# Noncolliding Macdonald walks with an absorbing wall

Leonid Petrov

###### Abstract

The branching rule is one of the most fundamental properties of the Macdonald
symmetric polynomials. It expresses a Macdonald polynomial as a nonnegative
linear combination of Macdonald polynomials with smaller number of variables.
Taking a limit of the branching rule under the principal specialization when
the number of variables goes to infinity, we obtain a Markov chain of $m$
noncolliding particles with negative drift and an absorbing wall at zero. The
chain depends on the Macdonald parameters $(q,t)$ and may be viewed as a
discrete deformation of the Dyson Brownian motion. The trajectory of the
Markov chain is equivalent to a certain Gibbs ensemble of plane partitions
with an arbitrary cascade front wall.

In the Jack limit $q=t^{\beta/2}\to 1$ the absorbing wall disappears, and the
Macdonald noncolliding walks turn into the $\beta$-noncolliding random walks
studied by Huang [huang2021beta]. Taking $q=0$ (Hall–Littlewood degeneration)
and further sending $t\to 1$, we obtain a continuous time particle system on
$\mathbb{Z}_{\geq 0}$ with inhomogeneous jump rates and absorbing wall at
zero.

##  1 Introduction

###  1.1 Overview

The Dyson Brownian motion [dyson1962brownian] is a continuous stochastic
dynamics of $N$ particles on the one-dimensional line $\mathbb{R}$. The
particles evolve according to independent Brownian motions which are
conditioned to never collide. The noncolliding property may be also modeled as
Coulomb repelling. The Dyson Brownian motion arises as the dynamics of
eigenvalues from the standard Brownian motion on the space of complex
Hermitian matrices. As such, it has been heavily utilized towards universality
results for random matrix spectra [Johansson2001Universality],
[AndersonGuionnetZeitouniBook], [LY_RMT_Bull2011].

Within Integrable Probability, a number of discrete deformations of the Dyson
Brownian motion were introduced, starting from noncolliding Poisson and
Bernoulli random walks [konig2002non] (based on a classical formula of
[KMG59-Coincidence]) and followed by their Macdonald deformation depending on
two parameters $(q,t)$ which is defined in [BorodinCorwin2011Macdonald]. A
notable special case of the latter considered in [GorinShkolnikov2014] is the
Jack limit $t=q^{\beta/2}\to 1$, where $\beta>0$ is the beta parameter from
Random Matrix Theory. In the Jack limit, one obtains $\beta$-noncolliding
Poisson random walks (further studied in [huang2021beta]), and also their
multilevel versions. A scaling limit of the latter leads to the multilevel
Dyson Brownian motion with the general $\beta$ parameter.

Each of the known discrete deformations of the Dyson Brownian motion is
powered by the Cauchy summation identity for some family of symmetric
polynomials $\left\\{P_{\lambda}\right\\}$ such as Schur (for noncolliding
Poisson and Bernoulli walks), Jack, or Macdonald polynomials. Here
$\left\\{P_{\lambda}\right\\}$ is one of these families of polynomials in $N$
variables. The $P_{\lambda}$’s form a linear basis in the space of symmetric
polynomials in $N$ variables as $\lambda$ runs over partitions
$\lambda=(\lambda_{1}\geq\ldots\geq\lambda_{N}\geq 0)$,
$\lambda_{i}\in\mathbb{Z}$, with $N$ parts.

The Cauchy identity is a fundamental property of many families of symmetric
polynomials, and is closely tied to their orthogonality with respect to a
suitable inner product. It provides a product-form expression for the sum
$\sum_{\lambda}b_{\lambda}P_{\lambda}(x_{1},\ldots,x_{N})P_{\lambda}(y_{1},\ldots,y_{N})$,
where $b_{\lambda}$ are certain explicit coefficients. In stochastic dynamics
of $N$ noncolliding particles, this Cauchy identity implies the normalization
to one property of the transition probability, which also involves an $N$-fold
summation over partitions $\lambda$.

Along with the Cauchy identity, most families of symmetric polynomials satisfy
a branching rule. This identity expresses $P_{\lambda}(x_{1},\ldots,x_{N})$ in
$N$ variables as a nonnegative linear combination of polynomials
$P_{\mu}(x_{1},\ldots,x_{N-1})$ with $N-1$ variables, where the sum runs over
$\mu$. For particular symmetric polynomials, such an expansion has clear
representation-theoretic meaning. For example, for Schur polynomials the
branching rule is behind the decomposition of a given irreducible
representation of the unitary group $U(N)$ when restricted to the subgroup
$U(N-1)$.

