}}=(e_{1},e_{2},\dots,e_{n})\in\\{0,1\\}^{n}$ is distributed
according to the transition probability (4). Then the following observable is
a holomorphic function of $z\in U$:

$\displaystyle\mathbb{E}\left[\phi^{+}(z)\prod_{j=1}^{n}\frac{b(z+\theta)-b(x_{j}+\theta
e_{j})}{b(z)-b(x_{j})}+\phi^{-}(z)\prod_{j=1}^{n}\frac{b(z)-b(x_{j}+\theta
e_{j})}{b(z)-b(x_{j})}\right].$ (6)

######  Remark 1.2.

The condition ${\bm{x}}\in\mathbb{W}_{\theta}^{n}$ is not crucial for the
validity of Theorem 1.1 and one can replace $x_{i}$ by other complex numbers
in $U$; yet, all the examples considered in our paper have this condition or
its limiting forms.

We give two proofs of Theorem 1.1 in Section 2.1. The first proof is a direct
verification that each possible (simple) pole of (6) has a zero residue. The
second proof identifies (6) with a ratio of two partition functions
$Z({\bm{x}})$ of the ensemble (4) — with $n+1$ particles and with $n$
particles — and notices that these partition functions are holomorphic
functions of $x_{i}$. The second proof suggests possible further
generalizations of Theorem 1.1, see Remark 2.1.

Special or limiting cases of the above dynamical loop equations were known
before. For Dyson Brownian motion equivalent expressions were implicitly used
in [4, 8, 72] to study the limiting empirical particle density and its
fluctuations. For $\beta$-nonintersecting Poisson random walks and uniformly
random lozenge tiling, they were developed by the second named author in [67,
68] to study the macroscopic fluctuations. Certain formulas, resembling our
dynamical loop equations, can be also found in the work of Nekrasov [87,
88]444There is no Markovian dynamics in these papers, however, certain
holomorphic observables on random ensembles of $n$ Young diagrams are being
studied as one of the examples. The probabilities in these ensembles depend on
a degree $2n$ polynomial and by choosing the polynomial appropriately, one can
force each diagram to be either empty or one box — resembling our
${\bm{e}}\in\\{0,1\\}^{n}$. In order to closer match our weights (4) with
certain specific $b(\cdot)$ and $\phi^{\pm}(\cdot)$ functions, one needs to
deal with (colored) theories with defects in the terminology of [87, 88].. The
first main contribution of this paper is to introduce the general form (4)
(see also (27)) of transition probability and analyze it by dynamical loop
equations.

It is also instructive to compare Theorem 1.1 with two-level discrete loop
equations of [40, 41]. An important distinction is that in our setup
${\bm{x}}$ is deterministic, while in [40, 41] it is random (of log-gas type).
We are going to significantly rely on this feature in the applications of
Theorem 1.1: it allows us to iterate the equations and reach joint
distributions at arbitrary times, rather than only at adjacent ones.

Proceeding to a broader context, we refer to [63] for an introduction to the
loop equations and historic overview. A general phenomena in the study of many
particle systems with strong mean field interactions is that the moments of a
large class of test functions admit an infinite system of equations that are
called the loop equations, Dyson-Schwinger equations, Master equations, or
Ward identities depending on the context. These equations are often
asymptotically closed in the limit of growing dimensions and can be used to
get delicate asymptotic information on the particle systems. This strategy has
been developed at the formal level (i.e. assuming that the asymptotic
expansions should exist) in physics starting from [3], see [22, 35, 36, 52,
53, 54] for more recent and much more detailed studies. The approach was put
on the rigorous grounds starting from [73] and we refer to [63] for an
outline, as well as many references on the related developments in the
mathematical literature. The theoretical physics literature made a curious
observation that the asymptotic expansions in these particle systems
(developed through the loop equations or by other means) satisfy the so-called
topological recursion, which then connects them to enumerative geometry and
generating functions for maps, cf. [101], [27], and [102]. It is yet to be
seen, whether connections to enumerative geometry can be also found in the
study of our dynamical loop equations.

In Section 4 we prove that under technical assumptions the dynamical loop
equations lead to Gaussian fluctuations with explicitly computable mean and
covariance. More precisely, take a small scaling parameter $\varepsilon$ and
assume that $n$ is inverse proportional to $\varepsilon$, i.e.
$n=\lfloor{\mathsf{n}}\varepsilon^{-1}\rfloor$ for some ${\mathsf{n}}>0$. We
encode the particle configurations ${\bm{x}}$ and ${\bm{x}}+{\bm{e}}$ by a
smoothed version of the empirical density:

$\displaystyle\rho(s;{\bm{x}})=\frac{1}{\theta}\sum_{i=1}^{n}\bm{1}(\varepsilon
x_{i}\leqslant s\leqslant\varepsilon
x_{i}+\varepsilon\theta),\quad\rho(s;{\bm{x}}+{\bm{e}})=\frac{1}{\theta}\sum_{i=1}^{n}\bm{1}(\varepsilon(x_{i}+e_{i})\leqslant
s\leqslant\varepsilon(x_{i}+e_{i})+\varepsilon\theta).$

Then under the transition probability (4), $\rho(s;{\bm{x}}+{\bm{e}})$ is a
random probability density. Its difference with $\rho(s;{\bm{x}})$ has the
following decomposition:

$\rho(s;{\bm{x}}+{\bm{e}})-\rho(s;{\bm{x}})=\text{``stochastic
term"}+\text{``deterministic term"}.$ (7)

In Theorem 4.5, under technical assumptions, by analyzing the loop equation
(6) we prove that as $\varepsilon\to 0$ the stochastic term is asymptotically
a Gaussian field and obtain asymptotic expansions for the deterministic term
and the covariance of the stochastic term. As a consequence, for any two
dimensional interacting particle system, if we can view it as a Markov process
with transition probabilities (4) (with potentially time-dependent functions
$b$ and $\phi^{\pm}$), then its asymptotic limit shape can be found from
deterministic terms in (7) and macroscopic fluctuations converge to a two-
dimensional Gaussian field, which is found from the stochastic terms in (7).
Note that technical assumptions of Theorem 4.5 still remain to be checked on
case-by-case basis; we demonstrate in Section 5.2 how this is done for
$(q;\kappa)$–distributions on lozenge tilings.

Our analysis of the decomposition (7) based on Theorem 1.1 has one important
difference from the numerous previous one-dimensional results based on the
loop equations for log-gas type systems, as in [23, 24, 79, 73, 18]. Those
texts relied on a two-step procedure, in which one needs to establish a priori
estimates on the concentration of the empirical density in the first step
before using the loop equations in the second step. Such estimates rely on a
different set of tools, such as variational problems or large deviations
techniques. In contrast, our approach does not require any a priori estimates
and extracts all asymptotic information directly from Theorem 1.1: since we
deal with elementary one-step transitions, we can rely on the determinstic
fact that ${\bm{x}}+{\bm{e}}$ is close to ${\bm{x}}$.

A reader might be wondering why we chose to deal with the smoothed empirical
density $\rho(s;{\bm{x}})$ rather than with a more standard empirical measure
$\varepsilon\sum_{i=1}^{n}\delta_{\varepsilon x_{i}}$. While we could have
dealt with the latter as well (at the expense of modifications of some of the
formulas), the former leads to “better” functions (polynomials, rational
functions, etc) before taking limits, which is convenient when we aim at
algebraic answers, as in the following Theorems 1.3 and 1.6. For instance, if
$x_{i}=-i\theta$, $1\leq i\leq n$, then

$\exp\left(\theta\int_{\mathbb{R}}\frac{\rho(s;{\bm{x}})}{z-s}{\rm
d}s\right)=\exp\left(\int^{0}_{-\varepsilon\theta n}\frac{{\rm
d}s}{z-s}\right)=\frac{z+\varepsilon\theta n}{z}.$

On the other hand, if we had used the standard empirical measure, then a
similar expression would have become a much more singular function of $z$.

###  1.3 Height fluctuations of $(q,\kappa)$-distributions on lozenge tilings

We illustrate the strength of Theorem 1.1 by applying it to the study of the
fluctuations of height functions of $(q,\kappa)$-distributions on lozenge
tilings introduced in [19], which were not accessible by previous methods.

We use the triangular grid with coordinate system $(t,x)$ based on two lattice
directions inclined to each other under $120$ degrees, as in the left panel of
Figure 2. $x$ increases in the vertical direction and $t$ increases in the
down–right direction. The lozenges are defined as unions of adjacent
elementary triangles of the grid and they have three types:

We are going to refer to the first type as _horizontal_ lozenges; they are
shown in red in our drawings.

Figure 2: Left panel: trapezoid domain with $N=8$, $T=12$, $a_{1}=-6$,
$b_{1}=-4$, $a_{2}=-2$, $b_{2}=4$; and $(t,x)$–coordinate system. Middle
panel: one possible lozenge tiling and the values for $\tilde{x}$ in (9).
Right panel: non-intersecting paths and values of the height function.

For any integers $N>0$, $T>0$, and $-T\leqslant
a_{1}<b_{1}<a_{2}<b_{2}<\dots<a_{r}<b_{r}\leqslant N$, with
$\sum_{i=1}^{r}(b_{i}-a_{i})=N$, we consider a trapezoidal domain $P$ with
right vertical side $[0,N]$ at horizontal coordinate $T$ and left vertical
sides $[a_{i},b_{i}]$ at horizontal coordinate $0$. Other sides of the domain
are segments going in down-right direction from points $(0,a_{i})$,
$i=1,\dots,r$; segment in up-left direction from the point $(T,N)$; segments
in up-right direction from points $(0,b_{i})$, $i=1,\dots,r$; and segment in
down-left direction from $(T,0)$. As a result, $P$ is a polygonal domain with
$3r+3$ sides. We refer to the left panel of Figure 2 for an example with
$r=2$.

We let $\Omega(P)$ denote the set of all lozenge tilings of the domain $P$;
one such tiling for a particular choice of the parameters is shown in the
middle panel of Figure 2. Uniformly random lozenge tilings of $P$ are now well
understood, see [57] for general overview and [96, 97, 58, 46, 47, 45, 29] for
more specific results, and our task is to go beyond these results and consider
more advanced measures on tilings. The $(q,\kappa)$-distribution on lozenge
tilings of $P$ assigns to any ${\mathcal{T}}\in\Omega(P)$ the probability

$\displaystyle\mathbb{P}({\mathcal{T}})=\frac{w({\mathcal{T}})}{\sum_{{\mathcal{T}}^{\prime}\in\Omega(P)}w({\mathcal{T}}^{\prime})},$
(8)

where the weights $w({\mathcal{T}})$ of a tiling is the product of weights of
its horizontal lozenges:

$\displaystyle
w({\mathcal{T}})=\prod_{{\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}}\in{\mathcal{T}}}w({\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}}),\qquad
w({\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}})=\kappa
q^{x-t/2}-\kappa^{-1}q^{-x+t/2}=\kappa
q^{\tilde{x}}-\kappa^{-1}q^{-\tilde{x}},$ (9)

and $(t,x)$ is the coordinate of the left corner of a horizontal lozenge
${\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}}\in{\mathcal{T}}$. The
number $\tilde{x}=x-t/2$ appearing (9) has a simple meaning: this is the
vertical coordinate counted so that its fixed level lines are horizontal, i.e.
by using the Cartesian coordinate system (rather than non-orthogonal $(t,x)$
coordinate system that we stick to), see the middle panel of Figure 2.

We need to make sure that the weights (8) are positive. There are three
possible restrictions on the parameters which guarantee positivity:

  1. 1.

imaginary case: $q$ is a positive real number, $\kappa$ is an arbitrary pure
imaginary complex number.

  2. 2.

real case: $q$ is a positive real number, $\kappa$ is a real number with
additional restrictions depending on the trapezoidal domain; $\kappa^{2}$
cannot lie inside the interval $[q^{-(N+b_{r}-1)},q^{T+a_{1}-1}]$ or
$[q^{T+a_{1}-1},q^{-(N+b_{r}-1)}]$ depending on whether $q>1$ or $q<1$.

  3. 3.

trigonometric case: $q$ and $\kappa$ are complex numbers of modulus $1$,
$q=e^{i\alpha},\kappa=e^{\mathbf{i}\beta}$, with additional restrictions
depending on the trapezoidal domain; $2\beta-\alpha(T+a_{1}-1)$ and
$2\beta+\alpha(N+b_{r}-1)$ must lie in the same interval $(2k\pi,2(k+1)\pi)$,
for some $k\in\mathbb{Z}$.

Figure 3 shows samples from $(q,\kappa)$–distribution on lozenge tilings of
hexagons for various values of the parameters. For special values of $q$ and
$\kappa$ the formulas (8),(9) can be simplified: $\kappa=\infty$ leads to the
weight $q^{\text{volume}}$ on tilings, where by volume we mean the total
volume enclosed under the stepped surface corresponding to the tiling (the
surface becomes visible by treating three types of lozenges as projections of
three sides of a unit cube, cf. the middle panel of Figure 2); $\kappa=0$
leads to the weight $q^{-\text{volume}}$; $q=1$ leads to the uniform measure
on tilings; in the limit $q\to 1$, $\kappa\to 1$ and after division by $q-1$,
the weight (9) turns into a linear function of $\tilde{x}$.

Figure 3: Samples of tilings of $100\times 100\times 100$ hexagon. Left panel:
imaginary case with $\kappa=\infty$, $q=0.97$. Center panel: real case with
$q=0.99$, $\kappa=q^{-110}$. Right panel: trigonometric case with
$\alpha=0.015$, $\beta=1.57$.

The weights (9) were first introduced in [19] for lozenge tilings of hexagons.
The same paper showed that tilings with such weights lead to rich algebraic
structures. In other words, in contrast to generic weights, (9) should be
treated as an exactly-solvable or integrable case: the analysis of [19]
revealed connections to $q$–Racah orthogonal polynomials; in addition, the
limit shape of the tilings was found to be encoded in an appropriate
coordinate system by a quadratic polynomial in two variables. These results
have led naturally to a question: is such integrability a specific feature of
the hexagons, or does it extend to tilings with weights (9) for more general
domains? Our results show that the latter is true.555One conceptual
explanation is as follows: the weights (8),(9) are not ad hoc, instead they
are deeply related to the combinatorics of the Koornwinder symmetric
polynomials, as we detail in Section 3.7.2. We now present three theorems
explaining how the macroscopic behavior of $(q,\kappa)$–random lozenge tilings
of $P$ is encoded in certain algebraic equations.

We introduce a small real parameter $\varepsilon\ll 1$ and assume that $r$ and
$\kappa$ stay fixed, while $q$, $N$, $T$, $a_{i},b_{i}$, $i=1,\dots,r$, depend
on $\varepsilon$ in such a way that as $\varepsilon$ goes to $0$ we have:

$\displaystyle\varepsilon N\to{\mathsf{N}},\quad\varepsilon
T\to{\mathsf{T}},\quad\varepsilon\ln(q)\to\ln({\mathsf{q}}),\quad\varepsilon
a_{i}\to{\mathsf{a}}_{i},\quad\varepsilon b_{i}\to{\mathsf{b}}_{i},\quad
1\leqslant i\leqslant r,$ (10)

with asymptotic parameters ${\mathsf{q}}$, ${\mathsf{N}}$, ${\mathsf{T}}$,
${\mathsf{a}}_{i},{\mathsf{b}}_{i}$, $i=1,\dots,r$ satisfying
$-{\mathsf{T}}\leqslant{\mathsf{a}}_{1}<{\mathsf{b}}_{1}<{\mathsf{a}}_{2}<{\mathsf{b}}_{2}<\dots<{\mathsf{a}}_{r}<{\mathsf{b}}_{r}\leqslant{\mathsf{N}}$,
and $\sum_{i=1}^{r}({\mathsf{b}}_{i}-{\mathsf{a}}_{i})={\mathsf{N}}$. We also
keep positivity constraints, so that in the real case
${\mathsf{q}},\kappa\in\mathbb{R}$ and $\kappa^{2}$ cannot lie inside the
interval
$[{\mathsf{q}}^{-({\mathsf{N}}+{\mathsf{b}}_{r}-1)},{\mathsf{q}}^{{\mathsf{T}}+{\mathsf{a}}_{1}-1}]$
or
$[{\mathsf{q}}^{{\mathsf{T}}+{\mathsf{a}}_{1}-1},{\mathsf{q}}^{-({\mathsf{N}}+{\mathsf{b}}_{r}-1)}]$
depending on whether ${\mathsf{q}}>1$ or ${\mathsf{q}}<1$. The conditions
imply that the rescaled trapezoidal domain $\varepsilon P$ converges to a
limiting domain ${\mathsf{P}}$.

It is convenient to state the asymptotic results on tilings in terms of their
_height functions._ We are going to use the following version of the height
function $h$ defined on the vertices of the triangular lattice inside $P$. Let
us ignore  lozenges in the tiling and trace only the lozenges of the remaining
two types  and  . Then these lozenges form $N$ _non-intersecting paths_
traversing $P$ from the left boundary to the right boundary, as shown in the
right panel of Figure 2. We define $h(t,x)$ as the total number of paths below
a given point $(t,x)$; this is originally defined for integer values of $t$
and $x$ and then extended to all real values by linear interpolation. Because
tilings are random, so is the function $h(t,x)$; however, its values on the
boundary of $P$ are deterministic: for instance, it equals $0$ on the two
bottom-most segments of the boundary of $P$ and it equals $N$ on the two
upper-most segments of the boundary of $P$.

A very general _variational principle for tilings_ of [37] implies that in the
regime (10) the height functions
$h(\varepsilon^{-1}{\mathsf{t}},\varepsilon^{-1}{\mathsf{x}})$ for random
lozenge tilings of $P$ converge in probability to a deterministic limit:

$\displaystyle\varepsilon
h(\varepsilon^{-1}{\mathsf{t}},\varepsilon^{-1}{\mathsf{x}})\rightarrow{\mathsf{h}}({\mathsf{t}},{\mathsf{x}}),\quad({\mathsf{t}},{\mathsf{x}})\in{{\mathsf{P}}}.$
(11)

The limiting height function ${\mathsf{h}}({\mathsf{t}},{\mathsf{x}})$ is the
minimizer of a certain surface tension integral, as noticed in [19, Section
2.4]. In general such minimization problems are hard to solve. However, in our
situation ${\mathsf{h}}({\mathsf{t}},{\mathsf{x}})$ is explicit and we compute
it for all trapezoidal domains $P$ in Theorems 1.3 and 1.6.

For a point $({\mathsf{t}},{\mathsf{x}})$ in the interior of ${{\mathsf{P}}}$,
we identify the gradient of the limiting height function $\nabla{\mathsf{h}}$
with the local densities of three types of lozenges:

$p_{{\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}}}({\mathsf{t}},{\mathsf{x}})=1-\partial_{{\mathsf{x}}}{\mathsf{h}}({\mathsf{t}},{\mathsf{x}}),\quad
p_{{\scalebox{0.15}{\,\includegraphics{lozenge_v_up.pdf}}}}({\mathsf{t}},{\mathsf{x}})=-\partial_{\mathsf{t}}{\mathsf{h}}({\mathsf{t}},{\mathsf{x}}),\quad
p_{{\scalebox{0.15}{\,\includegraphics{lozenge_v_down.pdf}}}}({\mathsf{t}},{\mathsf{x}})=\partial_{{\mathsf{x}}}{\mathsf{h}}({\mathsf{x}},{\mathsf{t}})+\partial_{\mathsf{t}}{\mathsf{h}}({\mathsf{t}},{\mathsf{x}}).$
(12)

A central feature of random lozenge tilings is that they exhibit boundary-
induced phase transitions. Depending on the shape of the domain, they can
admit frozen regions, where the associated height function is flat almost
deterministically, and liquid regions, where the height function appears more
rough and random and all three types of lozenges are asymptotically present.
More precisely, the liquid region is defined as

${\mathsf{L}}({\mathsf{P}}):=\\{({\mathsf{t}},{\mathsf{x}})\in{\mathsf{P}}:0<p_{{\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}}}({\mathsf{t}},{\mathsf{x}}),p_{{\scalebox{0.15}{\,\includegraphics{lozenge_v_up.pdf}}}}({\mathsf{t}},{\mathsf{x}}),p_{{\scalebox{0.15}{\,\includegraphics{lozenge_v_down.pdf}}}}({\mathsf{t}},{\mathsf{x}})<1\\},$
(13)

and the rest of the domain is called the frozen region. In particular, in
Figure 3 we clearly see frozen regions near the boundaries of the hexagon: in
each such region only one type of lozenges is present. Our first result is an
explicit rational parameterization of the _arctic curve_ separating frozen and
liquid regions.