Note that the branching rule is often dual to the Pieri rule expressing the
product $f\hskip 1.0ptP_{\lambda}$ (for a special choice of $f$ like
$x_{1}+\ldots+x_{N}$) as a linear combination of $P_{\nu}$’s in the same
number of variables. In the present paper we do not explicitly use this
duality, and keep the branching rule perspective.

The goal of the present work is to construct and explore noncolliding random
walks arising from the branching rule instead of the Cauchy identity. We start
at the level of Macdonald polynomials with parameters $(q,t)\in(0,1)^{2}$, and
take a limit of the branching rule under the principal specialization
$(x_{1},\ldots,x_{N})=(1,t,\ldots,t^{N-1})$ as the number of variables $N$
goes to infinity. Using the resulting summation identity (formulated in
Theorem 1.1 later in the Introduction), we define a new discrete-time Markov
process $\Upsilon_{m}$ of $m$ distinct ordered particles in $\mathbb{Z}_{\geq
0}$ with negative drift and absorbing wall at zero (where
$m\in\mathbb{Z}_{\geq 1}$ is assumed fixed). The presence of the wall means
that the process almost surely reaches its only absorbing state
$(m-1,m-2,\ldots,2,1,0)$.

Trajectories of $\Upsilon_{m}$ may be identified with lozenge tilings or plane
partitions with certain explicit boundary conditions depending on the initial
configuration in $\Upsilon_{m}$. We show that the probability measure on plane
partitions coming from $\Upsilon_{m}$ has a Gibbs characterization via the so-
called Boltzmann factors which are ratios of probability weights of two
configurations differing by an elementary transformation. We explicitly
compute these Boltzmann factors in the general Macdonald case. In the
particular case $t=q$, the Gibbs probability weight of a plane partition is
proportional simply to $q^{\mathsf{vol}}$, where $\mathsf{vol}$ is the sum of
the entries of the plane partition. See Section 5 in the text for details.

We consider a number of degenerations of our process $\Upsilon_{m}$ leading to
known deformations of the Dyson Brownian motion mentioned above. All these
degenerations correspond to specializing the parameters $(q,t)$ in such a way
that the Macdonald polynomials turn into another well-known family of
symmetric polynomials:

  * •

(Schur polynomials) Setting $t=q$, we get a simpler Markov process
$\Upsilon_{m}^{\mathrm{Schur}}$ of $m$ noncolliding particles on
$\mathbb{Z}_{\geq 0}$ with an absorbing wall at $0$. To the best of the
author’s knowledge, this process and the underlying normalization identity
(stating that the quantities (4.1) below sum to $1$) are also new. The process
$\Upsilon_{m}^{\mathrm{Schur}}$ looks similar to the translation invariant
$q$-noncolliding random walks on $\mathbb{Z}$ introduced and studied in
[BG2011non]. The normalization of transition probabilities in the latter
process can be traced back to the Cauchy identity. However, it does not seem
that our process $\Upsilon_{m}^{\mathrm{Schur}}$ can be scaled to that of
[BG2011non].

  * •

(Jack polynomials) Take $t=q^{\beta/2}\to 1$ (where $\beta>0$ is the parameter
coming from Random Matrix Theory), and simultaneously scale the coordinates of
the process $\Upsilon_{m}$ away from $0$. In this way we get a dynamics of $m$
particles on $\mathbb{Z}$ which is invariant under space translations of the
particles. This dynamics is closely related to the $\beta$-noncolliding
Poisson random walks studied in [GorinShkolnikov2014], [huang2021beta]. Thus,
we see that our new Macdonald noncolliding walks $\Upsilon_{m}$ generalize all
known noncolliding processes at the Jack level (with general random matrix
$\beta$ parameter). In the particular case $\beta=2$, we recover the Bernoulli
and Poisson walks conditioned to never collide which were studied in
[konig2002non]. Under Brownian scaling, it is known that the latter random
walks turn into the classical Dyson Brownian motion coming from Hermitian
random matrices.