Figure 4: Arctic curves via Theorem 1.3. Left panel:
${\mathsf{q}}=\tfrac{1}{10}$, $\kappa=\frac{\mathbf{i}}{10}$. Right panel:
${\mathsf{q}}=20$, $\kappa=\frac{\mathbf{i}}{10}$.

######  Theorem 1.3.

Consider the limit regime (10) and assume that we are in the real or imaginary
case, so that ${\mathsf{q}}$ is real and $\kappa$ is either real or purely
imaginary. For any $u\in{\mathbb{C}}$, let

$f_{0}(u)=\frac{({\mathsf{q}}^{{\mathsf{N}}}-{\mathsf{q}}^{u})(\kappa^{2}{\mathsf{q}}^{-{\mathsf{T}}}-{\mathsf{q}}^{-u})}{(\kappa^{2}{\mathsf{q}}^{{\mathsf{N}}}-{\mathsf{q}}^{-u})({\mathsf{q}}^{-{\mathsf{T}}}-{\mathsf{q}}^{u})}\prod_{i=1}^{r}\frac{({\mathsf{q}}^{{\mathsf{a}}_{i}}-{\mathsf{q}}^{u})(\kappa^{2}{\mathsf{q}}^{{\mathsf{b}}_{i}}-{\mathsf{q}}^{-u})}{(\kappa^{2}{\mathsf{q}}^{{\mathsf{a}}_{i}}-{\mathsf{q}}^{-u})({\mathsf{q}}^{{\mathsf{b}}_{i}}-{\mathsf{q}}^{u})}.$
(14)

and

$\displaystyle
U(u)=\frac{f_{0}(u){\mathsf{q}}^{-u}-\kappa^{2}{\mathsf{q}}^{u}}{1-f_{0}(u)},\qquad
V(u)=\frac{{\mathsf{q}}^{-u}-f_{0}(u)\kappa^{2}{\mathsf{q}}^{u}}{1-f_{0}(u)}.$
(15)

The arctic curve separating frozen and liquid regions can be parameterized as
$({\mathsf{t}}(u),{\mathsf{x}}(u))$ found from:

${\mathsf{q}}^{\mathsf{t}}=\frac{V^{\prime}(u)}{U^{\prime}(u)},\qquad{\mathsf{q}}^{\mathsf{x}}=\frac{{\mathsf{w}}\pm\sqrt{{\mathsf{w}}^{2}-4\kappa^{2}{\mathsf{q}}^{-{\mathsf{t}}}}}{2\kappa^{2}{\mathsf{q}}^{-{\mathsf{t}}}},\quad\text{
where }\quad{\mathsf{w}}=V(u)-\frac{U(u)}{{\mathsf{q}}^{\mathsf{t}}},$ (16)

and the parameter $u$ is chosen in such a way that
${\mathsf{q}}^{-u}+\kappa^{2}{\mathsf{q}}^{u}$ belongs to the real line
${\mathbb{R}}$.

######  Remark 1.4.

The choice of the sign in (16) arises from the solution of the quadratic
equation in ${\mathsf{q}}^{\mathsf{x}}$:

$\kappa^{2}{\mathsf{q}}^{-{\mathsf{t}}}{\mathsf{q}}^{\mathsf{x}}+{\mathsf{q}}^{-{\mathsf{x}}}={\mathsf{w}}.$

When $\kappa$ is purely imaginary, we should choose $+$ in (16) in order to
guarantee the positivity of ${\mathsf{q}}^{\mathsf{x}}$. When $\kappa$ is
real, the two choices lead to two curves, which are images of each other under
the map
$({\mathsf{x}},{\mathsf{t}})\mapsto(-{\mathsf{x}}+{\mathsf{t}}-2\log_{\mathsf{q}}\kappa,{\mathsf{t}})$.
One of them is the desired arctic curve, while the second one is outside the
polygon ${\mathsf{P}}$.

######  Remark 1.5.

We are not going to detail the trigonometric case here and in the following
theorems. Our approach still applies and, algebraically, all the formulas
should remain the same; however, various small features (e.g., the choice of
the curve on the complex plane where the parameter $u$ belongs in Theorem 1.3)
need to be changed.

Examples of the arctic curves drawn using Theorem 1.3 are shown in Figure 4.

Inside the liquid region, following the general philosophy of [76], [57,
Lectures 9,10] we encode the triplet of local densities by the complex slope
$f$, which is a complex number with positive imaginary part and such that the
triangle $(0,1,f)$ has angles $(\pi
p_{{\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}}},\pi
p_{{\scalebox{0.15}{\,\includegraphics{lozenge_v_up.pdf}}}},\pi
p_{{\scalebox{0.15}{\,\includegraphics{lozenge_v_down.pdf}}}})$, as in Figure
5:

$\displaystyle\arg f=\pi
p_{{\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}}},\quad\arg(f-1)=\pi(p_{{\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}}}+p_{{\scalebox{0.15}{\,\includegraphics{lozenge_v_down.pdf}}}}).$
(17)

Figure 5: Triangle $(0,1,f)$ has angles $(\pi
p_{{\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}}},\pi
p_{{\scalebox{0.15}{\,\includegraphics{lozenge_v_up.pdf}}}},\pi
p_{{\scalebox{0.15}{\,\includegraphics{lozenge_v_down.pdf}}}})$.

In the next theorem, we state the convergence of the height function, and
provide an explicit characterization of the complex slope for the
$(q,\kappa)$-distributions on lozenge tilings of trapezoid domains.

######  Theorem 1.6.

Consider the limit regime (10) and assume that we are in the real or imaginary
case. Then the height function $h(x,t)$ of random tilings converges in
probability,

$\displaystyle\sup_{({\mathsf{t}},{\mathsf{x}})\in{{\mathsf{P}}}}|\varepsilon
h(\varepsilon^{-1}{\mathsf{t}},\varepsilon^{-1}{\mathsf{x}})-{\mathsf{h}}({\mathsf{t}},{\mathsf{x}})|\to
0.$ (18)

Further, using $U(u)$ and $V(u)$ from (15), consider an equation on an unknown
function $u=u({\mathsf{t}},{\mathsf{x}})$

$\displaystyle{\mathsf{q}}^{-{\mathsf{x}}}+\kappa^{2}{\mathsf{q}}^{{\mathsf{x}}-{\mathsf{t}}}=V(u)-\frac{U(u)}{{\mathsf{q}}^{\mathsf{t}}}.$
(19)

Then $({\mathsf{t}},{\mathsf{x}})\in{\mathsf{P}}$ belongs to the liquid region
${\mathsf{L}}({\mathsf{P}})$ defined by (12) and (13) if and only if (19) has
a solution such that ${\mathsf{q}}^{-u}+\kappa^{2}{\mathsf{q}}^{u}$ is non-
real. In the latter case the complex slope (17) at
$({\mathsf{t}},{\mathsf{x}})$ is given by

$\displaystyle
f_{\mathsf{t}}({\mathsf{x}})=\frac{({\mathsf{q}}^{-u}+\kappa^{2}{\mathsf{q}}^{u})-({\mathsf{q}}^{-{\mathsf{x}}}+\kappa^{2}{\mathsf{q}}^{{\mathsf{x}}})}{({\mathsf{q}}^{-u}+\kappa^{2}{\mathsf{q}}^{u})-({\mathsf{q}}^{-{\mathsf{x}}+{\mathsf{t}}}+\kappa^{2}{\mathsf{q}}^{{\mathsf{x}}-{\mathsf{t}}})}.$
(20)

######  Remark 1.7.

For any $({\mathsf{t}},{\mathsf{x}})\in{\mathsf{L}}({\mathsf{P}})$ we show in
Propositions 5.18 and 5.19 that (19) has four solutions with non-real
${\mathsf{q}}^{u}$: $u({\mathsf{t}},{\mathsf{x}})$;
$-u({\mathsf{t}},{\mathsf{x}})-2\log_{\mathsf{q}}\kappa$; and their complex
conjugates. When we plug them into (20), we get two values for
$f_{\mathsf{t}}({\mathsf{x}})$, which are complex conjugate of each other.
This uniquely determines $f_{\mathsf{t}}({\mathsf{x}})$, since by our
definitions the complex slope $f$ is in the upper-half plane.

######  Remark 1.8.

As explained in [19, Section 2.4], the variational problem for the limit shape
implies that the complex slope $f$ satisfies a version of the complex Burgers
equation inside the liquid region:

$\frac{\partial_{\mathsf{t}}f}{f}-\frac{\partial_{\mathsf{x}}f}{1-f}=\ln({\mathsf{q}})\frac{\kappa^{2}{\mathsf{q}}^{{\mathsf{x}}-{\mathsf{t}}}+{\mathsf{q}}^{-{\mathsf{x}}}}{\kappa^{2}{\mathsf{q}}^{{\mathsf{x}}-{\mathsf{t}}}-{\mathsf{q}}^{-{\mathsf{x}}}},\qquad({\mathsf{t}},{\mathsf{x}})\in{\mathsf{L}}({\mathsf{P}}).$
(21)

On the other hand, (20), (19), and (15) imply666Expressing $f_{0}(u)$ through
$U(u)$ and $V(u)$ via (15) and equating the results, one gets a linear
relation between $U(u)$ and $V(u)$. Combining with (19), one gets a system of
two linear equations on $U(u)$ and $V(u)$, whose solution gives (22).

$U(u)={\mathsf{q}}^{\mathsf{t}}\frac{f{\mathsf{q}}^{-{\mathsf{x}}}-\kappa^{2}{\mathsf{q}}^{{\mathsf{x}}-{\mathsf{t}}}}{1-f},\qquad\qquad
V(u)=\frac{{\mathsf{q}}^{-{\mathsf{x}}}-f\kappa^{2}{\mathsf{q}}^{{\mathsf{x}}-{\mathsf{t}}}}{1-f}.$
(22)

The right-hand sides of (22) are the first integrals of the Burgers equation
(21). Definition of $U(u)$ and $V(u)$ of (14) (15) then implies that there is
an algebraic relation between these two first integrals. This generalizes the
algebraicity observation of [76] in the $\kappa=0$ case.

The existence of the non-random limit in (18) can be alternatively deduced
from the variational principle for random tilings of [37], see discussion in
[19, Section 2.4]. The novel part of Theorem 1.6 is the explicit description
of the limit ${\mathsf{h}}({\mathsf{t}},{\mathsf{x}})$ through (20). In the
particular case when $r=1$ and the tiled domain is a hexagon, Theorems 1.3 and
1.6 coincide with [19, Theorem 2.1], but for more general polygons with $r>1$
the formulas are new. If we set $q=1$, then these results degenerate into the
formulas of [96] for uniformly random tilings of trapezoids $P$.

Our next theorem describes the fluctuations of the height function
$h(\varepsilon^{-1}{\mathsf{t}},\varepsilon^{-1}{\mathsf{x}})$ around its
expectation. Let
$\mathbb{H}^{+}=\\{z\in\mathbb{C}\mid\mathop{\mathrm{Im}}(z)>0\\}$ be the
upper half-plane. The Gaussian free field on $\mathbb{H}^{+}$ with zero
boundary conditions is a random generalized Gaussian field $\mathbf{GFF}(z)$,
$z\in\mathbb{H}^{+}$, which has mean $0$ and covariance:

$\mathbb{E}\mathbf{GFF}(z)\mathbf{GFF}(w)=K(z,w)=-\frac{1}{2\pi}\ln\left|\frac{z-w}{z-\bar{w}}\right|.$
(23)

Due to the singularity of (23) at $z=w$, the values of $\mathbf{GFF}(z)$ at
points are not-well defined, but we can make sense of the integrals of
$\mathbf{GFF}(z)$ against (sufficiently smooth) test–measures, see [99],
[105], [9], [57, Lecture 11] for more details.

We also need a bijection between $\mathbb{H}^{+}$ and the liquid region
${\mathsf{L}}({\mathsf{P}})$.

######  Lemma 1.9.

For each $({\mathsf{t}},{\mathsf{x}})\in{\mathsf{L}}({\mathsf{P}})$, define
$\bar{\Omega}({\mathsf{t}},{\mathsf{x}})={\mathsf{q}}^{-u}+\kappa^{2}{\mathsf{q}}^{u}$,
where $u$ is a non-real solution to (19), chosen so that
$\mathop{\mathrm{Im}}(\bar{\Omega}({\mathsf{t}},{\mathsf{x}}))>0$. Then
$\bar{\Omega}$ is a bijection between ${\mathsf{L}}({\mathsf{P}})$ and
$\mathbb{H}^{+}$.

######  Remark 1.10.

Continuing Remark 1.7, we note that by Propositions 5.18 and 5.19,
$\bar{\Omega}$ is uniquely determined by (19) and
$\mathop{\mathrm{Im}}(\bar{\Omega}({\mathsf{t}},{\mathsf{x}}))>0$. The choice
of $u$ guaranteeing $\mathop{\mathrm{Im}}(f_{\mathsf{t}}({\mathsf{x}}))>0$
might differ by conjugation from the choice of $u$ guaranteeing
${\mathop{\mathrm{Im}}(\bar{\Omega}({\mathsf{t}},{\mathsf{x}}))>0}$.

######  Theorem 1.11.

Consider the limit regime (10) and assume that we are in the real or imaginary
case. Then inside the liquid region,
$({\mathsf{t}},{\mathsf{x}})\in{\mathsf{L}}({\mathsf{P}})$, we have

$\lim_{\varepsilon\to
0}\sqrt{\pi}\bigl{(}h(\varepsilon^{-1}{\mathsf{t}},\varepsilon^{-1}{\mathsf{x}})-\mathbb{E}[h(\varepsilon^{-1}{\mathsf{t}},\varepsilon^{-1}{\mathsf{x}})]\bigr{)}=\bar{\Omega}\text{-pullback
of GFF in }\mathbb{H}^{+},$

in the sense of convergence of the joint moments for pairings with appropriate
test-measures.

We refer to Theorem 6.15 for a more precise statement of Theorem 1.11.

If $q=1$, then $(q,\kappa)$-distributions turn into the uniform distribution
on lozenge tilings of $P$ and Theorem 1.11 degenerates into the results of
[97, 29]. For general $(q,\kappa)$ and $r=1$, i.e. for the hexagons, Theorem
1.11 was conjectured in [39, Conjecture 8.4.1]. The same article also contains
a partial result establishing Gaussian fluctuations along a single vertical
section by using discrete loop equations for log-gases; see also [26] for
another approach. For $\kappa=0$ and $q\neq 1$, i.e. for the
$q^{\text{volume}}$–weighted lozenge tilings Theorem 1.11 verifies for the
trapezoidal domains $P$ a general conjecture of Kenyon and Okounkov, cf. [76]
and [57, Lectures 11-12]. The only previous Gaussian free field type result
for $q^{\text{volume}}$–weighted tilings is [2] dealing with plane partitions,
which can be obtained from tilings of the hexagons by sending side lengths to
infinity while keeping $q$ fixed. For general values of the $(q,\kappa)$
parameters, we are led to a conjecture:

######  Conjecture 1.12.

The macroscopic fluctuations of the height function in the liquid region for
$(q,\kappa)$–distributed random lozenge tilings of arbitrary domains are
asymptotically described by the Gaussian Free Field in the complex structure
given by either of the first integrals (22) of the complex Burgers equation
(21).

We refer to [57, Lectures 11-12] for the detailed explanations and heuristics
underlying Conjecture 1.12 in the special case of the uniform measure, $q=1$.
The formulas (22) imply that the first integrals are holomorphic functions of
$\Omega({\mathsf{t}},{\mathsf{x}})$ in Theorem 1.11. Hence, these first
integrals define the same complex structure in the liquid region as $\Omega$
(or its complex conjugate $\bar{\Omega}$, depending on the choice of $u$;
conjugation of the complex structure does not change the covariance of GFF)
and Theorem 1.11 verifies Conjecture 1.12 for trapezoidal domains $P$. What
Conjecture 1.12 says is that our result should extend from trapezoids to more
general domains.

As we will see in Section 3.7, the $(q,\kappa)$-distribution (8) on lozenge
tilings can be viewed as a Markov process of nonintersecting paths, and the
transition probability is given in the form (4). Our analysis of
$(q,\kappa)$–distributions relies on Theorem 4.5, which uses the dynamic loop
equations of Theorem 1.1 to obtain an asymptotic expansion for the one-step
transition probabilities. We remark that checking the assumptions of Theorem
4.5 takes efforts and we devote Section 5.2 to it. Eventually, the
deterministic part of the expansion (7) leads to Theorems 1.3 and 1.6 after
careful analysis and massaging of the formulas in Section 5. The form of the
stochastic part implies that the height fluctuations converge to a Gaussian
field, described by a stochastic evolution equation; we analyze this equation
and prove that it outputs the Gaussian Free Field in Section 6.

In addition to the dynamical loop equations, the second key ingredient of the
proofs is the use of the _complex characteristic flow_ which we introduce. The
flow is related to the characteristic curves of the first order PDE (21), but
has an important difference. Namely, in (21) (and its relatives for uniform
and $q^{\text{volume}}$-weighted tilings in [76]), the variables
$({\mathsf{t}},{\mathsf{x}})$ in the equation are real and belong to the
liquid region ${\mathsf{L}}({\mathsf{P}})$; this complicates the analysis,
because the shape of the liquid region is not known a priory. In our approach,
we observe that there is a way to extend the definition of the complex slope
(17) from real ${\mathsf{x}}$ to complex $z$, see (178), thus arriving at the
doubly complex slope. The advantage of the latter is that it decouples from
the notion of the liquid region, but the connection to the first order PDE
(21) is preserved (with ${\mathsf{x}}$ replaced by complex $z$). We further
investigate all the main observables of the random tilings along the complex
curves $z^{\mathsf{t}}$, ${\mathsf{t}}\geqslant 0$, which arise from the
characteristics of the doubly complex version of (21), see Section 5.4. We
discover that all the formulas are dramatically simplified along these complex
curves, which eventually allows us to arrive at the explicit answers of
Theorems 1.3, 1.6, and 1.11. In random matrix setting, the complex
characteristic flow has been previously used in [72, 8, 71, 69] to analyze the
complex Burgers equation, which describes the evolution of the Stieltjes
transform of the empirical eigenvalue density. Comparing with those analyses,
our complex Burgers equation (21) has extra symmetries (217) and the
characteristic flow has singular behavior, which makes the analysis more
involved.

##  2 Dynamical Loop Equation

In this section we first give the proof of Theorem 1.1. Then we apply the
particle-hole duality to produce a version of the transition probability (4)
for descending transitions and derive the corresponding dynamical loop
equations.

###  2.1 Proof of Theorem 1.1

###### First proof.

The only possible singularities of (6) are simple poles at $z=x_{i}$,
$i=1,\dots,n$. Thus, our plan is to fix any $1\leqslant i\leqslant n$ and
check that the residue of (6) at $z=x_{i}$ vanishes. The first term in (6) has
a residue at $z=x_{i}$ if and only if $e_{i}=0$. The residue is given by

$\displaystyle\sum_{{\bm{e}}:e_{i}=0}\mathbb{P}({\bm{x}}+{\bm{e}}|{\bm{x}})\phi^{+}(x_{i})\frac{b(x_{i}+\theta)-b(x_{i})}{b^{\prime}(x_{i})}\prod_{j:j\neq
i}\frac{b(x_{i}+\theta)-b(x_{j}+\theta e_{j})}{b(x_{i})-b(x_{j})}.$ (24)

The second term has a residue at $z=x_{i}$ if and only if $e_{i}=1$. The
residue is given by

$\displaystyle\sum_{{\bm{e}}:e_{i}=1}\mathbb{P}({\bm{x}}+{\bm{e}}|{\bm{x}})\phi^{-}(x_{i})\frac{b(x_{i})-b(x_{i}+\theta)}{b^{\prime}(x_{i})}\prod_{j:j\neq
i}\frac{b(x_{i})-b(x_{j}+\theta e_{j})}{b(x_{i})-b(x_{j})}.$ (25)

For any pair ${\bm{e}}^{0}=(e_{1},e_{2},\dots,e_{i-1},0,e_{i+1},\dots,e_{n})$
and ${\bm{e}}^{1}=(e_{1},e_{2},\dots,e_{i-1},1,e_{i+1},\dots,e_{n})$, the
ratio of the transition probabilities
$\mathbb{P}({\bm{x}}+{\bm{e}}^{1}|{\bm{x}})$ and
$\mathbb{P}({\bm{x}}+{\bm{e}}^{0}|{\bm{x}})$ is given by

$\displaystyle\frac{\mathbb{P}({\bm{x}}+{\bm{e}}^{1}|{\bm{x}})}{\mathbb{P}({\bm{x}}+{\bm{e}}^{0}|{\bm{x}})}=\frac{\phi^{+}(x_{i})}{\phi^{-}(x_{i})}\prod_{j:j\neq
i}\frac{b(x_{i}+\theta)-b(x_{j}+\theta e_{j})}{b(x_{i})-b(x_{j}+\theta
e_{j})}.$ (26)

Plugging (26) into (24) and (25), we see that the residues at ${\bm{e}}^{0}$
and ${\bm{e}}^{1}$ cancel out:

$\mathbb{P}({\bm{x}}+{\bm{e}}^{0}|{\bm{x}})\phi^{+}(x_{i})\frac{b(x_{i}+\theta)-b(x_{i})}{b^{\prime}(x_{i})}\prod_{j\neq
i}\frac{b(x_{i}+\theta)-b(x_{j}+\theta e_{j})}{b(x_{i})-b(x_{j})}\\\
+\mathbb{P}({\bm{x}}+{\bm{e}}^{1}|{\bm{x}})\phi^{-}(x_{i})\frac{b(x_{i})-b(x_{i}+\theta)}{b^{\prime}(x_{i})}\prod_{j:j\neq
i}\frac{b(x_{i})-b(x_{j}+\theta e_{j})}{b(x_{i})-b(x_{j})}=0.$

Summing over
$(e_{1},e_{2},\dots,e_{i-1},e_{i+1},\dots,e_{n})\in\\{0,1\\}^{n-1}$, we
conclude that the residue of (6) at $z=x_{i}$ vanishes. ∎

###### Second proof.

For any complex numbers $x_{1},\dots,x_{n}\in U$, we denote the partition
function

$Z_{n}(x_{1},\dots,x_{n})=\sum_{{\bm{e}}\in\\{0,1\\}^{n}}\prod_{1\leqslant
i<j\leqslant n}\frac{b(x_{i}+\theta e_{i})-b(x_{j}+\theta
e_{j})}{b(x_{i})-b(x_{j})}\prod_{i=1}^{n}\phi^{+}(x_{i})^{e_{i}}\phi^{-}(x_{i})^{1-e_{i}}.$

We claim that $Z_{n}(x_{1},\dots,x_{n})$ is a holomorphic function on $U^{n}$.
Indeed, the only possible singularities of $Z_{n}$ are on the lines
$x_{i}=x_{j}$. On the other hand, directly from the definition, the function

$F(x_{1},\dots,x_{n}):=Z_{n}(x_{1},\dots,x_{n})\prod_{1\leqslant i<j\leqslant
n}(b(x_{i})-b(x_{j}))$

is skew-symmetric in $x_{1},\dots,x_{n}$; hence, it vanishes on the lines
$x_{i}-x_{j}$ and

$\frac{F(x_{1},\dots,x_{n})}{\prod_{i<j}(x_{i}-x_{j})}$

has no singularities. Therefore,

$Z_{n}(x_{1},\dots,x_{n})=\frac{F(x_{1},\dots,x_{n})}{\prod_{i<j}(x_{i}-x_{j})}\cdot\prod_{1\leqslant
i<j\leqslant n}\frac{x_{i}-x_{j}}{b(x_{i})-b(x_{j})}$

also has no singularities.

The expectation (6) can be identified with a ratio of two partition functions
$\frac{Z_{n+1}(x_{1},\dots,x_{n},z)}{Z_{n}(x_{1},\dots,x_{n})}$ and its
holomorphicity in $z$ follows from the holomorphicity of the partition
functions. ∎

######  Remark 2.1.

One can introduce more advanced weights, which would lead to holomorphic
partition functions: in Theorem 1.1 we are allowing the increments $e_{i}$ of
$x_{i}$ to be in $\\{0,1\\}$, but more generally one can allow $m$ complex
increments: $e_{i}\in\\{v_{1},\dots,v_{m}\\}$. Given $m$ holomorphic functions
$\phi_{1},\dots,\phi_{m}$, this would lead to the factor $\phi_{i}(x_{i})$ in
$\prod_{i=1}^{n}$ part of the distribution generalizing (4) whenever
$e_{i}=v_{i}$ and to $m^{n}$ possible transitions. The argument for the
holomorphicity of the partition function and of the corresponding observable
$\frac{Z_{n+1}(x_{1},\dots,x_{n},z)}{Z_{n}(x_{1},\dots,x_{n})}$ remains the
same, leading to an extension of Theorem 1.1 with (6) being replaced by
expectation of the sum of $m$ (rather than two) terms.

###  2.2 Particle-Hole Duality

In this section, we study descending transitions. Take two signatures
${\bm{\lambda}}\in\mathbb{GT}_{n+1}$ and ${\bm{\mu}}\in\mathbb{GT}_{n}$, with
interlacing condition ${\bm{\mu}}\prec{\bm{\lambda}}$, as in (1). Similarly to
(3), we encode them by particle systems

$\displaystyle x_{i}=\lambda_{i}-(i-1)\theta,\quad 1\leqslant i\leqslant
n+1,\qquad y_{i}=\mu_{i}-(i-1)\theta,\quad 1\leqslant i\leqslant n.$

Then ${\bm{x}}=(x_{1},x_{2},\dots,x_{n+1})\in\mathbb{W}_{\theta}^{(n+1)}$ and
${\bm{y}}=(y_{1},y_{2},\dots,y_{n})\in\mathbb{W}_{\theta}^{(n)}$, and the
interlacing condition ${\bm{\mu}}\prec{\bm{\lambda}}$ is equivalent to
$y_{i}\in\\{x_{i+1}+\theta,x_{i+1}+1+\theta,x_{i+1}+2+\theta,\dots,x_{i}-1,x_{i}\\}$
for $1\leqslant i\leqslant n$.

Given ${\bm{x}}=(x_{1},x_{2},\dots,x_{n+1})\in\mathbb{W}_{\theta}^{(n+1)}$, we
denote the lattice

$\displaystyle\mathbb{L}({\bm{x}})=\bigcup_{i=1}^{n}\\{x_{i+1}+\theta,x_{i+1}+1+\theta,x_{i+1}+2+\theta,\dots,x_{i}-1,x_{i}\\}.$

We fix two functions $b(z)$, $w(z)$ and consider the following transition
probability:

$\displaystyle\mathbb{P}({\bm{y}}|{\bm{x}})=\frac{1}{Z({\bm{x}})}\prod_{1\leqslant
i<j\leqslant
n}\bigl{(}b(y_{i})-b(y_{j})\bigr{)}\,\prod_{i=1}^{n}\left[w(y_{i})\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell>y_{i}\end{smallmatrix}}\frac{1}{b(\ell)-b(y_{i})}\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell<y_{i}\end{smallmatrix}}\frac{1}{b(y_{i})-b(\ell)}\right],$ (27)

where ${\bm{y}}=(y_{1},y_{2},\dots,y_{n})\in\mathbb{W}_{\theta}^{n}$ with
$y_{i}\in\\{x_{i+1}+\theta,x_{i+1}+1+\theta,x_{i+1}+2+\theta,\dots,x_{i}-1,x_{i}\\}$
for $1\leqslant i\leqslant n$.

We remark that the transition probability (27) is obtained from the ascending
transition (4) by the particle-hole duality or, equivalently, by transposition
of the involved Young diagrams, i.e. by mirroring Figure 1 with respect to the
vertical axis. More precisely, starting from (27) we do the following three
steps:

  * •

We replace the particle configurations ${\bm{x}}$ and ${\bm{y}}$, by the
holes, i.e. by the complementary configurations
$\mathbb{L}({\bm{x}})\setminus\\{x_{1},x_{2},\dots,x_{n}\\}$ and
$\mathbb{L}({\bm{x}})\setminus\\{y_{1},y_{2},\dots,y_{n}\\}$;

  * •

We divide all coordinates by $\theta$;

  * •

We replace $\theta$ by $\theta^{-1}$.

Applying these operations to (27) we get (4), as can be seen through the
identities (where we denote the configurations of holes as
${\bm{x}}^{\prime}=\mathbb{L}({\bm{x}})\setminus{\bm{x}}$ and
${\bm{y}}^{\prime}=\mathbb{L}({\bm{x}})\setminus{\bm{y}}$):

$\displaystyle\begin{split}\prod_{i<j}\bigl{(}b(x^{\prime}_{i})-b(x^{\prime}_{j})\bigr{)}&=\frac{\prod_{i<j}\bigl{(}b(x_{i})-b(x_{j})\bigr{)}\prod_{\ell>\ell^{\prime}\in\mathbb{L}({\bm{x}})}\bigl{(}b(\ell)-b(\ell^{\prime})\bigr{)}}{\prod_{i}(\prod_{\mathbb{L}({\bm{x}})\ni\ell>x_{i}}\bigl{(}b(\ell)-b(x_{i})\bigr{)}\prod_{\mathbb{L}({\bm{x}})\ni\ell<x_{i}}\bigl{(}b(x_{i})-b(\ell)\bigr{)}}\\\
\prod_{i<j}\bigl{(}b(y^{\prime}_{i})-b(y^{\prime}_{j})\bigr{)}&=\frac{\prod_{i<j}\bigl{(}b(y_{i})-b(y_{j})\bigr{)}\prod_{\ell>\ell^{\prime}\in\mathbb{L}({\bm{x}})}\bigl{(}b(\ell)-b(\ell^{\prime})\bigr{)}}{\prod_{i}(\prod_{\mathbb{L}({\bm{x}})\ni\ell>y_{i}}\bigl{(}b(\ell)-b(y_{i})\bigr{)}\prod_{\mathbb{L}({\bm{x}})({\bm{x}})\ni\ell<x_{i}}\bigl{(}b(y_{i})-b(\ell)\bigr{)}}.\end{split}$
(28)

Because of this connection, Theorem 1.1 can be recast for the ascending
process. We record the result in the following theorem, supplied with a self-
contained proof.

######  Theorem 2.2.

Choose an open set $U\subset\mathbb{C}$, a parameter $\theta>0$, a particle
configuration
${{\bm{x}}=(x_{1},x_{2},\dots,x_{n+1})\in\mathbb{W}_{\theta}^{n+1}}$ such that
$[x_{n+1},x_{1}]\subset U$, two holomorphic functions $\phi^{+}(z)$,
$\phi^{-}(z)$ on $U$ and a conformal (i.e., holomorphic and injective)
function $b(z)$ on $U$. Assume that the random $n$–tuple ${\bm{y}}$ is
distributed according to the transition probability (27) with weight function
$w(z)$ satisfying

$\displaystyle\frac{w(z+1)}{w(z)}=\frac{\phi^{+}(z)}{\phi^{-}(z)}.$ (29)

Then the following observable is a holomorphic function of $z\in U$:

$\displaystyle\mathbb{E}\left[\phi^{+}(z)\dfrac{\prod\limits_{i=1}^{n+1}\bigl{(}b(z)-b(x_{i})\bigr{)}}{\prod\limits_{i=1}^{n}\bigl{(}b(z)-b(y_{i})\bigr{)}}+\phi^{-}(z)\prod\limits_{\ell\in\mathbb{L}({\bm{x}})}\dfrac{\bigl{(}b(z+1)-b(\ell)\bigr{)}}{\bigl{(}b(z)-b(\ell)\bigr{)}}\cdot\frac{\prod\limits_{i=1}^{n+1}\bigl{(}b(z)-b(x_{i})\bigr{)}}{\prod\limits_{i=1}^{n}\bigl{(}b(z+1)-b(y_{i})\bigr{)}}\right].$
(30)

###### Proof.

The possible singularities of the first term in (30) are simple poles at the
points
$z=m\in\\{x_{i+1}+\theta,x_{i+1}+1+\theta,x_{i+1}+2+\theta,\dots,x_{i}-1\\}$;
they arise whenever $m=y_{i}$ for any $1\leqslant i\leqslant n$. Note that we
excluded $m=x_{i}$: there is no pole at such point due to the factor
$b(z)-b(x_{i})$ in the numerator.

The possible singularities of the second term in (30) are also simple poles at
the points
$z=m\in\\{x_{i+1}+\theta,x_{i+1}+1+\theta,x_{i+1}+2+\theta,\dots,x_{i}-1\\}$;
they arise whenever $m=y_{i}-1$ for any $1\leqslant i\leqslant n$. Note that
we excluded $m=x_{i+1}+\theta-1$: there is no pole at such point due to the
factor $b(z+1)-b(\ell)$, $\ell=x_{i+1}+\theta$, in the numerator; also note
that the denominators in the first product in the second term in (30) do not
produce additional poles, since for each zero of $b(z)-b(\ell)$ in the
denominator, we have a matching zero in the numerator coming either from
$b(z+1)-b(\ell+1)$ or from $b(z)-b(x_{i})$.

In the rest of the proof we fix $i=1,\dots,n$ and check that the residues of
two terms in (30) at a point
$z=m\in\\{x_{i+1}+\theta,x_{i+1}+1+\theta,x_{i+1}+2+\theta,\dots,x_{i}-1\\}$
differ by a sign and, hence, cancel out. For the first term, the residue is
given by

$\displaystyle\sum_{{\bm{y}}:y_{i}=m}\mathbb{P}({\bm{y}}|{\bm{x}})\cdot\phi^{+}(m)\cdot\frac{b(m)-b(x_{i})}{b^{\prime}(m)}\cdot\frac{\prod\limits_{j\neq
i}\bigl{(}b(m)-b(x_{j})\bigr{)}}{\prod\limits_{j\neq
i}\bigl{(}b(m)-b(y_{j})\bigr{)}},$ (31)

while for the second term the residue is given by

$\displaystyle\sum_{{\bm{y}}:y_{i}=m+1}\mathbb{P}({\bm{y}}|{\bm{x}})\phi^{-}(m)\frac{b(x_{i})-b(m)}{b^{\prime}(m)}\prod_{\ell\in\mathbb{L}({\bm{x}})\setminus\\{m,m+1\\}}\frac{b(m+1)-b(\ell)}{b(m)-b(\ell)}\frac{\prod\limits_{j\neq
i}\bigl{(}b(m)-b(x_{j})\bigr{)}}{\prod\limits_{j\neq
i}\bigl{(}b(m+1)-b(y_{j})\bigr{)}}.$ (32)

For any pair of particle configurations
${\bm{y}}^{1}=(y_{1},y_{2},\dots,y_{i-1},m+1,y_{i+1},\dots,y_{n})$ and
${\bm{y}}^{0}=(y_{1},y_{2},\dots,y_{i-1},m,y_{i+1},\dots,y_{n})$, the ratio of
the transition probabilities $\mathbb{P}({\bm{y}}={\bm{y}}^{1}|{\bm{x}})$ and
$\mathbb{P}({\bm{y}}={\bm{y}}^{0}|{\bm{x}})$ is given by

$\displaystyle\frac{\mathbb{P}({\bm{y}}={\bm{y}}^{0}|{\bm{x}})}{\mathbb{P}({\bm{y}}={\bm{y}}^{1}|{\bm{x}})}=\frac{w(m)}{w(m+1)}\prod_{\ell\in\mathbb{L}({\bm{x}})\setminus\\{m,m+1\\}}\frac{b(m+1)-b(\ell)}{b(m)-b(\ell)}\prod_{j\neq
i}\frac{b(m)-b(y_{j})}{b(m+1)-b(y_{j})}.$ (33)

We recall from (29) that $w(m)/w(m+1)=\phi^{-}(m)/\phi^{+}(m)$. Plugging (33)
into the expressions of (32) and (31), we see that the residuals at
${\bm{y}}^{0}$ and ${\bm{y}}^{1}$ cancel out:

$0=\mathbb{P}({\bm{y}}={\bm{y}}^{0}|{\bm{x}})\phi^{+}(m)\frac{b(m)-b(x_{i})}{b^{\prime}(m)}\frac{\prod\limits_{j\neq
i}\bigl{(}b(m)-b(x_{j})\bigr{)}}{\prod\limits_{j\neq
i}\bigl{(}b(m)-b(y_{j})\bigr{)}}\\\
+\mathbb{P}({\bm{y}}={\bm{y}}^{1}|{\bm{x}})\phi^{-}(m)\frac{b(x_{i})-b(m)}{b^{\prime}(m)}\prod_{\ell\in\mathbb{L}({\bm{x}})\setminus\\{m,m+1\\}}\frac{b(m+1)-b(\ell)}{b(m)-b(\ell)}\frac{\prod\limits_{j\neq
i}\bigl{(}b(m)-b(x_{j})\bigr{)}}{\prod\limits_{j\neq
i}\bigl{(}b(m+1)-b(y_{j})\bigr{)}}.$

Summing the last identity over
$(y_{1},y_{2},\dots,y_{i-1},y_{i+1},\dots,y_{n})$ with
$y_{j}\in\\{x_{j+1}+\theta,x_{j+1}+1+\theta,\dots,x_{j}\\}$, we conclude that
the sum of (32) and (31) vanishes and (30) has no residue and, hence, no
singularity at $z=m$. ∎

##  3 Examples of Dynamical Loop Equations

In this section, we show that many well-known stochastic dynamics, including
nonintersecting Bernoulli/Poisson random walks [59, 78, 67], $\beta$–corners
processes [60, 16], Dyson Brownian motion [48], measures on Gelfand-Tsetlin
patterns [60, 17, 97], Macdonald processes [14], and
$(q,\kappa)$-distributions on lozenge tilings [19] can be transformed to the
forms of (4), (27), and their degenerations. As a corollary, we derive the
dynamical loop equations for all these systems.