  * •

(Hall–Littlewood polynomials) Setting $q=0$, further sending $t\to 1$ and
taking a Poisson-type limit from discrete time to continuous, we arrive at a
new particle system $\Upsilon_{m}^{\mathrm{cont}}$ on $\mathbb{Z}_{\geq 0}$
with an absorbing wall at zero which evolves as follows. To each particle
$\mathsf{x}_{1}>\ldots>\mathsf{x}_{m}\geq 0$ we assign an independent
exponential clock of rate $i(\mathsf{x}_{i}-\mathsf{x}_{i+1}-1)$, where, by
agreement, $\mathsf{x}_{m+1}=-1$. When the clock of $\mathsf{x}_{i}$ rings, we
additionally select an index $j\in\left\\{1,\ldots,i\right\\}$ uniformly at
random, and all the particles
$\mathsf{x}_{i},\mathsf{x}_{i-1},\ldots,\mathsf{x}_{j}$ simultaneously jump to
the left by $1$. The process $\Upsilon_{m}^{\mathrm{cont}}$ almost surely
reaches its absorbing state $(m-1,\ldots,1,0)$. A more detailed investigation
of this particle system will be performed elsewhere.

In the next Section 1.2 we describe in detail our most general Markov
processes $\Upsilon_{m}$ arising at the Macdonald level.

###  1.2 Macdonald noncolliding walks

Throughout the paper we assume that $(q,t)$ are real numbers belonging to
$(0,1)$. We need some notation. Recall that the $q$-Pochhammer symbols are
given by

$(z;q)_{k}\coloneqq(1-z)(1-zq)\ldots(1-zq^{k-1}),\qquad k\in\mathbb{Z}_{\geq
0},$

and $(z;q)_{\infty}\coloneqq\prod_{i=0}^{\infty}(1-zq^{i})$ is a convergent
infinite product because $|q|<1$.

For $\vec{\mathsf{x}}=(\mathsf{x}_{1}>\ldots\mathsf{x}_{m}\geq 0)$, denote the
$(t,q)$-deformed Vandermonde product by

$V_{t,q}(\vec{\mathsf{x}}):=\prod_{1\leq i<j\leq
m}\frac{\left(q^{j-i-1}t^{\mathsf{x}_{i}-\mathsf{x}_{j}-j+i+1};t\right)_{\infty}}{(q^{j-i}t^{\mathsf{x}_{i}-\mathsf{x}_{j}-j+i+1};t)_{\infty}}.$
(1.1)

When $t=q\to 1$, $V_{t,q}$ turns into the usual Vandermonde
$V(\vec{\mathsf{x}})=\prod_{1\leq i<j\leq m}(\mathsf{x}_{i}-\mathsf{x}_{j})$.
Moreover, when $\mathsf{x}_{i}=\mathsf{x}_{i+1}$ for some $i$, one readily
sees that $V_{t,q}(\vec{\mathsf{x}})$ vanishes.

Let $\mathsf{y}=(\mathsf{y}_{1}>\ldots\mathsf{y}_{m}\geq 0)$ be such that

$\mathsf{y}_{i}-\mathsf{x}_{i}\in\left\\{-1,0\right\\}\qquad\textnormal{for
all $1\leq i\leq m$}.$ (1.2)

Now we can define the main object of the present paper:

$\begin{split}&\Upsilon_{m}(\vec{\mathsf{x}},\vec{\mathsf{y}})\coloneqq
t^{-\binom{m}{2}}\hskip
1.0pt\frac{V_{t,q}(\vec{\mathsf{y}})}{V_{t,q}(\vec{\mathsf{x}})}\prod_{\begin{subarray}{c}1\leq
i<j\leq m\\\
\mathsf{y}_{i}=\mathsf{x}_{i},\,\mathsf{y}_{j}=\mathsf{x}_{j}-1\end{subarray}}\frac{(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}+1}q^{j-i-1})(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}}q^{j-i+1})}{(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}+1}q^{j-i})(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}}q^{j-i})}\\\
&\hskip
240.0pt\times\prod_{i\colon\mathsf{y}_{i}=\mathsf{x}_{i}}t^{\mathsf{x}_{i}}\prod_{i\colon\mathsf{y}_{i}=\mathsf{x}_{i}-1}(t^{m-i}-q^{m-i}t^{\mathsf{x}_{i}}).\end{split}$
(1.3)

From (1.2) one readily sees that the infinite products in
${V_{t,q}(\vec{\mathsf{y}})}/{V_{t,q}(\vec{\mathsf{x}})}$ cancel out in such a
way that (1.3) is always a rational function of $q,t$. Moreover, for $0<q,t<1$
the quantities (1.3) are nonnegative. One of our main results is the sum to
one identity for the $\Upsilon_{m}$’s:

######  Theorem 1.1.

With the above notation, for any
$\vec{\mathsf{x}}=(\mathsf{x}_{1}>\ldots\mathsf{x}_{m}\geq 0)$ we have

$\sum_{\begin{subarray}{c}\vec{\mathsf{y}}=(\mathsf{y}_{1}>\ldots\mathsf{y}_{m}\geq
0)\\\ \mathsf{y}_{i}-\mathsf{x}_{i}\in\left\\{-1,0\right\\}\ \textnormal{for
all $1\leq i\leq
m$}\end{subarray}}\Upsilon_{m}(\vec{\mathsf{x}},\vec{\mathsf{y}})=1.$

Theorem 1.1 implies that the quantities
$\Upsilon_{m}(\vec{\mathsf{x}},\vec{\mathsf{y}})$ may be viewed as transition
probabilities of a discrete time Markov chain of $m$ ordered distinct
particles on $\mathbb{Z}_{\geq 0}$ in which at each step, each particle either
stays, or moves to the left by $1$. Eventually with probability $1$ the chain
reaches the absorbing state $(m-1,m-2,\ldots,2,1,0)$. We call the Markov chain
$\Upsilon_{m}$ the _Macdonald noncolliding walks_ with an absorbing wall at
zero.

We prove Theorem 1.1 in Section 3 by obtaining the transition probabilities
$\Upsilon_{m}(\vec{\mathsf{x}},\vec{\mathsf{y}})$ as a limit of certain ratios
of Macdonald polynomials evaluated at the principal specializations
$(1,t,t^{2},\ldots,t^{N-1})$, as the number of variables goes to infinity. The
fact that before the limit these ratios sum to one is equivalent to the
branching rule for the Macdonald polynomials.

###  1.3 Outline

In Section 2 we review the definition of Macdonald symmetric polynomials
together with all the required formulas. In Section 3 we perform the main
limit transition, and obtain the Macdonald noncolliding walks $\Upsilon_{m}$.
In Section 4 we consider various degenerations of our dynamics when the
Macdonald parameters $(q,t)$ are specialized in a certain way. More precisely,
we look at the dynamics at $t=q$ (when the Macdonald polynomials reduce to the
Schur polynomials), as $q=t^{\alpha}\to 1$ (reduction to the Jack
polynomials), and as $q=0$ (when Macdonald polynomials become the
Hall–Littlewood polynomials). Moreover, in the latter case we see that sending
$t\to 1$ leads to a new continuous time Markov chain on $m$ particles with
inhomogeneous jump rates. In Section 5 we give a Gibbs characterization of the
probability measure on the space of trajectories of our noncolliding walks by
means of the so-called Boltzmann factors which are ratios of probability
weights of two trajectories differing by an elementary transformation.

###  1.4 Acknowledgments

I am grateful to Alexei Borodin, Grigori Olshanski, and Mikhail Tikhonov for
fruitful discussions. The work was partially supported by the NSF grant
DMS-1664617, and the Simons Collaboration Grant for Mathematicians 709055.
This material is based upon work supported by the National Science Foundation
under Grant No. DMS-1928930 while LP participated in the program “Universality
and Integrability in Random Matrix Theory and Interacting Particle Systems”
hosted by the Mathematical Sciences Research institute in Berkeley,
California, during the Fall 2021 semester.

##  2 Review of Macdonald polynomials

Here we collect the necessary notation and results around Macdonald symmetric
polynomials. We follow [Macdonald1995, Chapter VI].

###  2.1 Definition

Let $N\geq 1$. Macdonald symmetric polynomials $P_{\lambda}$ in $N$ variables
$x_{1},\ldots,x_{N}$ are indexed by partitions
$\lambda=(\lambda_{1}\geq\ldots\geq\lambda_{N}\geq 0)$,
$\lambda_{i}\in\mathbb{Z}$, with $N$ parts. Denote the set of these partitions
by $\mathbb{Y}(N)$. The $P_{\lambda}$’s depend on two parameters
$q,t\in[0,1)$. For each fixed $(q,t)$, they form a basis in the space of
symmetric polynomials in $N$ variables when $\lambda$ runs over
$\mathbb{Y}(N)$. The shortest definition of the $P_{\lambda}$’s is through the
first Macdonald $q$-difference operator acting in the $x_{i}$’s:

$D_{1}\coloneqq\sum_{i=1}^{N}\biggl{(}\prod_{1\leq j\leq N\colon j\neq
i}\frac{x_{j}-tx_{i}}{x_{j}-x_{i}}\biggr{)}T_{q;x_{i}},\qquad
T_{q;x}f(x)\coloneqq f(qx).$