###  3.1 Nonintersecting Bernoulli random walks

For $n\geqslant 1$, we consider $n$ independent Bernoulli random walks on
$\mathbb{Z}$, with each walker jumping by $1$ in the positive direction with
probability $p\in(0,1)$, or staying with the complementary probability $1-p$
at each time step, and condition them to never intersect. If we denote the
configuration of walkers at time $t$ as
${\bm{x}}(t)=(x_{1}(t)>x_{2}(t)>\dots>x_{n}(t))\in\mathbb{Z}^{n}$, then for
any ${\bm{e}}=\\{0,1\\}^{n}$, the transition probability
$\mathbb{P}({\bm{x}}(t+1)={\bm{x}}+{\bm{e}}|{\bm{x}}(t)={\bm{x}})$ is given by

$\displaystyle\mathbb{P}({\bm{x}}(t+1)={\bm{x}}+{\bm{e}}|{\bm{x}}(t)={\bm{x}})=\frac{V({\bm{x}}+{\bm{e}})}{V({\bm{x}})}\prod_{i=1}^{n}p^{e_{i}}(1-p)^{1-e_{i}},$
(34)

where $V({\bm{x}})$ is the Vandermonde determinant

$\displaystyle V({\bm{x}})=\prod_{1\leqslant i<j\leqslant n}(x_{j}-x_{i}).$
(35)

The computation leading to (34) can be found in [78]. Recently, local
statistics of nonintersecting Bernoulli random walks were studied in [59].
There are two natural generalizations of (34). First, we can consider a
$\theta$ version and, second, we can allow the jump probability $p$ to depend
on $x_{i}$. Hence, for general $\theta>0$, we fix $\theta=\beta/2$ and define
the _$\theta$ -nonintersecting Bernoulli random walk_ as a discrete time
Markov process
${\bm{x}}(t)=(x_{1}(t),x_{2}(t),\dots,x_{n}(t))\in\mathbb{W}^{n}_{\theta}$, as
in (2), with the transition probability
$\mathbb{P}({\bm{x}}(t+1)={\bm{x}}+{\bm{e}}|{\bm{x}}(t)={\bm{x}})$ given by

$\displaystyle\mathbb{P}({\bm{x}}(t+1)={\bm{x}}+{\bm{e}}|{\bm{x}}(t)={\bm{x}})=\frac{1}{Z({\bm{x}})}\frac{V({\bm{x}}+\theta{\bm{e}})}{V({\bm{x}})}\prod_{i=1}^{n}\phi^{+}(x_{i})^{e_{i}}\phi^{-}(x_{i})^{1-e_{i}}.$
(36)

######  Corollary 3.1.

Fix any particle configuration
${\bm{x}}=(x_{1}>x_{2}>\dots>x_{n})\in\mathbb{W}_{\theta}^{n}$ and consider
transition probability (36), with weight functions $\phi^{+}(z),\phi^{-}(z)$
analytic in a neighborhood of $[x_{n},x_{1}]$. Then the following observable
is analytic in a neighborhood of $[x_{n},x_{1}]$

$\displaystyle\mathbb{E}\left[\phi^{+}(z)\prod_{j=1}^{n}\frac{z-x_{j}+\theta(1-e_{j})}{z-x_{j}}+\phi^{-}(z)\prod_{j=1}^{n}\frac{z-x_{j}-\theta
e_{j}}{z-x_{j}}\right].$

###### Proof.

We take $b(z)=z$ in Theorem 1.1. ∎

###  3.2 Nonintersecting Poisson random walks

Taking the limit as $p\rightarrow 0$ and scaling to the continuous time as
$\tau=\lfloor{p^{-1}t}\rfloor$, for $t>0$, turns the nonintersecting Bernoulli
random walk (34) into the nonintersecting Poisson random walk. This is a
continuous time dynamics obtained by conditioning $n$ independent Poisson
random walks to never intersect. This Markov chain
${\bm{x}}(t)=(x_{1}(t)>x_{2}(t)>\dots>x_{n}(t))\in\mathbb{Z}^{n}$ has jump
rates

$\displaystyle\mathbb{P}({\bm{x}}(t+{\rm
d}t)={\bm{y}}|{\bm{x}}(t)={\bm{x}})=\begin{cases}\frac{V({\bm{x}}+{\bm{e}}_{j})}{V({\bm{x}})}{\rm
d}t+\operatorname{O}({\rm d}t^{2}),&{\bm{y}}={\bm{x}}+{\bm{e}}_{j},\\\ 1-n{\rm
d}t+\operatorname{O}({\rm d}t^{2}),&{\bm{y}}={\bm{x}},\\\
\operatorname{O}({\rm d}t^{2}),&\text{otherwise}.\end{cases}$ (37)

In other words, the nonintersecting Poisson random walk is a continuous time
Markov process, with generator

$\displaystyle{\mathcal{L}}^{n}f({\bm{x}})=\sum_{j=1}^{n}\frac{V({\bm{x}}+{\bm{e}}_{j})}{V({\bm{x}})}(f({\bm{x}}+{\bm{e}}_{j})-f({\bm{x}})),\quad
1\leqslant j\leqslant n.$ (38)

This nonintersecting process appeared in various settings, cf. [78, 92, 15,
61]. More generally, we can also consider a general $\theta$ version (as in
[60, 67]) and add spatial inhomogeneities in $x$. Thus, we fix $\theta>0$ and
define the _$\theta$ -nonintersecting Poisson random walk_, denoted by
${\bm{x}}(t)=(x_{1}(t),x_{2}(t),\dots,x_{n}(t))\in\mathbb{W}^{n}_{\theta}$, as
a discrete time Markov process with generator given by

$\displaystyle{\mathcal{L}}_{\theta}^{n}f({\bm{x}})=\sum_{j=1}^{n}\frac{V({\bm{x}}+\theta{\bm{e}}_{j})}{V({\bm{x}})}\phi(x_{j})\bigl{(}f({\bm{x}}+{\bm{e}}_{j})-f({\bm{x}})\bigr{)}.$
(39)

The following statement can be obtained as a limit of Corollary 3.1; we also
provide a direct proof.

######  Corollary 3.2.

Take a particle configuration
${\bm{x}}=(x_{1}>x_{2}>\dots>x_{n})\in\mathbb{W}_{\theta}^{n}$ and consider
the generator (39), with weight function $\phi(z)$ analytic in a complex
neighborhood of $[x_{n},x_{1}+1]$. Let $G({\bm{x}})$ be the Stieltjes
transform of the particle configuration ${\bm{x}}$:

$G({\bm{x}})=\sum_{i=1}^{n}\frac{1}{z-x_{i}}.$

Then the following observable is analytic in the same neighborhood of
$[x_{n},x_{1}+1]$

$\phantom{{}={}}\theta{\mathcal{L}}_{\theta}^{n}G({\bm{x}})+\phi(z)\left(\prod_{j=1}^{n}\frac{z-x_{j}+\theta}{z-x_{j}}-\prod_{j=1}^{n}\frac{z-1-x_{j}+\theta}{z-1-x_{j}}\right).$
(40)

###### Proof.

Let us show that the following expression in complex variable $u$ has no
singularities:

$\displaystyle\sum_{j=1}^{n}\left[\frac{V({\bm{x}}+\theta{\bm{e}}_{j})}{V({\bm{x}})}\phi(x_{j})\frac{u-x_{j}-\theta}{u-x_{j}}\right]+\phi(z)\prod_{j=1}^{n}\frac{u-x_{j}+\theta}{u-x_{j}}.$
(41)

Indeed, the sum might have simple poles at $u=x_{i}$, $1\leqslant i\leqslant
n$. However, the residue at such a pole is

$\displaystyle-\theta\frac{V({\bm{x}}+\theta{\bm{e}}_{i})}{V({\bm{x}})}\phi(x_{i})+\theta\phi(x_{i})\prod_{j:j\neq
i}\frac{x_{i}-x_{j}+\theta}{x_{i}-x_{j}}=0.$

Therefore (41) is analytic in a neighborhood of $[x_{n},x_{1}]$. Subtracting
(41) at $u=z$ and $u=z-1$ and noting

$\theta{\mathcal{L}}_{\theta}^{n}G({\bm{x}})=\sum_{j=1}^{n}\frac{V({\bm{x}}+\theta{\bm{e}}_{j})}{V({\bm{x}})}\phi(x_{j})\left(\frac{z-x_{j}-\theta}{z-x_{j}}-\frac{z-1-x_{j}-\theta}{z-1-x_{j}}\right),$

we conclude that (40) has no singularities. ∎

The dynamical loop equation for the $\theta$-nonintersecting Poisson random
walk with trivial weight $\phi(z)=1$ was discovered in [67, Corollary 2.10],
and used to show that the macroscopic fluctuations of this process with
general initial data converge to the Gaussian free field.

###  3.3 Dyson Brownian motion

Consider nonintersecting Poisson random walk (37). Its rescaled version
${\varepsilon^{1/2}({\bm{x}}(\varepsilon^{-1}t)-\varepsilon^{-1}t)}$,
$t\geqslant 0$, converges as $\varepsilon\to 0$ to the Dyson Brownian motion
with $\beta=2$, which is a solution to the following stochastic differential
equations:

$\displaystyle{\rm d}x_{i}(t)={\rm d}B_{i}(t)+\sum_{j:j\neq i}\frac{{\rm
d}t}{x_{i}(t)-x_{j}(t)},\quad 1\leqslant i\leqslant n,$ (42)

with independent standard Brownian motions $B_{i}(t)$. A generalization of
(42) involves replacing $\theta=\frac{\beta}{2}=1$ by an arbitrary positive
real number and adding a potential. The resulting diffusion is a solution to
the SDE

$\displaystyle{\rm d}x_{i}(t)=\sqrt{\frac{2}{\beta}}{\rm
d}B_{i}(t)+\sum_{j:j\neq i}\frac{{\rm
d}t}{x_{i}(t)-x_{j}(t)}+W^{\prime}(x_{i}(t)){\rm d}t,\quad 1\leqslant
i\leqslant n.$ (43)

Similarly to (42), it should be possible to obtain (43) from (39) by a
diffusive scaling limit (for $\beta=2\theta\geqslant 1$ and no potential this
was proven in [60, Theorem 3.2], see also [67, Theorem 1.2]). The solution to
(43) is called the $\beta$-Dyson Brownian Motion with potential $W$; it
originates in the work of Dyson from 1960s on the evolution of eigenvalues of
random matrices, cf. [48]. Recently the Dyson Brownian motion (43) played a
central role in the three-step approach to the proofs of the universal
asymptotic behavior of local statistics of eigenvalues for various ensembles
of random matrices in a series of works [81, 80, 50, 51, 49, 1], developed by
Erdős, Yau and their collaborators.

By taking the diffusive scaling limit from the dynamical loop equations of
Corollaries 3.1 and 3.2, we get the following dynamical loop equation for
$\beta$-Dyson Brownian motion with analytic potential. We also present a
direct proof.

######  Corollary 3.3.

Consider the $\beta$-Dyson Brownian motion (43), suppose that the potential
$W(z)$ analytic in a complex neighborhood of the real axis, and take any
$0\leqslant t_{1}<t_{2}$. Let

$\displaystyle m_{n}(z;t)=\sum_{i=1}^{n}\frac{1}{z-x_{i}(t)}.$

Then the following stochastic integral is an analytic function of $z$ in the
same neighborhood

$\displaystyle\int_{t_{1}}^{t_{2}}$ $\displaystyle\left({\rm
d}m_{n}(z;t)-\sqrt{\frac{2}{\beta}}\sum_{i=1}^{n}\frac{{\rm
d}B_{i}(t)}{(z-x_{i}(t))^{2}}\right.$
$\displaystyle\left.+\left(\partial_{z}\frac{m_{n}^{2}(z;t)}{2}+\partial_{z}[m_{n}(z;t)W^{\prime}(z)]+\frac{\beta-2}{2\beta}\partial^{2}_{z}m_{n}(z;t)\right)\right){\rm
d}t.$

###### Proof.

By Ito’s formula and (43), the integrand simplifies to

$\displaystyle\int_{t_{1}}^{t_{2}}\sum_{i=1}^{n}\frac{W^{\prime}(x_{i})-W^{\prime}(z)-(x_{i}(t)-z)W^{\prime\prime}(z)}{(z-x_{i}(t))^{2}}{\rm
d}t.$ (44)

Analyticity of $W(z)$ implies that the expression

$\frac{W^{\prime}(x)-W^{\prime}(z)-(x-z)W^{\prime\prime}(z)}{(z-x)^{2}}$

is not singular at $z=x$. Hence, (44) is analytic. ∎

Special versions of Corollary 3.3 (for trivial potential $W\equiv 0$) were
used in [72] and [33, 62] (see also [4, Section 4.3]) to prove that the
macroscopic fluctuations of the $\beta$-Dyson Brownian motion converge to a
Gaussian field, which can be further identified with the Gaussian free field,
cf. [13, 67]. We remark that the loop equations in [33, 62, 4] are different
from those in Corollary 3.3. Instead of the Stieltjes transforms, polynomial
test functions are used, and the corresponding loop equations are used to
study the evolution of polynomial functionals of Gaussian large random
matrices. For convex potential, it has been proven in [103] that the
$\beta$-Dyson Brownian motion converge to a Gaussian field.

###  3.4 Measures on Gelfand-Tsetlin Patterns

A Gelfand-Tsetlin pattern of depth $T$ is an interlacing sequences of
signatures

$\displaystyle\emptyset\prec{\bm{\lambda}}(1)\prec{\bm{\lambda}}(2)\prec\dots\prec{\bm{\lambda}}(T-1)\prec{\bm{\lambda}}(T).$
(45)

The patterns were originally introduced in the context of the branching rules
for restrictions of irreducible representations of the unitary group $U(N)$
onto the subgroup $U(N-1)$. The characters of these representations are Schur
symmetric polynomials and, therefore, the branching rule is equivalent to the
following combinatorial formula for the Schur polynomials:

$\displaystyle
s_{\bm{\lambda}}(u_{1},u_{2},\dots,u_{T})=\sum_{\emptyset\prec{\bm{\lambda}}(1)\prec{\bm{\lambda}}(2)\prec\dots\prec{\bm{\lambda}}(T-1)\prec{\bm{\lambda}}(T)={\bm{\lambda}}}u_{1}^{|{\bm{\lambda}}(1)|}u_{2}^{|{\bm{\lambda}}(2)|-|{\bm{\lambda}}(1)|}\dots
u_{T}^{|{\bm{\lambda}}(T)|-|{\bm{\lambda}}(T-1)|},$ (46)

where $|{\bm{\lambda}}(n)|$ is the sum of coordinates of a signature
$|{\bm{\lambda}}(n)|\in\mathbb{GT}_{n}$, $n=1,2,\dots,T$. We would like to
study the uniform measure on all Gelfand-Tsetlin patterns of depth $T$ with
the fixed top row ${\bm{\lambda}}(T)$ and refer to [21] and [57, Lecture 22]
for more details on the interplay between Gelfand-Tsetlin patterns, random
tilings, and symmetric functions. The total number of the Gelfand-Tsetlin
patterns with a fixed top row can be computed by combining (46) with
evaluation formulas for the Schur functions (known as Weyl dimension formulas
in the representation-theoretic literature):

$\displaystyle\sum_{\emptyset\prec{\bm{\lambda}}(1)\prec{\bm{\lambda}}(2)\prec\dots\prec{\bm{\lambda}}(T)={\bm{\lambda}}}1=s_{\bm{\lambda}}(1,1,\dots,1)=\frac{V(\lambda_{1},\lambda_{2}-1,\lambda_{3}-2,\dots,\lambda_{T}-(T-1))}{V(1,2,\dots,T)},$
(47)

where $V$ is the Vandermonde determinant (35). We can view the uniform measure
on all Gelfand-Tsetlin patterns as a Markov process on signatures with
descending transitions from level $T$ down to level $1$. The transition
probability from $(n+1)$-st level to the $n$-level is computed using (47) to
be

$\displaystyle\mathbb{P}({\bm{\lambda}}(n)={\bm{\mu}}|{\bm{\lambda}}(n+1)={\bm{\lambda}})=\frac{s_{\bm{\mu}}(\bm{1})}{s_{\bm{\lambda}}(\bm{1})}=\frac{1}{n!}\frac{V(\mu_{1},\mu_{2}-1,\dots,\mu_{n}-(n-1))}{V(\lambda_{1},\lambda_{2}-1,\dots,\lambda_{n+1}-n)},$
(48)

where
${\bm{\lambda}}=(\lambda_{1},\lambda_{2},\dots,\lambda_{n+1})\in\mathbb{GT}_{n+1}$,
${\bm{\mu}}=(\mu_{1},\mu_{2},\dots,\mu_{n})\in\mathbb{GT}_{n}$ and
${\bm{\mu}}\prec{\bm{\lambda}}$. We encode ${\bm{\lambda}}$ and ${\bm{\mu}}$
as particle systems ${\bm{x}}$ and ${\bm{y}}$, respectively as in Section 2.2
with $\theta=1$: $x_{i}=\lambda_{i}-(i-1)$, $1\leqslant i\leqslant n+1$, and
$y_{i}=\mu_{i}-(i-1)$, $1\leqslant i\leqslant n$. We can rewrite the
transition probability (48) as

$\displaystyle\mathbb{P}({\bm{y}}|{\bm{x}})=\frac{1}{n!}\frac{V({\bm{y}})}{V({\bm{x}})}.$
(49)

This matches (27), with $\theta=1$, $b(z)=z$,
$y_{i}\in\\{x_{i+1}+1,x_{i+1}+2,x_{i+1}+2,\dots,x_{i}\\}$, $1\leqslant
i\leqslant n$, and

$\displaystyle
w(z)=\Gamma(z-x_{n+1})\Gamma(x_{1}-z+1),\qquad\phi^{+}(z)=z-x_{n+1},\quad\phi^{-}(z)=x_{1}-z.$

More generally, we consider the Jack-deformed Gibbs distributions on Gelfand-
Tsetlin patterns, as introduced by [60, Definition 2.12]. For general
$\theta>0$, the transition probability is

$\displaystyle\mathbb{P}({\bm{\lambda}}(n)={\bm{\mu}}|{\bm{\lambda}}(n+1)={\bm{\lambda}})=\frac{J_{{\bm{\lambda}}/{\bm{\mu}}}(1;\theta)J_{\bm{\mu}}(\bm{1};\theta)}{J_{\bm{\lambda}}(\bm{1};\theta)},\qquad{\bm{\mu}}\in\mathbb{GT}_{n},\quad{\bm{\lambda}}\in\mathbb{GT}_{n+1},\quad{\bm{\mu}}\prec{\bm{\lambda}},$
(50)

where $J_{\lambda}$ is the Jack symmetric polynomial, and
$J_{{\bm{\lambda}}/{\bm{\mu}}}$ is its skew version. At $\theta=1$,
$J_{\lambda}$ turns into the Schur polynomial $s_{\lambda}$ and
$J_{{\bm{\lambda}}/{\bm{\mu}}}(1,1)$ is the indicator of
${\bm{\mu}}\prec{\bm{\lambda}}$; hence, we get back to (48).

We encode the signatures ${\bm{\lambda}},{\bm{\mu}}$ as particle systems
${\bm{x}},{\bm{y}}$, given by $x_{i}=\lambda_{i}-(i-1)\theta$, $1\leqslant
i\leqslant n+1$, and $y_{i}=\mu_{i}-(i-1)\theta$, $1\leqslant i\leqslant n$.
This turns the transition probability (50) into a special case of (27); we
recode this statement in the following claim, whose proof is postponed till
Appendix A.

######  Claim 3.4.

The transition probability (50) in terms of ${\bm{x}}$ and ${\bm{y}}$ is
proportional to

$\displaystyle\mathbb{P}({\bm{y}}|{\bm{x}})\propto\prod_{1\leqslant
i<j\leqslant n}(y_{i}-y_{j})\prod_{1\leqslant i\leqslant j\leqslant
n}\frac{\Gamma(y_{i}-x_{j+1})\Gamma(x_{i}-y_{j}+\theta)}{\Gamma(y_{i}-x_{j+1}-\theta+1)\Gamma(x_{i}-y_{j}+1)},$
(51)

which is identified with a special case of (27), with $b(z)=z$ and weight
$w(z)={\Gamma(x_{1}-z+\theta)}{\Gamma(z-x_{n+1})}$.

Theorem 2.2 gives the dynamical loop equation for transition probabilities
generalizing (51).

######  Corollary 3.5.

Fix ${\bm{x}}=(x_{1},x_{2},\dots,x_{n+1})\in\mathbb{W}_{\theta}^{n+1}$ and
consider the following transition probability

$\displaystyle\mathbb{P}({\bm{y}}|{\bm{x}})\propto\prod_{1\leqslant
i<j\leqslant n}(y_{i}-y_{j})\prod_{1\leqslant i\leqslant j\leqslant
n}\frac{\Gamma(y_{i}-x_{j+1})\Gamma(x_{i}-y_{j}+\theta)}{\Gamma(y_{i}-x_{j+1}-\theta+1)\Gamma(x_{i}-y_{j}+1)}\prod_{i=1}^{n}w(y_{i}),$
(52)

where $y_{i}\in\\{x_{i+1}+\theta,x_{i+1}+\theta+1,\dots,x_{i}\\}$, $1\leqslant
i\leqslant n$, and the weight function $w(z)$ satisfies

$\displaystyle\frac{w(z+1)}{w(z)}=\frac{\phi^{+}(z)}{\phi^{-}(z)}.$

If we assume that $\phi^{+}(z),\phi^{-}(z)$ are holomorphic in a complex
neighborhood of $[x_{n+1}+\theta,x_{1}]$, then the following observable is
holomorphic in the same neighborhood:

$\displaystyle\mathbb{E}\left[\phi^{+}(z)\frac{\prod\limits_{i=1}^{n+1}(z-x_{i})}{\prod\limits_{i=1}^{n}(z-y_{i})}-\phi^{-}(z)\frac{\prod\limits_{i=1}^{n+1}(z-x_{i}-\theta+1)}{\prod\limits_{i=1}^{n}(z-y_{i}+1)}\right].$
(53)

###### Proof.

Using Claim 3.4 and Theorem 2.2 with $b(z)=z$, we obtain analyticity of the
following observable:

$\mathbb{E}\left[\phi^{+}(z)\cdot(z-x_{n+1})\dfrac{\prod\limits_{i=1}^{n+1}\bigl{(}z-x_{i}\bigr{)}}{\prod\limits_{i=1}^{n}\bigl{(}z-y_{i}\bigr{)}}\right.\\\
\left.+\phi^{-}(z)\cdot(x_{1}-z+\theta-1)\frac{\prod\limits_{i=1}^{n+1}\bigl{(}z-x_{i}\bigr{)}}{\prod\limits_{i=1}^{n}\bigl{(}z+1-y_{i}\bigr{)}}\prod\limits_{\begin{smallmatrix}\ell\in\\{x_{i+1}+\theta,x_{i+1}+\theta+1,\dots,x_{i}\\}\\\
i=1,\dots,n\end{smallmatrix}}\dfrac{\bigl{(}z+1-\ell\bigr{)}}{\bigl{(}z-\ell\bigr{)}}\right].$
(54)

Computing the telescoping product over $\ell$ and cancelling the factors, we
convert (54) into

$(z-x_{n+1})\cdot\mathbb{E}\left[\phi^{+}(z)\dfrac{\prod\limits_{i=1}^{n+1}\bigl{(}z-x_{i}\bigr{)}}{\prod\limits_{i=1}^{n}\bigl{(}z-y_{i}\bigr{)}}-\phi^{-}(z)\frac{\prod\limits_{i=1}^{n+1}\bigl{(}z-x_{i}-\theta+1\bigr{)}}{\prod\limits_{i=1}^{n}\bigl{(}z+1-y_{i}\bigr{)}}\right].$

The last expression differs from (53) by $(z-x_{n+1})$ factor, hence, we have
shown so far only that (53) is holomorphic except, possibly, at $z=x_{n+1}$.
However, neither of the terms under expectation in (53) have a singularity at
$z=x_{n+1}$. Thus, the sum is holomorphic also at $z=x_{n+1}$. ∎

###  3.5 $\beta$–Corners processes

Let us consider again the uniform measure on Gelfand-Tsetlin patterns (47),
(48) and rescale the discrete integer lattice to the continuous space via
${\bm{x}}=\lfloor{\tilde{\bm{x}}/\delta}\rfloor$, $\delta\to 0$. After the
limit, the transition probability from the particle configuration
$\tilde{\bm{x}}=(\tilde{x}_{1}>\tilde{x}_{2}>\dots>\tilde{x}_{n+1})\in{\mathbb{R}}^{n+1}$
at level $n+1$ to the particle configuration
$\tilde{\bm{y}}=(\tilde{y}_{1}>\tilde{y}_{2}>\dots>\tilde{y}_{n})\in{\mathbb{R}}^{n}$
at level $n$ is given by

$\displaystyle\mathbb{P}(\tilde{\bm{y}}|\tilde{\bm{x}})=\frac{1}{n!}\frac{V(\tilde{\bm{y}})}{V(\tilde{\bm{x}})},$
(55)

with additional requirement that $\tilde{\bm{y}}$ interlaces with
$\tilde{\bm{x}}$, i.e. $\tilde{x}_{i+1}<\tilde{y}_{i}<\tilde{x}_{i}$ for
$1\leqslant i\leqslant n$. The same limit can be performed for each $\theta>0$
in transition probabilities (51) and (52), resulting in

$\mathbb{P}(\tilde{\bm{y}}|\tilde{\bm{x}})\propto\prod_{1\leqslant i\leqslant
j\leqslant n}(\tilde{y}_{i}-\tilde{y}_{j})\prod_{1\leqslant i\leqslant
j\leqslant
n}(\tilde{y}_{i}-\tilde{x}_{i+1})^{\theta-1}(\tilde{x}_{i}-\tilde{y}_{j})^{\theta-1}\prod_{i=1}^{n}e^{W(\tilde{y}_{i})},$
(56)

again with interlacing $\tilde{\bm{x}}$ and $\tilde{\bm{y}}$. The
distributions of the form (56) for specific choices of $W(y)$ and their
discrete approximations were recently introduced in [60, 16] under the name of
$\beta$–corners processes with $\beta=2\theta$, see also [55]. As shown in
[91] with ideas going back as far as [56], at $\beta=1,2,4$, the transition
probabilities of this type appear in joint distributions of eigenvalues of
corners of self-adjoint real/complex/quaternionic matrices with uniformly
random eigenvectors.

Here is the dynamical loop equation for the $\beta$–corners process with
analytic potential.

######  Corollary 3.6.

Fix
$\tilde{\bm{x}}=(\tilde{x}_{1}>\tilde{x}_{2}>\dots>\tilde{x}_{n+1})\in{\mathbb{R}}^{n+1}$
and consider the following transition probability:

$\displaystyle\begin{split}&\mathbb{P}(\tilde{\bm{y}}|\tilde{\bm{x}})\propto\prod_{1\leqslant
i<j\leqslant n}(\tilde{y}_{i}-\tilde{y}_{j})\prod_{1\leqslant i\leqslant
j\leqslant n}(\tilde{y}_{i}-\tilde{x}_{j+1})^{\theta-1}\prod_{1\leqslant
i\leqslant j\leqslant
n}(\tilde{x}_{i}-\tilde{y}_{j})^{\theta-1}\prod_{i=1}^{n}e^{W(\tilde{y}_{i})},\\\
&\tilde{\bm{y}}=(\tilde{y}_{1}>\tilde{y}_{2}>\dots>\tilde{y}_{n})\in{\mathbb{R}}^{n},\quad\tilde{x}_{i+1}<\tilde{y}_{i}<\tilde{x}_{i},\quad
1\leqslant i\leqslant n.\end{split}$

Suppose that $\theta>0$ and the weight function $W(z)$ is holomorphic in a
complex neighborhood of $[\tilde{x}_{n+1},\tilde{x}_{1}]$. Then the following
function of $\tilde{z}$ is holomorphic in the same neighborhood:

$\mathbb{E}\left[\frac{\prod\limits_{j=1}^{n+1}(\tilde{z}-\tilde{x}_{j})}{\prod\limits_{j=1}^{n}(\tilde{z}-\tilde{y}_{j})}\left(W^{\prime}(\tilde{z})+\sum_{j=1}^{n+1}\frac{\theta-1}{\tilde{z}-\tilde{x}_{j}}+\sum_{j=1}^{n}\frac{1}{\tilde{z}-y_{j}}\right)\right].$
(57)

###### Proof.

Let us compute the scaling limit for the dynamical loop equation (53), setting
${\bm{x}}=\lfloor{\tilde{\bm{x}}/\delta}\rfloor$,
${\bm{y}}=\lfloor{\tilde{\bm{y}}/\delta}\rfloor$, $z=\tilde{z}/\delta$,
$w(z)=\exp(W(\delta z))$. We can further set

$\phi^{-}(z)=1,\qquad\phi^{+}(z)=\frac{w(z+1)}{w(z)}=e^{\delta
W^{\prime}(\delta z)+\operatorname{O}(\delta^{2})}.$

Multiplying (53) by $\delta^{-1}$, for $\tilde{z}$ outside
$[\tilde{x}_{n+1},\tilde{x}_{1}]$ we get

$\delta^{-1}\mathbb{E}\left[\frac{\prod\limits_{j=1}^{n+1}(\tilde{z}-\tilde{x}_{j})}{\prod\limits_{j=1}^{n}(\tilde{z}-\tilde{y}_{j})}\left(\exp\bigl{(}W(\tilde{z}+\delta)-W(\tilde{z})\bigr{)}-\frac{\prod\limits_{j=1}^{n+1}(\tilde{z}-\tilde{x}_{j}+\delta(1-\theta))}{\prod\limits_{j=1}^{n+1}(\tilde{z}-\tilde{x}_{j})}\frac{\prod\limits_{j=1}^{n}(\tilde{z}-\tilde{y}_{j})}{\prod\limits_{j=1}^{n}(\tilde{z}-\tilde{y}_{j}+\delta)}\right)\right]\\\
=\mathbb{E}\left[\frac{\prod\limits_{j=1}^{n+1}(\tilde{z}-\tilde{x}_{j})}{\prod\limits_{j=1}^{n}(\tilde{z}-\tilde{y}_{j})}\left(W^{\prime}(\tilde{z})+\sum_{j=1}^{n+1}\frac{\theta-1}{\tilde{z}-\tilde{x}_{j}}+\sum_{j=1}^{n}\frac{1}{\tilde{z}-\tilde{y}_{j}}+\operatorname{O}(\delta)\right)\right].$
(58)

Let us denote through $f_{\delta}(\tilde{z})$ the expression of (58) and by
$f_{0}(\tilde{z})$ its $\delta\to 0$ limit, which matches (57). Choose a
positively oriented complex contour $\gamma$ enclosing
$[\tilde{x}_{n+1},\tilde{x}_{1}]$, such that $f_{\delta}(\tilde{z})$ is
holomorphic inside $\gamma$ by Corollary 3.5, and use the Cauchy integral
formula to write for $\tilde{z}$ inside $\gamma$:

$f_{\delta}(\tilde{z})=\frac{1}{2\pi\mathbf{i}}\int_{\gamma}\frac{f_{\delta}(u)}{u-\tilde{z}}{\rm
d}u.$

Using the computation (58), we send $\delta\to 0$ in the last identity and get
the Cauchy integral formula for $f_{0}(\tilde{z})$, which implies that
$f_{0}(\tilde{z})$ is holomorphic. ∎

###  3.6 Macdonald processes

We use the following notations:

$\displaystyle(a;q)_{\infty}=\prod_{i=1}^{\infty}(1-aq^{i-1}),\quad
f(u)=\frac{(tu;q)_{\infty}}{(qu;q)_{\infty}},\quad\Gamma_{q}(x)=(1-q)^{1-x}\frac{(q;q)_{\infty}}{(q^{x};q)_{\infty}}.$
(59)

####  3.6.1 Macdonald polynomials and specializations

The next class of Markov chains is built out of the specializations of
Macdonald polynomials. We refer to [82, Section VI] and [14] for definitions
and properties of Macdonald symmetric functions and use them as a black box in
this section. Macdonald symmetric functions $P$ and $Q$ are indexed by
partitions and implicitly depend on two parameters $q,t\in(0,1)$. The
coefficients for the symmetric functions are in $\mathbb{Q}[q,t]$. Macdonald
symmetric functions $P_{\bm{\lambda}}({\bm{x}};q,t)$ and
$Q_{\bm{\lambda}}({\bm{x}};q,t)$ are elements of the algebra $\Lambda$ of the
symmetric functions in infinitely many variables $(x_{i})_{i=1}^{\infty}$
uniquely determined by the following two properties:

  1. 1.

$P_{\bm{\lambda}}$, $|\bm{\lambda}|=m$, can be expressed in terms of the
monomial symmetric functions via a strictly upper unitriangular transition
matrix:

$P_{\bm{\lambda}}=m_{\bm{\lambda}}+\sum_{{\bm{\mu}}<{\bm{\lambda}}\in\mathbb{Y}_{m}}R_{{\bm{\lambda}}{\bm{\mu}}}P_{{\bm{\mu}}},$

where $R_{{\bm{\lambda}}{\bm{\mu}}}$ are functions of $q,t$ and
${\bm{\mu}}<{\bm{\lambda}}$ is comparison in the dominance order on the set
$\mathbb{Y}_{m}$ of all partitions of $m$ (equivalently, Young diagrams with
$m$ boxes).

  2. 2.

They are pairwise orthogonal with respect to the scalar product defined on the
power sums via

$\langle
p_{\bm{\lambda}},p_{\bm{\mu}}\rangle_{q,t}=\delta_{{\bm{\lambda}}{\bm{\mu}}}z_{\bm{\lambda}}(q,t),\qquad
p_{\lambda}=\prod_{i=1}^{\infty}p_{\lambda_{i}},\qquad
z_{\bm{\lambda}}(q,t)=\prod_{i\geqslant
1}i^{m_{i}}(m_{i})!\prod_{i=1}^{\ell({\bm{\lambda}})}\frac{1-q^{\lambda_{i}}}{1-t^{\lambda_{i}}},$

where ${\bm{\lambda}}=1^{m_{1}}2^{m_{2}}\dots$, i.e. $m_{i}$ is the
multiplicity of $i$ in ${\bm{\lambda}}$, $\ell({\bm{\lambda}})$ is the number
of rows, and $p_{k}=(x_{1})^{k}+(x_{2})^{k}+\dots$, $k\geqslant 1$.

We further define $Q_{\bm{\lambda}}=\frac{P_{\bm{\lambda}}}{\langle
P_{\bm{\lambda}},P_{\bm{\lambda}}\rangle_{q,t}}.$ Finally, the skew Macdonald
polynomials $P_{{\bm{\lambda}}/{\bm{\mu}}}$ and
$Q_{{\bm{\lambda}}/{\bm{\mu}}}$ are defined through the expansions:

$\displaystyle P_{\bm{\lambda}}(x_{1},x_{2},\dots,y_{1},y_{2},\dots;q,t)$
$\displaystyle=\sum_{{\bm{\mu}}}P_{{\bm{\lambda}}/{\bm{\mu}}}(x_{1},x_{1},\dots;q,t)P_{\bm{\mu}}(y_{1},y_{2},\dots;q,t),$
$\displaystyle Q_{\bm{\lambda}}(x_{1},x_{2},\dots,y_{1},y_{2},\dots;q,t)$
$\displaystyle=\sum_{{\bm{\mu}}}Q_{{\bm{\lambda}}/{\bm{\mu}}}(x_{1},x_{1},\dots;q,t)Q_{\bm{\mu}}(y_{1},y_{2},\dots;q,t).$

Computations in the algebra of symmetric functions $\Lambda$ can be converted
into numeric identities by means of _specializations_ , which are algebra
homomorphism from $\Lambda$ to the set of complex numbers. Specialization
$\rho$ is uniquely determined by its values on any set of algebraic generators
of $\Lambda$ and we use $(p_{k})_{k=1}^{\infty}$ as such generators. The value
of $\rho$ on a symmetric function $f$ is denoted $f(\rho)$. Given two
specializations $\rho,\rho^{\prime}$, we define their union
$(\rho,\rho^{\prime})$ through the formula:

$p_{k}(\rho,\rho^{\prime})=p_{k}(\rho)+p_{k}(\rho^{\prime}),\quad k\geqslant
1.$

A specialization $\rho$ is called _Macdonald nonnegative_ if its values on all
(skew) Macdonald symmetric functions are non-negative, i.e., if for all
partitions $\bm{\lambda}$ and $\bm{\mu}$,

$\displaystyle P_{{\bm{\lambda}}/{\bm{\mu}}}(\rho;q,t)\geqslant 0,$

The description of Macdonald nonnegative specializations was conjectured in
[77] and proven in [83]:

######  Theorem 3.7 ([83]).

For any fixed $q,t\in(0,1)$, Macdonald nonnegative specializations can be
parameterized by triplets $(\bm{\alpha}=\\{\alpha_{i}\\}_{i\geqslant
1},\bm{\beta}=\\{\beta_{i}\\}_{i\geqslant 1},\gamma)$ of nonnegative numbers
satisfying $\sum_{i=1}^{\infty}(\alpha_{i}+\beta_{i})<\infty$. The
specialization $\rho$ corresponding to a triplet
$(\bm{\alpha},\bm{\beta},\gamma)$ is defined by for $k\geqslant 2$

$p_{1}(\rho)=\sum_{i=1}^{\infty}\alpha_{i}+\frac{1-q}{1-t}\left(\gamma+\sum_{i=1}^{\infty}\beta_{i}\right),\qquad
p_{k}(\rho)=\sum_{i=1}^{\infty}(\alpha_{i})^{k}+(-1)^{k-1}\frac{1-q^{k}}{1-t^{k}}\sum_{i=1}^{\infty}(\beta_{i})^{k}.$

Following [14, Section 2.3], we create Markov chains out of the Macdonald-
positive specializations:

######  Definition 3.8.

Given two specializations $\rho$ and $\rho^{\prime}$ we define the ascending
transition through

$\mathbb{P}({\bm{\lambda}}\mid{\bm{\mu}})=\frac{1}{\Pi(\rho;\rho^{\prime})}\frac{P_{\bm{\lambda}}(\rho;q,t)}{P_{\bm{\mu}}(\rho;q,t)}Q_{{\bm{\lambda}}/{\bm{\mu}}}(\rho^{\prime};q,t),$
(60)

where ${\bm{\lambda}}$ and ${\bm{\mu}}$ are partitions with
${\bm{\mu}}\subset{\bm{\lambda}}$ and $\Pi(\rho;\rho^{\prime})$ is the result
of applying $\rho$ to the $x_{i}$ variables and $\rho^{\prime}$ to the $y_{j}$
variables in the infinite product

$\Pi=\prod_{i,j\geqslant
1}\frac{(tx_{i}y_{j};q)_{\infty}}{(x_{i}y_{j};q)_{\infty}}.$

######  Definition 3.9.

Given two specializations $\rho$ and $\rho^{\prime}$ we define the descending
transition through

$\mathbb{P}({\bm{\mu}}\mid{\bm{\lambda}})=\frac{P_{\bm{\mu}}(\rho;q,t)}{P_{\bm{\lambda}}(\rho,\rho^{\prime};q,t)}Q_{{\bm{\lambda}}/{\bm{\mu}}}(\rho^{\prime};q,t),$

where ${\bm{\lambda}}$ and ${\bm{\mu}}$ are partitions with
${\bm{\mu}}\subset{\bm{\lambda}}$.