The operator $D_{1}$ preserves the space of symmetric polynomials in
$x_{1},\ldots,x_{N}$, and its eigenfunctions are the Macdonald polynomials:

$D_{1}P_{\lambda}(x_{1},\ldots,x_{N}\mid
q,t)=\bigl{(}q^{\lambda_{1}}t^{N-1}+q^{\lambda_{2}}t^{N-2}+\ldots+q^{\lambda_{N}}\bigr{)}P_{\lambda}(x_{1},\ldots,x_{N}\mid
q,t),\qquad\lambda\in\mathbb{Y}(N).$ (2.1)

For generic $(q,t)$, the eigenvalues in (2.1) for different $\lambda$ are
distinct, so the $P_{\lambda}$’s are determined uniquely up to normalization.
The normalization is specified by

$P_{\lambda}(x_{1},\ldots,x_{N}\mid
q,t)=x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\ldots
x_{N}^{\lambda_{N}}+\textnormal{lexicographically lower terms},$

where the lower terms depend on $q,t$.

In the case $q=t$, the polynomials $P_{\lambda}$ reduce to the well-known
Schur symmetric polynomials $s_{\lambda}$, which admit the following explicit
determinantal formula:

$P_{\lambda}(x_{1},\ldots,x_{N}\mid
q,q)=s_{\lambda}(x_{1},\ldots,x_{N})=\frac{\det\bigl{[}x_{i}^{\lambda_{j}+N-j}\bigr{]}_{i,j=1}^{N}}{\det\bigl{[}x_{i}^{N-j}\bigr{]}_{i,j=1}^{N}}.$
(2.2)

For generic $(q,t)$, there are no known formulas for Macdonald polynomials
which are as compact as (2.2).

###  2.2 Principal specialization

Figure 1: Young diagram $\lambda=(5,5,5,2,1,1)$. Highlighted are the arm, leg,
coarm, and coleg of the box $\square=(3,2)$. We have $|\lambda|=19$ and
$a(\square)=3$, $l(\square)=1$, $a^{\prime}(\square)=1$,
$l^{\prime}(\square)=2$.

When the variables $x_{i}$ are specialized to a finite geometric progression
with the ratio $t$ (the second Macdonald parameter), the polynomials
$P_{\lambda}$ admit an explicit product formula in terms of the Young diagram
corresponding to the partition $\lambda$. Recall that the Young diagram
$\lambda$ is a collection of $1\times 1$ boxes in the plane with $\lambda_{i}$
boxes in row $i$, see Figure 1 for an illustration. The principal
specialization takes the form [Macdonald1995, (VI.6.11), (VI.6.11′)]:

$P_{\lambda}(1,t,\ldots,t^{N-1}\mid
q,t)=t^{n(\lambda)}\prod_{\square\in\lambda}\frac{1-q^{a^{\prime}(\square)}t^{N-l^{\prime}(\square)}}{1-q^{a(\square)}t^{l(\square)+1}},$
(2.3)

where the product is over all boxes of the Young diagram $\lambda$,

$n(\lambda):=\sum_{i}(i-1)\lambda_{i},$ (2.4)

and the arms, legs, coarms, colegs of the box $\square=(i,j)$ are defined,
respectively, as

$a(\square)=\lambda_{i}-j,\qquad l(\square)=\lambda_{j}^{\prime}-i,\qquad
a^{\prime}(\square)=j-1,\qquad l^{\prime}(\square)=i-1.$ (2.5)

Here $\lambda_{j}^{\prime}$ are the column lengths in the Young diagram, see
Figure 1.

###  2.3 Branching

Let us first recall the Pieri coefficients for Macdonald polynomials
[Macdonald1995, (VI.6.24.ii)], [BorodinCorwin2011Macdonald, (2.11)]. They
depend on a pair of partitions $\mu,\lambda$ which _interlace_ , namely,

$\lambda_{1}\geq\mu_{1}\geq\lambda_{2}\geq\mu_{2}\geq\ldots$ (2.6)

(notation $\mu\prec\lambda$), and are defined as

$\psi_{\lambda/\mu}=\psi_{\lambda/\mu}(q,t)\coloneqq\prod_{1\leq
i<j\leq\ell(\mu)}\frac{\mathsf{f}(q^{\mu_{i}-\mu_{j}}t^{j-i})\hskip
1.0pt\mathsf{f}(q^{\lambda_{i}-\lambda_{j+1}}t^{j-i})}{\mathsf{f}(q^{\lambda_{i}-\mu_{j}}t^{j-i})\hskip
1.0pt\mathsf{f}(q^{\mu_{i}-\lambda_{j+1}}t^{j-i})},\qquad\mathsf{f}(u):=\frac{(tu;q)_{\infty}}{(qu;q)_{\infty}}.$
(2.7)