####  3.6.2 Loop equations for the ascending process

For particular choices of $\rho$ and $\rho^{\prime}$, the ascending process of
Definition 3.8 fits into our formalism of dynamic loop equations. Namely, we
fix $n=1,2,\dots$ and set in the notations of Theorem 3.7

$\rho:\,\alpha_{i}=t^{i-1},\,1\leqslant i\leqslant
n;\qquad\rho^{\prime}:\beta_{1}=b,$ (61)

with all other parameters set to $0$.

######  Claim 3.10.

The transition probability of Definition 3.8 under the specializations (61) is
non-degenerate only for partitions ${\bm{\lambda}},{\bm{\mu}}$ with at most
$n$ parts, i.e.
${\bm{\lambda}}=(\lambda_{1}\geqslant\lambda_{2}\geqslant\dots\geqslant\lambda_{n})$
and ${\bm{\mu}}=(\mu_{1}\geqslant\mu_{2}\geqslant\dots\geqslant\mu_{n})$.
Further, if we set $t=q^{\theta}$ and identify

${\bm{x}}=(x_{1},x_{2},\dots,x_{n})\in\mathbb{W}_{\theta}^{n},\quad
x_{i}=\mu_{i}-\theta(i-1),\quad x_{i}+e_{i}=\lambda_{i}-\theta(i-1),\quad
1\leqslant i\leqslant n,$ (62)

then $e_{i}\in\\{0,1\\}$ and the transition probability is proportional to

$\displaystyle\mathbb{P}({\bm{x}}+{\bm{e}}|{\bm{x}})\propto
b^{\sum_{i=1}^{n}e_{i}}\prod_{1\leqslant i<j\leqslant n}\frac{q^{x_{i}+\theta
e_{i}}-q^{x_{j}+\theta e_{j}}}{q^{x_{i}}-q^{x_{j}}},$ (63)

which is a special case of (4), with $b(z)=q^{z}$ and
$\phi^{+}(z)=b,\phi^{-}(z)=1$.

By taking $q\rightarrow 1$, Macdonald symmetric polynomials recover the Jack
symmetric polynomials with parameter $\theta$. The global fluctuations of
these measures related to Jack symmetric polynomials have been previously
studied in [42, 85, 70] by different methods. We postpone the proof of Claim
3.10 till Appendix A. Theorem 1.1 gives the dynamical loop equation for
transition probabilities generalizing the ascending Macdonald process (63).

######  Corollary 3.11.

Fix ${\bm{x}}=(x_{1}>x_{2}>\dots>x_{n})\in\mathbb{W}_{\theta}^{n}$, and
consider the following transition probability

$\displaystyle\mathbb{P}({\bm{x}}+{\bm{e}}|{\bm{x}})\propto\prod_{1\leqslant
i<j\leqslant n}\frac{q^{x_{i}+\theta e_{i}}-q^{x_{j}+\theta
e_{j}}}{q^{x_{i}}-q^{x_{j}}}\prod_{i=1}^{n}\phi^{+}(x_{i})^{e_{i}}\phi^{-}(x_{i})^{1-e_{i}},\quad{\bm{e}}\in\\{0,1\\}^{n}.$

Suppose that the weighs $\phi^{+}(z),\phi^{-}(z)$ are holomorphic functions in
a neighborhood of $[x_{n},x_{1}]$. Then the following observable is
holomorphic in the same neighborhood

$\displaystyle\mathbb{E}\left[\phi^{+}(z)\prod_{j=1}^{n}\frac{q^{z+\theta}-q^{x_{j}+\theta
e_{j}}}{q^{z}-q^{x_{i}}}+\phi^{-}(z)\prod_{j=1}^{n}\frac{q^{z}-q^{x_{j}+\theta
e_{j}}}{q^{z}-q^{x_{j}}}\right].$

###### Proof.

This is Theorem 1.1 with $b(z)=q^{z}$. ∎

####  3.6.3 Loop equations for the descending process

For particular choices of $\rho$ and $\rho^{\prime}$, the descending process
of Definition 3.9 also fits into our formalism of dynamic loop equations.
Namely, we fix $n=1,2,\dots$ and set in the notations of Theorem 3.7

$\rho:\,\alpha_{i}=t^{i-1},\,1\leqslant i\leqslant
n;\qquad\rho^{\prime}:\alpha_{1}=t^{n},$ (64)

with all other parameters setting to $0$.

######  Claim 3.12.

The transition probability of Definition 3.9 under the specializations (64) is
non-degenerate only for partitions ${\bm{\lambda}}$ with at most $(n+1)$
parts, i.e.
${\bm{\lambda}}=(\lambda_{1}\geqslant\lambda_{2}\geqslant\dots\geqslant\lambda_{n+1})$,
and ${\bm{\mu}}$ with at most $n$ parts
${\bm{\mu}}=(\mu_{1}\geqslant\mu_{2}\geqslant\dots\geqslant\mu_{n})$, and such
that ${\bm{\mu}}\prec{\bm{\lambda}}$. Further, if we set $t=q^{\theta}$ and
denote

$\displaystyle{\bm{x}}=(x_{1},x_{2},\dots,x_{n+1})\in\mathbb{W}_{\theta}^{n+1},$
$\displaystyle x_{i}=\lambda_{i}-\theta(i-1),\quad$ $\displaystyle 1\leqslant
i\leqslant n+1,$
$\displaystyle{\bm{y}}=(y_{1},y_{2},\dots,y_{n})\in\mathbb{W}_{\theta}^{n},$
$\displaystyle y_{i}=\mu_{i}-\theta(i-1),\quad$ $\displaystyle 1\leqslant
i\leqslant n,$

then the transition probability is proportional to

$\displaystyle\prod_{1\leqslant i<j\leqslant
n}(q^{-y_{i}}-q^{-y_{j}})\prod_{1\leqslant i\leqslant j\leqslant
n}\frac{\Gamma_{q}(y_{i}-x_{j+1})}{\Gamma_{q}(y_{i}-x_{j+1}+1-\theta)}\frac{\Gamma_{q}(x_{i}-y_{j}+\theta)}{\Gamma_{q}(x_{i}-y_{j}+1)}\prod_{i=1}^{n}q^{-((\theta-1)(n-i)+\theta)y_{i}},$
(65)

which is a special case of (27), with $b(z)=q^{-z}$ and weight

$w(z)=q^{-z^{2}/2+z(x_{n+1}+1/2)}{\Gamma_{q}(x_{1}-z+\theta)}{\Gamma_{q}(z-x_{n+1})}.$
(66)

We postpone the proof of Claim 3.12 till Appendix A. Note that for special
values of $\theta$, the weights (65) can be dramatically simplified. For
instance, at $\theta=1$ all $\Gamma_{q}$ factors cancel out and the
$\prod_{1\leqslant i\leqslant j\leqslant n}$ part disappears from the formula.
For $\theta=2$, the formula is also simple:

$\mathbb{P}({\bm{y}}|{\bm{x}})\propto\prod_{1\leqslant i<j\leqslant
n}(q^{-y_{i}}-q^{-y_{j}})\prod_{1\leqslant i\leqslant j\leqslant
n}\bigl{[}(1-q^{y_{i}-x_{j+1}-1})(1-q^{x_{i}-y_{j}+1})\bigr{]}\prod_{i=1}^{n}q^{-(n-i+2)y_{i}}.$
(67)

Theorem 2.2 gives the dynamical loop equation for transition probabilities
generalizing (65).

######  Corollary 3.13.

Fix ${\bm{x}}=(x_{1},x_{2},\dots,x_{n+1})\in\mathbb{W}_{\theta}^{n+1}$ and
consider the following transition probability

$\displaystyle\prod_{1\leqslant i<j\leqslant
n}(q^{-y_{i}}-q^{-y_{j}})\prod_{1\leqslant i\leqslant j\leqslant
n}\frac{\Gamma_{q}(y_{i}-x_{j+1})}{\Gamma_{q}(y_{i}-x_{j+1}+1-\theta)}\frac{\Gamma_{q}(x_{i}-y_{j}+\theta)}{\Gamma_{q}(x_{i}-y_{j}+1)}\prod_{i=1}^{n}q^{-((\theta-1)(n-i)+\theta)y_{i}}w(y_{i}),$

with
$y_{i}\in\\{x_{i+1}+\theta,x_{i+1}+1+\theta,x_{i+1}+2+\theta,\dots,x_{i}-1,x_{i}\\}$
for $1\leqslant i\leqslant n$. Suppose that

$\displaystyle\frac{w(z+1)}{w(z)}=\frac{\phi^{+}(z)}{\phi^{-}(z)},$

with $\phi^{+}(z)$ and $\phi^{-}(z)$ holomorphic in a complex neighborhood of
$[x_{n+1},x_{1}]$. Then the following observable is holomorphic in the same
neighborhood:

$\displaystyle\mathbb{E}\left[\phi^{+}(z)\frac{\prod\limits_{j=1}^{n+1}(q^{-z}-q^{-x_{j}})}{\prod\limits_{j=1}^{n}(q^{-z}-q^{-y_{j}})}-q^{(\theta-1)(n+1)}\phi^{-}(z)\frac{\prod\limits_{j=1}^{n+1}(q^{-z}-q^{-x_{i}+1-\theta})}{\prod\limits_{j=1}^{n}(q^{-z-1}-q^{-y_{j}})}\right].$

###### Proof.

Using Claim 3.12 and Theorem 2.2 with $b(z)=q^{-z}$, we obtain the
holomorphicity of

$\mathbb{E}\left[(1-q^{z-x_{n+1}})\phi^{+}(z)\dfrac{\prod\limits_{i=1}^{n+1}\bigl{(}q^{-z}-q^{-x_{i}}\bigr{)}}{\prod\limits_{i=1}^{n}\bigl{(}q^{-z}-q^{-y_{i}}\bigr{)}}\right.\\\
+\left.(1-q^{x_{1}-z+\theta-1})q^{z-x_{n+1}}\phi^{-}(z)\prod\limits_{\ell\in\mathbb{L}({\bm{x}})}\dfrac{\bigl{(}q^{-z-1}-q^{-\ell}\bigr{)}}{\bigl{(}q^{-z}-q^{-\ell}\bigr{)}}\cdot\frac{\prod\limits_{i=1}^{n+1}\bigl{(}q^{-z}-q^{-x_{i}}\bigr{)}}{\prod\limits_{i=1}^{n}\bigl{(}q^{-z-1}-q^{-y_{i}}\bigr{)}}\right].$
(68)

Let us simplify the product over $\ell$ in the second term:

$\prod_{i=1}^{n}\,\prod_{\ell\in\\{x_{i+1}+\theta,x_{i+1}+1+\theta,\dots,x_{i}-1,x_{i}\\}}\dfrac{q^{-1}\bigl{(}q^{-z}-q^{1-\ell}\bigr{)}}{\bigl{(}q^{-z}-q^{-\ell}\bigr{)}}=\prod_{i=1}^{n}\left[q^{-(x_{i}-x_{i+1}+1-\theta)}\frac{q^{-z}-q^{-x_{i+1}+1-\theta}}{q^{-z}-q^{-x_{i}}}\right].$

Therefore, (68) is

$(1-q^{z-x_{n+1}})\mathbb{E}\left[\phi^{+}(z)\dfrac{\prod\limits_{i=1}^{n+1}\bigl{(}q^{-z}-q^{-x_{i}}\bigr{)}}{\prod\limits_{i=1}^{n}\bigl{(}q^{-z}-q^{-y_{i}}\bigr{)}}\right.\\\
-\left.q^{(\theta-1)(n+1)}\phi^{-}(z)\frac{\prod\limits_{i=1}^{n+1}(q^{-z}-q^{-x_{i}+1-\theta})}{\prod\limits_{i=1}^{n}\bigl{(}q^{-z-1}-q^{-y_{i}}\bigr{)}}\right].$

It remains to note that neither of the terms under expectation has a
singularity at $z=x_{n+1}$ and, hence, we can divide by $(1-q^{z-x_{n+1}})$
and the result is still holomorphic. ∎

###  3.7 $(q,\kappa)$-distributions on tilings and Koornwinder polynomials

In this section we first explain how $(q,\kappa)$–distributions on tilings of
Section 1.3 fit into the formalism of dynamic loop equations. Then we show
that these distributions are particular cases of more general Markov chains
which are obtained from quasi-branching rules for Koornwinder symmetric
polynomials.

####  3.7.1 Weighted lozenge tilings

We deal with random lozenge tilings of a trapezoid, as in Figures 2 and 6. The
probability of each tiling ${\mathcal{T}}$ is proportional to the weight given
in terms of  lozenges, as in (8), (9):

$\displaystyle
w({\mathcal{T}})=\prod_{{\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}}\in{\mathcal{T}}}w({\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}}),\qquad
w({\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}})=\kappa
q^{x-t/2}-\kappa^{-1}q^{-x+t/2}=\kappa
q^{\tilde{x}}-\kappa^{-1}q^{-\tilde{x}}.$ (69)

We further identify a tiling with a collection of $N$ non-intersecting random
walks ${\bm{x}}(t)=(x_{1}(t)>\dots>x_{n}(t))$, $0\leqslant t\leqslant T$, by
tracing the trajectories of  and  lozenges, as in Figure 6. The trapezoid is
uniquely determined by specifying $N$, $T$, arbitrary initial configuration
${\bm{x}}(0)$, and prefixed ending configuration
${\bm{x}}(T)=(N-1,N-2,\dots,0)$. The measure (69) leads to a Markov chain
structures on ${\bm{x}}(t)$ with explicit transition probabilities.

Figure 6: Tracking positions of particles in non-intersecting paths, tilings
turn into a Markov chain in $\mathbb{W}^{N}$.

######  Theorem 3.14.

$\\{{\bm{x}}(t)\\}_{0\leqslant t\leqslant T}$ corresponding to the
$(q,\kappa)$-distributions on lozenge tilings with weights (69) is a Markov
chain with transition probability proportional to

$\displaystyle\mathbb{P}({\bm{x}}(t+1)={\bm{x}}+{\bm{e}}|{\bm{x}}(t)={\bm{x}})\propto\prod_{1\leqslant
i<j\leqslant
N}\frac{b_{t}(x_{i}+e_{i})-b_{t}(x_{j}+e_{j})}{b_{t}(x_{i})-b_{t}(x_{j})}\prod_{i=1}^{N}\phi_{t}^{+}(x_{i})^{e_{i}}\phi_{t}^{-}(x_{i})^{1-e_{i}},$
(70)

where ${\bm{e}}\in\\{0,1\\}^{N}$ and

$b_{t}(x)=q^{-x}+\kappa^{2}q^{x-t},$
$\phi_{t}^{+}(x)=q^{T+N-1-t}(1-q^{x-N+1})(1-\kappa^{2}q^{x-T+1}),\qquad\phi_{t}^{-}(x)=-(1-q^{x+T-t})(1-\kappa^{2}q^{x+N-t}).$

We remark that the weights $\phi_{t}^{\pm}(x)$ encode a part of the boundary
of the tiled trapezoid. More precisely, $\phi_{t}^{+}(x)$ has a zero at
$x=N-1$ and $\phi_{t}^{-}(x)$ has a zero at $t-x=T$. Therefore, if at time $t$
a particle is at $N-1$, it will stay; if a particle is at $t-T$, it will jump
one unit to the right.

###### Proof of Theorem 3.14.

Up to shift $x\mapsto x+S$, this is [19, Proposition 4.2]. Let us, however,
present a sketch of the proof. The Markov property that “given present, future
and past are independent” immediately follows from the definition of the
system. Further, let $Z(x_{1},\dots,x_{N};T)$ denote the partition function,
i.e. this is the total sum of weights (69) over the tilings corresponding to
non-intersecting paths ${\bm{x}}(t)=(x_{1}(t)>\dots>x_{n}(t))$, $0\leqslant
t\leqslant T$, with ${\bm{x}}(0)=(x_{1},\dots,x_{N})$ and
${\bm{x}}(T)=(N-1,N-2,\dots,0)$. We can express

$\mathbb{P}({\bm{x}}(t+1)={\bm{x}}+{\bm{e}}|{\bm{x}}(t)={\bm{x}})=\frac{Z({\bm{x}}+{\bm{e}};T-t-1)}{Z({\bm{x}};T-t)}\prod_{{\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}}}\left(\kappa
q^{x({\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}})-t/2}-\kappa^{-1}q^{-x({\scalebox{0.15}{\includegraphics{lozenge_hor.pdf}}})+t/2}\right),$
(71)

where the product is taken over all horizontal lozenges on the vertical line
with ordinate $t+1$; the positions of these lozenges complement the
configuration ${\bm{x}}+{\bm{e}}$. The key observation is that
$Z(x_{1},\dots,x_{N};T)$ has an explicit product formula: this computation
relying on explicit evaluation of certain determinants is present in [19,
Appendix]; another way to get the same answer is through explicit evaluation
of principally-specialized Koornwinder polynomials, as we explain in Theorem
3.19 below. Plugging the result into (71), we arrive at (70). ∎

Theorem 1.1 gives the dynamical loop equation for transition probabilities
(70).

######  Corollary 3.15.

Fix ${\bm{x}}=(x_{1}>x_{2}>\dots>x_{n})\in\mathbb{W}^{n}$ and $0\leqslant
t<T$. Consider transition probability (70). Then the following observable is
holomorphic in a complex neighborhood of $[x_{n},x_{1}]$,

$\mathbb{E}\left[q^{T+N-1-t}(1-q^{z-N+1})(1-\kappa^{2}q^{z-T+1})\prod_{j=1}^{n}\frac{b_{t}(z+\theta)-b_{t}(x_{j}+\theta
e_{j})}{b_{t}(z)-b_{t}(x_{j})}\right.\\\
-\left.(1-q^{z+T-t})(1-\kappa^{2}q^{z+N-t})\prod_{j=1}^{n}\frac{b_{t}(z)-b_{t}(x_{j}+\theta
e_{j})}{b_{t}(z)-b_{t}(x_{j})}\right],$

where $b_{t}(z)=q^{-z}+\kappa^{2}q^{z-t}$.

####  3.7.2 Quasi-branching of Koornwinder symmetric polynomials

Following the notations in [98, Definition 2], we denote
$K_{{\bm{\lambda}}}^{(n)}(x_{1},\dots,x_{n};q,t;t_{0},t_{1},t_{2},t_{3})$ the
Koorwinder symmetric (Laurent) polynomial in $n$ variables
$x_{1},\dots,x_{n}$. We explained in Sections 3.4 and 3.6 how branching rules
for Schur, Jack, and Macdonald polynomials lead to measures on Gelfand-Tsetlin
patterns, which fit into our formalism of dynamical loop equations. Generally
speaking, the branching rules for Koornwinder polynomials are much more
complicated, and do not lead to any simple measures on Gelfand–Tsetlin
patterns. However, when we deal with a special principal specialization, there
is a dramatic simplification known as _quasi-branching rule_. The following
result is [98, Theorem 5.21].

######  Theorem 3.16.

For any partition ${\bm{\lambda}}\in\mathbb{GT}_{n+1}$ with
$\lambda_{n+1}\geqslant 0$, we have

$\frac{K_{{\bm{\lambda}}}^{(n+1)}(x_{1},\dots,x_{n},t_{0};\,q,t;\,t_{0},t_{1},t_{2},t_{3})}{K_{{\bm{\lambda}}}^{(n+1)}(q^{n}t_{0},q^{n-1}t_{0},\dots,t_{0};\,q,t;\,t_{0},t_{1},t_{2},t_{3})}\\\
=\sum_{{\bm{\mu}}\prec{\bm{\lambda}}}\psi_{{\bm{\lambda}}\setminus{\bm{\mu}}}^{(i)}(t^{n+1};\,q,t,t^{n}\sqrt{t_{0}t_{1}t_{2}t_{3}/qt})\frac{K_{\bm{\mu}}^{(n)}(x_{1},\dots,x_{n};\,q,t;\,t_{0}t,t_{1},t_{2},t_{3})}{K_{\bm{\mu}}^{(n)}(q^{n}t_{0},q^{n-1}t_{0},\dots,qt_{0};\,q,t;\,t_{0}t,t_{1},t_{2},t_{3})},$
(72)

where the coefficients are given by

$\displaystyle\begin{split}\psi_{{\bm{\lambda}}\setminus{\bm{\mu}}}^{(i)}(u;q,t,s)&=(u/t)^{|{\bm{\lambda}}|-|{\bm{\mu}}|}t^{n({\bm{\mu}})-n({\bm{\lambda}})}\frac{C_{\bm{\lambda}}^{0}(s^{2}qt/u;q,t)C_{\bm{\mu}}^{0}(u/t;q,t)}{C_{\bm{\lambda}}^{0}(u;q,t)C_{\bm{\mu}}^{0}(s^{2}qt/u;q,t)}\\\
&\prod_{(i,j)\in{\bm{\lambda}}\atop\lambda_{j}^{\prime}=\mu_{j}^{\prime}}\frac{1-q^{\lambda_{i}+j-1}t^{-\lambda_{j}^{\prime}-i+3}s^{2}}{1-q^{\mu_{i}-j+1}t^{\mu_{j}^{\prime}-i}}\prod_{(i,j)\in{\bm{\lambda}}\atop\lambda_{j}^{\prime}\neq\mu_{j}^{\prime}}\frac{1-q^{\lambda_{i}-j}t^{\lambda_{j}^{\prime}-i+1}}{1-q^{\mu_{i}+j}t^{-\mu_{j}^{\prime}-i+1}s^{2}}\\\
&\prod_{(i,j)\in{\bm{\mu}}\atop\lambda_{j}^{\prime}=\mu_{j}^{\prime}}\frac{1-q^{\lambda_{i}-j+1}t^{\lambda_{j}^{\prime}-i}}{1-q^{\mu_{i}+j-1}t^{2-\mu_{j}^{\prime}-i}s^{2}}\prod_{(i,j)\in{\bm{\mu}}\atop\lambda_{j}^{\prime}\neq\mu_{j}^{\prime}}\frac{1-q^{\lambda_{i}+j}t^{2-\lambda_{j}^{\prime}-i}s^{2}}{1-q^{\mu_{i}-j}t^{\mu_{j}^{\prime}-i+1}},\end{split}$
(73)

with $|{\bm{\lambda}}|=\sum_{i=1}^{n+1}\lambda_{i}$,
$n(\bm{\lambda})=\sum_{i=1}^{n+1}\lambda_{i}(\lambda_{i}-1)/2$, and similarly
for ${\bm{\mu}}$. Also

$C_{\bm{\mu}}^{0}(x;q,t)=\prod_{(i,j)\in{\bm{\mu}}}(1-q^{j-1}t^{1-i}x)=\prod_{i\geqslant
1}(t^{1-i}x;q)_{\mu_{i}}.$

By taking principle specialization on both sides of (72), i.e.
$x_{i}=q^{n+1-i}t_{0}$ with $1\leqslant i\leqslant n$, we get a probability
measure over Gelfand-Tsetlin patterns, with the transition probability given
by

$\mathbb{P}({\bm{\mu}}|{\bm{\lambda}})=\psi_{{\bm{\lambda}}\setminus{\bm{\mu}}}^{(i)}(t^{n+1};q,t,t^{n}\sqrt{t_{0}t_{1}t_{2}t_{3}/qt}),\qquad{\bm{\lambda}}\in\mathbb{GT}_{n+1},\,\lambda_{n+1}\geqslant
0,\,{\bm{\mu}}\in\mathbb{GT}_{n},\,{\bm{\mu}}\prec{\bm{\lambda}}.$ (74)

We denote $t=q^{\theta}$ for some $\theta>0$, and encode ${\bm{\lambda}}$ and
${\bm{\mu}}$ as particle systems ${\bm{x}}$ and ${\bm{y}}$, with

$\displaystyle{\bm{x}}=(x_{1},x_{2},\dots,x_{n+1})\in\mathbb{W}_{\theta}^{n+1},$
$\displaystyle x_{i}=\lambda_{i}-\theta(i-1),\quad$ $\displaystyle 1\leqslant
i\leqslant n+1,$
$\displaystyle{\bm{y}}=(y_{1},y_{2},\dots,y_{n+1})\in\mathbb{W}_{\theta}^{n},$
$\displaystyle y_{i}=\mu_{i}-\theta(i-1),\quad$ $\displaystyle 1\leqslant
i\leqslant n.$

The interlacing condition ${\bm{\mu}}\prec{\bm{\lambda}}$ becomes
$x_{i+1}+\theta\leqslant y_{i}\leqslant x_{i}$ for $1\leqslant i\leqslant n$.
The transition probability (74) becomes a special case of (27), as Claim 3.17
(whose proof is postponed till Appendix A) shows:

######  Claim 3.17.

Let $q^{v/2}=t^{n}\sqrt{t_{0}t_{1}t_{2}t_{3}/qt}$. We can rewrite the
transition probability (74) as

$\displaystyle\begin{split}\mathbb{P}({\bm{y}}|{\bm{x}})&\propto\prod_{1\leqslant
i<k\leqslant n}(q^{-y_{i}}-q^{-y_{k}})\prod_{1\leqslant i\leqslant k\leqslant
n}\frac{\Gamma_{q}(y_{i}-x_{k+1})\Gamma_{q}(x_{i}-y_{k}+\theta)}{\Gamma_{q}(x_{i}-y_{k}+1)\Gamma_{q}(y_{i}-x_{k+1}-\theta+1)}\\\
&\times\prod_{1\leqslant i\leqslant k\leqslant
n}(1-q^{y_{i}+y_{k}+v})\left[\prod_{i=1}^{n}\prod_{k=1}^{n+1}\frac{\Gamma_{q}(y_{i}+x_{k}+v+\theta)}{\Gamma_{q}(y_{i}+x_{k}+1+v)}\right]\prod_{i=1}^{n}q^{-((\theta-1)(n-i)+\theta)y_{i}},\end{split}$
(75)

which is a special case of (27), with $b(z)=q^{-z}+q^{z+v}$ and weight

$\displaystyle
w(z)=q^{-z^{2}/2+z(x_{n+1}+1/2)}{\Gamma_{q}(x_{1}-z+\theta)}{\Gamma_{q}(z-x_{n+1})}\frac{\Gamma_{q}(z+x_{1}+v+\theta)}{\Gamma_{q}(z+x_{n+1}+v+1)}.$

######  Corollary 3.18.

Fix ${\bm{x}}=(x_{1},x_{2},\dots,x_{n+1})\in\mathbb{W}_{\theta}^{n+1}$ and
consider the following transition probability

$\displaystyle\begin{split}\mathbb{P}({\bm{y}}|{\bm{x}})&\propto\prod_{1\leqslant
i<k\leqslant n}(q^{-y_{i}}-q^{-y_{k}})\prod_{1\leqslant i\leqslant k\leqslant
n}\frac{\Gamma_{q}(y_{i}-x_{k+1})\Gamma_{q}(x_{i}-y_{k}+\theta)}{\Gamma_{q}(x_{i}-y_{k}+1)\Gamma_{q}(y_{i}-x_{k+1}-\theta+1)}\\\
&\times\prod_{1\leqslant i\leqslant k\leqslant
n}(1-q^{y_{i}+y_{k}+v})\left[\prod_{i=1}^{n}\prod_{k=1}^{n+1}\frac{\Gamma_{q}(y_{i}+x_{k}+v+\theta)}{\Gamma_{q}(y_{i}+x_{k}+1+v)}\right]\prod_{i=1}^{n}q^{-((\theta-1)(n-i)+\theta)y_{i}}w(y_{i}),\end{split}$
(76)

with
$y_{i}\in\\{x_{i+1}+\theta,x_{i+1}+1+\theta,x_{i+1}+2+\theta,\dots,x_{i}-1,x_{i}\\}$
for $1\leqslant i\leqslant n$. Suppose that

$\displaystyle\frac{w(z+1)}{w(z)}=\frac{\phi^{+}(z)}{\phi^{-}(z)},$

with $\phi^{+}(z)$ and $\phi^{-}(z)$ holomorphic in a complex neighborhood of
$[x_{n+1},x_{1}]$. Then the following observable is holomorphic in the same
neighborhood:

$\displaystyle\mathbb{E}\left[\phi^{+}(z)\frac{\prod\limits_{j=1}^{n+1}(q^{-z}+q^{z+v}-q^{-x_{j}}-q^{x_{j}+v})}{\prod\limits_{j=1}^{n}(q^{-z}+q^{z+v}-q^{-y_{j}}-q^{y_{i}+v})}\right.$
$\displaystyle\left.-q^{(\theta-1)(n+1)}\phi^{-}(z)\frac{\prod\limits_{j=1}^{n+1}(q^{-z}+q^{z+v}-q^{-x_{i}+1-\theta}-q^{x_{i}+\theta-1+v})}{\prod\limits_{j=1}^{n}(q^{-z-1}+q^{z+1+v}-q^{-y_{j}}-q^{y_{j}+v})}\right].$

We omit the proof, as it is very similar to Corollaries 3.5 and 3.13.

####  3.7.3 Koornwinder polynomials at $q=t$ and tilings

In this section we explain that $q=t$ case of the measures of Section 3.7.2
matches the measures on tilings of Section 3.7.1 after the particle-hole
involution. We start from a statement of independent interest.

######  Theorem 3.19.

For any
${\bm{\lambda}}=(\lambda_{1}\geqslant\lambda_{2}\geqslant\dots\geqslant\lambda_{n})\in\mathbb{GT}_{n}$,
the following identity of Laurent polynomials in $\sigma$ holds:

$\sum_{\emptyset={\bm{\lambda}}^{(0)}\prec\dots\prec{\bm{\lambda}}^{(n)}={\bm{\lambda}}}\prod_{k=1}^{n-1}\prod_{i=1}^{k}\left(\sigma
q^{\lambda^{(k)}_{i}+\frac{k+1}{2}-i}-\frac{1}{\sigma
q^{\lambda^{(k)}_{i}+\frac{k+1}{2}-i}}\right)=Z_{n}\prod_{i<j}\bigl{[}x(\lambda_{i}-i)-x(\lambda_{j}-j)\bigr{]},$
(77)

where

$x(\lambda_{i}-i)=\sigma q^{\lambda_{i}-i+\frac{n+1}{2}}+\frac{1}{\sigma
q^{\lambda_{i}-i+\frac{n+1}{2}}},$
$Z_{n}=(-1)^{-n(n-1)/2}\prod_{i=1}^{n}\frac{q^{(i-\frac{n+1}{2})i}}{(q;q)_{i-1}}.$

######  Remark 3.20.

An equivalent form of the same statement is

$\sum_{\emptyset={\bm{\lambda}}^{(0)}\prec\dots\prec{\bm{\lambda}}^{(n)}={\bm{\lambda}}}\prod_{k=1}^{n-1}\prod_{i=1}^{k}\frac{\sigma
q^{\lambda^{(k)}_{i}+\frac{k+1}{2}-i}-\frac{1}{\sigma
q^{\lambda^{(k)}_{i}+\frac{k+1}{2}-i}}}{\sigma
q^{\frac{k+1}{2}-i}-\frac{1}{\sigma q^{\frac{k+1}{2}-i}}}=\prod_{1\leqslant
i<j\leqslant n}\frac{x(\lambda_{i}-i)-x(\lambda_{j}-j)}{x(-i)-x(-j)}.$ (78)

On one hand, the sum in Theorem 3.19 is readily identified with a sum over
lozenge tilings, where $\lambda_{i}^{(k)}-i$ encode the positions of
horizontal lozenges and the weights match (69) after a change of notations; by
particle-hole involution it also gives rise to (70), as we explained in the
proof of Theorem 3.14. On the other hand, the proof777An alternative proof is
hidden in [19, Appendix]. A possibility of our present proof can be hinted by
[10, Theorem 7.5.1]. of Theorem 3.19 can be obtained by combining the $q=t$
version of Theorem 3.16 with known explicit evaluations of Koornwinder
polynomials at principal specialization; we detail this in Appendix A.

Theorem 3.19 has an interesting corollary of representation-theoretic flavor:

######  Corollary 3.21.

Take a signature
${\bm{\lambda}}=(\lambda_{1}\geqslant\lambda_{2}\geqslant\dots\geqslant\lambda_{n}\geqslant
0)$. We have888$\mathrm{Dim}_{SO(2n)}(\lambda_{1},\dots,\lambda_{n})$ is the
dimension of the irreducible representation of group $SO(2n)$ spelled out in
the same notation at
https://en.wikipedia.org/wiki/Representations_of_classical_Lie_groups or in
[20, Section 2.1].

$\sum_{\emptyset={\bm{\lambda}}^{(0)}\prec{\bm{\lambda}}^{(1)}\prec\dots\prec{\bm{\lambda}}^{(n)}={\bm{\lambda}}}\prod_{k=1}^{n-1}\prod_{i=1}^{k}\frac{\lambda^{(k)}_{i}+\frac{n+k}{2}-i}{\frac{n+k}{2}-i}\\\
=\prod_{1\leqslant i<j\leqslant
n}\frac{\lambda_{i}-i-(\lambda_{j}-j)}{j-i}\cdot\prod_{1\leqslant i<j\leqslant
n}\frac{2n+\lambda_{i}-i+\lambda_{j}-j}{2n-j-i}=\mathrm{Dim}_{SO(2n)}(\lambda_{1},\dots,\lambda_{n}).$
(79)

###### Proof.

Take a signature
${\bm{\lambda}}=(\lambda_{1}\geqslant\lambda_{2}\geqslant\dots\geqslant\lambda_{n})\in\mathbb{GT}_{n}$.
We claim that for any $\mathfrak{s}$,

$\sum_{\emptyset={\bm{\lambda}}^{(0)}\prec{\bm{\lambda}}^{(1)}\prec\dots\prec{\bm{\lambda}}^{(n)}={\bm{\lambda}}}\prod_{k=1}^{n-1}\prod_{i=1}^{k}\frac{\mathfrak{s}+\lambda^{(k)}_{i}+\frac{k+1}{2}-i}{\mathfrak{s}+\frac{k+1}{2}-i}\\\
=\prod_{1\leqslant i<j\leqslant
n}\frac{\lambda_{i}-i-(\lambda_{j}-j)}{(j-i)}\cdot\prod_{1\leqslant
i<j\leqslant
n}\frac{2\mathfrak{s}+n+1+\lambda_{i}-i+\lambda_{j}-j}{2\mathfrak{s}+n+1-j-i}.$
(80)

Indeed, notice that

$x(a)-x(b)=\frac{1}{\sigma q^{a+\frac{n+1}{2}}}\left(\sigma
q^{a+\frac{n+1}{2}}-\sigma q^{b+\frac{n+1}{2}}\right)\left(\sigma
q^{a+\frac{n+1}{2}}-\frac{1}{\sigma q^{b+\frac{n+1}{2}}}\right).$

Setting $\sigma=q^{\mathfrak{s}}$ and sending $q\to 1$, (78) turns into (80).
In particular, taking $\mathfrak{s}=\frac{n-1}{2}$ in (80) and demanding that
$\lambda_{n}\geqslant 0$, we get (79). ∎

######  Remark 3.22.

Grigori Olshanski noticed that (79) and (80) can be deduced from the
(multiplicity-free) branching rules for restrictions of irreducible
representations of $SO(2n)$ onto the subgroup $SO(2n-1)$, thus, avoiding the
machinery of Koornwinder polynomials. The plan is as follows. We first notice
that (80) for one particular value of $\mathfrak{s}$ implies the same identity
for all $\mathfrak{s}$: indeed, shifting by integer values of $\mathfrak{s}$
we are getting the same statement for shifted $\lambda_{i}$; on the other
hand, we are proving a polynomial identity, hence, its validity at infinitely
many integer shifts of a single $\mathfrak{s}$ implies the validity at all
$\mathfrak{s}$.

Next, we are checking (80) by induction in $n$ for a particular value of
$\mathfrak{s}$. This check is reduced to the statement that the dimension of
an irreducible representation of $SO(2n)$ is the sum of dimensions of
irreducible representations of $SO(2n-1)$ into which it splits.

It would be interesting to produce a direct representation-theoretic proof of
the identity (79).

##  4 Analysis of Dynamical Loop Equation

###  4.1 Statement of the asymptotic theorem

The aim of present Section 4 is to analyze the transition probability
$\mathbb{P}({\bm{y}}={\bm{x}}+{\bm{e}}|{\bm{x}})=a_{\bm{e}}$ of (4):

$\displaystyle a_{\bm{e}}=\frac{1}{Z({\bm{x}})}\prod_{1\leqslant i<j\leqslant
n}\frac{b(x_{i}+\theta e_{i})-b(x_{j}+\theta
e_{j})}{b(x_{i})-b(x_{j})}\prod_{i=1}^{n}\phi^{+}(x_{i})^{e_{i}}\phi^{-}(x_{i})^{1-e_{i}},\quad{\bm{e}}\in\\{0,1\\}^{n},$
(81)

where ${\bm{x}}\in\mathbb{W}_{\theta}^{n}$, as defined in (2); in particular,
$x_{i}\geqslant x_{i+1}+\theta$ for $i=1,2,\dots,n-1$. The prefactor
$\frac{1}{Z({\bm{x}})}$ is chosen so that $\sum_{{\bm{e}}}a_{\bm{e}}=1$. We
encode the particle configurations ${\bm{x}}$, ${\bm{y}}$ in (81) by their
(smoothed) empirical densities:

$\displaystyle\rho(s;{\bm{x}})=\frac{1}{\theta}\sum_{i=1}^{n}\bm{1}(\varepsilon
x_{i}\leqslant s\leqslant\varepsilon
x_{i}+\varepsilon\theta),\qquad\rho(s;{\bm{y}})=\frac{1}{\theta}\sum_{i=1}^{n}\bm{1}(\varepsilon
y_{i}\leqslant s\leqslant\varepsilon y_{i}+\varepsilon\theta).$ (82)

We impose several conditions on the asymptotic behavior of various ingredients
of (81) as $n\to\infty$:

######  Assumption 4.1.

Fix constants ${\mathsf{N}}>0$ and $\mathbf{l},\mathbf{r}$. We introduce a
small parameter $\varepsilon={\mathsf{N}}/n\ll 1$, and assume the particle
configuration ${\bm{x}}=(x_{1},x_{2},\dots,x_{n})\in\mathbb{W}_{\theta}^{n}$
satisfies $\mathbf{l}\leqslant\varepsilon x_{n}\leqslant\varepsilon
x_{1}\leqslant\mathbf{r}-\varepsilon$. We further assume

  1. 1.

There exists an open complex neighborhood $\Lambda$ of
$[\mathbf{l},\mathbf{r}]$, such that for $z=\varepsilon\xi$,

$\displaystyle b(\xi)={\mathsf{b}}(z)\quad\text{ and
}\quad\phi^{\pm}(\xi)={\varphi}^{\pm}(z),\quad z\in\Lambda.$ (83)

where ${\mathsf{b}}(z)$ and $\varphi^{\pm}(z)$ are holomorphic functions on
$\Lambda$. In addition, we require ${\mathsf{b}}(z)$ to be conformal
(injective and biholomorphic) and
${\mathsf{b}}(\bar{z})=\overline{{\mathsf{b}}(z)}$.

  2. 2.

The functions ${\mathsf{b}}(z)$, $[\partial_{z}{\mathsf{b}}(z)]^{-1}$, and
$\varphi^{\pm}(z)$ are uniformly bounded, namely there exists a universal999By
“universal” we mean not depending on $n$, $\varepsilon$, or
$(x_{1},\dots,x_{n})$. constant $C>0$ such that

$\displaystyle|{\mathsf{b}}(z)|\leqslant
C,\quad|\partial_{z}{\mathsf{b}}(z)|\geqslant 1/C,\
\quad|\varphi^{\pm}(z)|\leqslant C,\quad z\in\Lambda.$ (84)

######  Remark 4.2.

In the rest of this section, we view ${\mathsf{N}},\mathbf{l},\mathbf{r},C$ in
Assumption 4.1 as fixed constants. On the other hand, in our applications the
functions ${\mathsf{b}},\varphi^{\pm}$ might depend on $\varepsilon$.

######  Remark 4.3.

Injectivity of ${\mathsf{b}}$ implies that it is a (real) monotone function on
$[\mathbf{l},\mathbf{r}]$. Without loss of generality, throughout this section
we silently assume that ${\mathsf{b}}(x)$ is an increasing function of
$x\in[\mathbf{l},\mathbf{r}]$.

Throughout this section we analyze (81) in two separate situations:

  1. 1.

Probabilistic case. The weight functions $\varphi^{\pm}$ in (83) are real
analytic and positive on $(\mathbf{l},\mathbf{r})$:

$\varphi^{\pm}(\bar{z})=\overline{\varphi^{\pm}(z)},\quad
z\in\Lambda;\quad\text{ and }\quad\varphi^{\pm}(x)>0,\quad
x\in(\mathbf{l},\mathbf{r}).$ (85)

In this situation (81) is a probability measure.

  2. 2.

Complex case. The weight functions $\varphi^{\pm}$ might be complex, but we
additionally assume

$\displaystyle\sum_{\bm{e}}|a_{\bm{e}}|\leqslant C,$ (86)

for a universal constant $C$.

While eventually we are only interested in the former case, the latter appears
in the intermediate steps of our analysis: in the proofs we deform the weights
$\phi^{\pm}$ in (81) so that (81) becomes a complex-valued measure.

Our last assumption for the asymptotic analysis of (81) involves the following
functions defined in terms of the empirical density (82) and the functions
$\phi^{\pm}(z)$ of Assumption 4.1:

${\mathcal{B}}(z)={\mathcal{G}}(z)\varphi^{+}(z)+\varphi^{-}(z),\qquad\qquad{\mathcal{G}}(z)=\exp\left[\theta\int_{\mathbf{l}}^{\mathbf{r}}\frac{{\mathsf{b}}^{\prime}(z)\rho(s;{\bm{x}})}{{\mathsf{b}}(z)-{\mathsf{b}}(s)}{\rm
d}s\right].$ (87)

Note that ${\mathcal{B}}(z)$ is a holomorphic function for
$z\in\Lambda\setminus[\mathbf{l},\mathbf{r}]$.

######  Assumption 4.4.

For any compact set
$S\subset\bigl{(}\Lambda\setminus[\mathbf{l},\mathbf{r}]\bigr{)}$ there exists
a small universal constant $c=c(S)>0$ such that for $z\in S$ we have
$c<|{\mathcal{B}}(z)|<c^{-1}$. Moreover, for any closed contour
$\omega\subset\bigl{(}\Lambda\setminus[\mathbf{l},\mathbf{r}]\bigr{)}$, we
have

$\displaystyle\frac{1}{2\pi\mathbf{i}}\oint_{\omega}\frac{\partial_{z}{\mathcal{B}}(z)}{{\mathcal{B}}(z)}{\rm
d}z=0,$ (88)

which implies that there exists a well-defined single-valued branch of the
function $\ln{\mathcal{B}}(z)$ in $\Lambda\setminus[\mathbf{l},\mathbf{r}]$.

The last assumption plays the same role in our asymptotic analysis as the non-
criticality condition in the analysis of the (discrete) $\beta$-ensembles, cf.
[18, 23, 24, 7, 25]. Here is our central asymptotic result:

######  Theorem 4.5.

Consider transition probability (81) with parameters satisfying Assumptions
4.1 and 4.4 for all small enough $\varepsilon$ and in the probabilistic case
(85). Then for any $z\in\Lambda\setminus[\mathbf{l},\mathbf{r}]$ we have as
$\varepsilon\to 0$:

$\displaystyle\frac{1}{\varepsilon}\int_{\mathbf{l}}^{\mathbf{r}}\frac{{\mathsf{b}}^{\prime}(z)(\rho(s;{\bm{y}})-\rho(s;{\bm{x}}))}{{\mathsf{b}}(z)-{\mathsf{b}}(s)}{\rm
d}s$
$\displaystyle=\frac{1}{2\pi\mathbf{i}\theta}\oint_{{\omega_{-}}}\frac{\ln{\mathcal{B}}(w){\mathsf{b}}^{\prime}(w){\mathsf{b}}^{\prime}(z){\rm
d}w}{({\mathsf{b}}(w)-{\mathsf{b}}(z))^{2}}$
$\displaystyle+\frac{\varepsilon}{2\pi\mathbf{i}\theta}\oint_{{\omega_{-}}}{\mathcal{E}}^{(2)}(w,z)\frac{{\mathsf{b}}^{\prime}(w){\mathsf{b}}^{\prime}(z){\rm
d}w}{({\mathsf{b}}(w)-{\mathsf{b}}(z))^{2}}\,+\Delta{\mathcal{M}}(z)\,+\,\operatorname{O}\left(\varepsilon^{2}\right),$
(89)

where the contour ${\omega_{-}}\subset\Lambda$ encloses
$[\mathbf{l},\mathbf{r}]$, but not $z$, and

$\displaystyle{\mathcal{E}}^{(2)}(w,z)$
$\displaystyle=\frac{\varphi^{+}(w){\mathcal{G}}(w)}{{\mathcal{B}}(w)}\Biggl{(}\frac{\theta{\mathsf{b}}^{\prime}(w)}{{\mathsf{b}}(z)-{\mathsf{b}}(w)}+\frac{\theta}{2\pi\mathbf{i}}\oint_{\omega_{-}^{\prime}}\frac{\ln{\mathcal{B}}(u){\mathsf{b}}^{\prime}(w){\mathsf{b}}^{\prime}(u){\rm
d}u}{({\mathsf{b}}(u)-{\mathsf{b}}(w))^{2}}$
$\displaystyle+\frac{\theta^{2}}{2}\int_{\mathbf{l}}^{\mathbf{r}}\frac{{\mathsf{b}}^{\prime\prime}(w)({\mathsf{b}}(w)-{\mathsf{b}}(s))-({\mathsf{b}}^{\prime}(w))^{2}-{\mathsf{b}}^{\prime}(w){\mathsf{b}}^{\prime}(s)}{({\mathsf{b}}(w)-{\mathsf{b}}(s))^{2}}\rho(s;{\bm{x}}){\rm
d}s\Biggr{)}-\frac{\partial_{w}{\mathcal{B}}(w)}{2{\mathcal{B}}(w)},$

where the contour $\omega^{\prime}_{-}\subset\Lambda$ encloses
$[\mathbf{l},\mathbf{r}]$, but not $w$.

Moreover, $\Delta{\mathcal{M}}(z)$ are mean $0$ random variables such that
$\\{\varepsilon^{-1/2}\Delta{\mathcal{M}}(z)\\}_{z\in\Lambda\setminus[\mathbf{l},\mathbf{r}]}$
are asymptotically Gaussian with covariance given by

$\displaystyle\mathbb{E}\left[\frac{\Delta{\mathcal{M}}(z_{1})}{\varepsilon^{1/2}}\cdot\frac{\Delta{\mathcal{M}}(z_{2})}{\varepsilon^{1/2}}\right]=$
$\displaystyle\frac{1}{2\pi\mathbf{i}\theta}\oint_{{\omega_{-}}}\frac{{\mathcal{G}}(w)\varphi^{+}(w)}{{\mathcal{B}}(w)}\frac{{\mathsf{b}}^{\prime}(w){\mathsf{b}}^{\prime}(z_{1})}{({\mathsf{b}}(w)-{\mathsf{b}}(z_{1}))^{2}}\frac{{\mathsf{b}}^{\prime}(w){\mathsf{b}}^{\prime}(z_{2})}{({\mathsf{b}}(w)-{\mathsf{b}}(z_{2}))^{2}}{\rm
d}w$ $\displaystyle+o(1),$ (90)

where the contour ${\omega_{-}}\subset\Lambda$ encloses
$[\mathbf{l},\mathbf{r}]$, but not $z_{1},z_{2}$. The higher order joint
moments of
$\\{\varepsilon^{-1/2}\Delta{\mathcal{M}}(z)\\}_{z\in\Lambda\setminus[\mathbf{l},\mathbf{r}]}$
converge as $\varepsilon\to 0$ to the Gaussian joint moments.

The implicit constants in the $\operatorname{O}(\varepsilon^{2})$, in $o(1)$,
and in higher order joint moments are uniform in all the involved parameters
and particle configurations ${\bm{x}}$ satisfying Assumption 4.1 and 4.4, as
long as the constants $C$ and $c$ of Assumptions 4.1 and 4.4 are fixed and $z$
(or $z_{1},z_{2}$) belongs to a compact subset of
$\Lambda\setminus[\mathbf{l},\mathbf{r}]$.

######  Remark 4.6.

In the rest of this paper, when we write a quantity
${\mathcal{E}}(z;{\bm{x}})$ is uniformly small, it refers to that for any
compact subset $F\subset\Lambda\setminus[\mathbf{l},\mathbf{r}]$

$\displaystyle\lim_{\varepsilon\rightarrow 0}\sup_{\begin{smallmatrix}z\in
F,\\\ \text{${\bm{x}}$ satisfying Assumption \ref{a:asymp} and
\ref{a:stable}}\end{smallmatrix}}{\mathcal{E}}(z;{\bm{x}})\rightarrow 0,$ (91)

as long as the constants $C$ and $c$ of Assumptions 4.1 and 4.4 are fixed.

######  Remark 4.7.

When we apply Theorem 4.5 in our study of random lozenge tilings, the exact
expression for ${\mathcal{E}}^{(2)}(w,z)$ is never used. However, our method
of the proof of Theorem 4.5 provides this expression and, hence, we decided to
include it for potential future applications.

######  Remark 4.8.

Dividing (4.5) by $b^{\prime}(z)$ and changing the variables $u=b(z)$, we can
equivalently rewrite it as

$\displaystyle\frac{1}{\varepsilon}\int_{\mathbf{l}}^{\mathbf{r}}\frac{\rho(s;{\bm{y}})-\rho(s;{\bm{x}})}{u-{\mathsf{b}}(s)}{\rm
d}s=\frac{1}{2\pi\mathbf{i}\theta}\oint_{{\omega_{-}}}\frac{\ln{\mathcal{B}}(w){\mathsf{b}}^{\prime}(w){\rm
d}w}{({\mathsf{b}}(w)-u)^{2}}\,$
$\displaystyle+\,\frac{\varepsilon}{2\pi\mathbf{i}\theta}\oint_{{\omega_{-}}}{\mathcal{E}}^{(2)}(w,z)\frac{{\mathsf{b}}^{\prime}(w){\rm
d}w}{({\mathsf{b}}(w)-u)^{2}}\,$
$\displaystyle+\Delta{\mathcal{M}}^{\prime}(u)\,+\,\operatorname{O}\left(\varepsilon^{2}\right),$
(92)

with an appropriately covariance for the Gaussian process
${\mathcal{M}}^{\prime}(u)$. One advantage of the form (4.8) is that we can
take $u$ in it to be an arbitrary complex number outside
$[{\mathsf{b}}(\mathbf{l}),{\mathsf{b}}(\mathbf{r})]$; see also Remarks 4.12
and 4.15.

In general words, Theorem 4.5 shows that, conditional on ${\bm{x}}$, the
empirical density of the particle configuration ${\bm{y}}={\bm{x}}+{\bm{e}}$
(distributed with weights (81)) consists of two parts: a computable
deterministic part and a random fluctuation, which is asymptotically a
Gaussian field. However, instead of directly studying the difference of the
empirical densities, we study the difference of their centered and modified
Stieltjes transforms:

$\displaystyle\int_{\mathbf{l}}^{\mathbf{r}}\frac{{\mathsf{b}}^{\prime}(z)(\rho(s;{\bm{y}})-\rho(s;{\bm{x}}))}{{\mathsf{b}}(z)-{\mathsf{b}}(s)}{\rm
d}s.$ (93)

We remark that (93) is different from the standard definition of Stieltjes
transform. However, for any function $g(z)$ which is analytic in a
neighborhood of $[\mathbf{l},\mathbf{r}]$, we have

$\displaystyle\frac{1}{2\pi\mathbf{i}}\oint_{\omega}g(z)\left[\int_{\mathbf{l}}^{\mathbf{r}}\frac{{\mathsf{b}}^{\prime}(z)\rho(s;{\bm{y}})}{{\mathsf{b}}(z)-{\mathsf{b}}(s)}{\rm
d}s\right]{\rm d}z=\int_{\mathbf{l}}^{\mathbf{r}}g(s)\rho(s;{\bm{y}}){\rm
d}s,$ (94)

where $\omega$ is a contour enclosing $[\mathbf{l},\mathbf{r}]$. Hence,
Theorem 4.5 can be used to deduce the asymptotic behavior of the pairings of
$\rho(s;{\bm{y}})$ with arbitrary analytic test functions.

The rest of this section is devoted to the proof of Theorem 4.5. For a reader
familiar with the arguments in [18], some parts of the proof might look
similar, so let us emphasize two important differences. First, as a starting
point in [18] a concentration bound on the empirical measure was used; this
bound was produced by a separate method unrelated to the loop equations. In
our present approach such a bound is no longer needed and we rely on the
dynamical loop equation and nothing else. Second, an asymptotic expansion of
the discrete loop equations in [18] was directly leading to the expectation of
the centered Stieltjes transform, which could then be used to identify the
first term in an analogue of (4.5). This is different for our dynamic loop
equations: instead the expectation of exponentiated Stieltjes transform
appears. There is no direct connection between $\mathbb{E}\xi$ and
$\mathbb{E}\exp(\xi)$ for a random variable $\xi$, hence, this new feature
leads to a significant difference in the argument.

###  4.2 Plan of the proof for Theorem 4.5

In this subsection we present the general scheme for the proof of Theorem 4.5
omitting technical details which are going to be reconstructed in the
following subsections. We rely on the dynamical loop equation of Theorem 1.1,
which states that the expression

$\displaystyle\begin{split}{\mathcal{C}}(z):=\mathbb{E}\left[\varphi^{+}(z)\prod_{i=1}^{n}\frac{{\mathsf{b}}(z+\varepsilon\theta)-{\mathsf{b}}(\varepsilon
x_{i}+\varepsilon\theta e_{i})}{{\mathsf{b}}(z)-{\mathsf{b}}(\varepsilon
x_{i})}+\varphi^{-}(z)\prod_{i=1}^{n}\frac{{\mathsf{b}}(z)-{\mathsf{b}}(\varepsilon
x_{i}+\varepsilon\theta e_{i})}{{\mathsf{b}}(z)-{\mathsf{b}}(\varepsilon
x_{i})}\right]\end{split}$ (95)

is a holomorphic function of $z\in\Lambda$.

Step 1. We use (95) to analyze the following quantity (which directly appears
in its second term):

${\mathcal{A}}(z):=\mathbb{E}\left[\prod_{i=1}^{n}\frac{{\mathsf{b}}(z)-{\mathsf{b}}(\varepsilon
x_{i}+\varepsilon\theta e_{i})}{{\mathsf{b}}(z)-{\mathsf{b}}(\varepsilon
x_{i})}\right].$ (96)

We remark that under Assumption 4.1, ${\mathcal{A}}(z)$ is analytic on
$\Lambda\setminus[\mathbf{l},\mathbf{r}]$. This quantity encodes the
information about the measure $\rho(s;{\bm{y}})-\rho(s;{\bm{x}})$, which we
want to understand. Our approach is to use holomorphicity of (95) to express
${\mathcal{A}}(z)$ explicitly in terms of contour integrals of
${\mathcal{B}}(z)$ defined in (87). For that we expand ${\mathcal{C}}(z)$ as

$\displaystyle{\mathcal{C}}(z)={\mathcal{A}}(z){\mathcal{B}}(z)+\operatorname{O}(\varepsilon).$
(97)

Note that in (97) we know explicitly ${\mathcal{B}}(z)$, and in addition we
know that ${\mathcal{C}}(z)$ is holomorphic (but we do not have any formulas
for ${\mathcal{C}}(z)$). A priory, this does not give enough information to
extract anything about ${\mathcal{A}}(z)$, but the situation changes when we
impose Assumption 4.4. More precisely, adopting Assumption 4.4, we can
decompose ${\mathcal{B}}(z)$ into two parts

$\displaystyle{\mathcal{B}}(z)=e^{h_{+}({\mathsf{b}}(z))}e^{-h_{-}({\mathsf{b}}(z))},$
(98)

where $h_{+}({\mathsf{b}}(z))$ is analytic in a neighborhood of
$[\mathbf{l},\mathbf{r}]$, and $h_{-}(u)$ is analytic in a neighborhood of
$u=\infty$ with $h_{-}(u)=\operatorname{O}(1/u)$ as $u$ approaches $\infty$.
(98) is a version of the Wiener-Hopf decomposition, adapted to our setting. In
more detail, computing the difference of the integrals as the residue at
$w=z$, we have

$\displaystyle\ln{\mathcal{B}}(z)=\frac{1}{2\pi\mathbf{i}}\oint_{\omega_{+}}\frac{\ln{\mathcal{B}}(w){\mathsf{b}}^{\prime}(w){\rm
d}w}{{\mathsf{b}}(w)-{\mathsf{b}}(z)}-\frac{1}{2\pi\mathbf{i}}\oint_{\omega_{-}}\frac{\ln{\mathcal{B}}(w){\mathsf{b}}^{\prime}(w){\rm
d}w}{{\mathsf{b}}(w)-{\mathsf{b}}(z)},$ (99)

where a positively oriented contour $\omega_{+}\subset\Lambda$ encloses
$[\mathbf{l},\mathbf{r}]$ and $z$; a positively oriented contour
$\omega_{-}\subset\Lambda$ encloses $[\mathbf{l},\mathbf{r}]$ but not $z$. We
can rewrite (99) as an expansion (98) with

$h_{+}(u):=\frac{1}{2\pi\mathbf{i}}\oint_{\omega_{+}}\frac{\ln{\mathcal{B}}(w){\mathsf{b}}^{\prime}(w){\rm
d}w}{{\mathsf{b}}(w)-u},\qquad
h_{-}(u):=\frac{1}{2\pi\mathbf{i}}\oint_{\omega_{-}}\frac{\ln{\mathcal{B}}(w){\mathsf{b}}^{\prime}(w){\rm
d}w}{{\mathsf{b}}(w)-u}.$ (100)

Note that the particular choices of $\omega_{\pm}$ are not important in (99),
as long as $z$ is inside $\omega_{+}$ and outside $\omega_{-}$, while
$[\mathbf{l},\mathbf{r}]$ is inside both contours. However, in order for the