Here and below by $\ell(\mu)$ we denote the number of parts in $\mu$ which are
strictly positive, that is, $\mu_{\ell(\mu)}>0$, $\mu_{\ell(\mu)+1}=0$. Let
also $|\lambda|=\lambda_{1}+\ldots+\lambda_{\ell(\lambda)}$ denote the number
of boxes in the Young diagram $\lambda$.

######  Proposition 2.1 ([Macdonald1995, (VI.7.13’)]).

Let $\lambda\in\mathbb{Y}(N)$. We have

$P_{\lambda}(x_{1},\ldots,x_{N}\mid
q,t)=\sum_{\mu\colon\mu\prec\lambda}\psi_{\lambda/\mu}(q,t)\hskip
1.0ptP_{\mu}(x_{1},\ldots,x_{N-1}\mid q,t)\hskip
1.0ptx_{N}^{|\lambda|-|\mu|},$ (2.8)

where the sum is over $\mu\in\mathbb{Y}(N-1)$ which interlace with $\lambda$.

Note that for $q,t\in[0,1)$ the coefficients $\psi_{\lambda/\mu}$ (2.7) are
all nonnegative. Together with Proposition 2.1 this implies:

######  Corollary 2.2.

Let $q,t\in[0,1)$. Specializing the variables $x_{1},\ldots,x_{N}$ into
nonnegative real numbers makes the Macdonald polynomial
$P_{\lambda}(x_{1},\ldots,x_{N}\mid q,t)$ nonnegative.

More generally, for $N\geq K\geq 1$ define the skew Macdonald polynomials
$P_{\lambda/\mu}$ as the coefficients in the expansion:

$P_{\lambda}(x_{1},\ldots,x_{N}\mid
q,t)=\sum_{\mu\in\mathbb{Y}(K)}P_{\mu}(x_{1},\ldots,x_{K}\mid
q,t)P_{\lambda/\mu}(x_{K+1},\ldots,x_{N-1},x_{N}\mid q,t).$

Here we use the fact that the polynomials $P_{\mu}(x_{1},\ldots,x_{K})$ form a
basis in the space of symmetric functions in $K$ variables, and expand
$P_{\lambda}(x_{1},\ldots,x_{N})$ in this basis. The Pieri coefficient is
related to the skew Macdonald polynomial in one variable as follows:

$P_{\lambda/\mu}(x_{1}\mid q,t)=\begin{cases}\psi_{\lambda/\mu}(q,t)\hskip
1.0ptx_{1}^{|\lambda|-|\mu|},&\mu\prec\lambda;\\\
0,&\textnormal{otherwise}.\end{cases}$ (2.9)

Later we will also use the dual Pieri coefficients
$\psi_{\lambda/\mu}^{\prime}$ which are defined as [Macdonald1995,
(VI.6.24.iv)], [BorodinCorwin2011Macdonald, (2.12)]

$\psi_{\lambda/\mu}^{\prime}=\psi_{\lambda/\mu}^{\prime}(q,t)\coloneqq\prod_{\begin{subarray}{c}i<j\\\
\lambda_{i}=\mu_{i},\,\lambda_{j}=\mu_{j}+1\end{subarray}}\frac{(1-q^{\mu_{i}-\mu_{j}}t^{j-i-1})(1-q^{\lambda_{i}-\lambda_{j}}t^{j-i+1})}{(1-q^{\mu_{i}-\mu_{j}}t^{j-i})(1-q^{\lambda_{i}-\lambda_{j}}t^{j-i})}.$
(2.10)

for partitions $\lambda,\mu$ whose column lengths interlace, that is,
$\lambda_{1}^{\prime}\geq\mu_{1}^{\prime}\geq\lambda_{2}^{\prime}\geq\mu_{2}^{\prime}\geq\ldots$.
We have

$P_{\lambda^{\prime}/\mu^{\prime}}(x_{1}\mid
t,q)=\begin{cases}\psi^{\prime}_{\lambda/\mu}(q,t)\hskip
1.0ptx_{1}^{|\lambda|-|\mu|},&\mu^{\prime}\prec\lambda^{\prime};\\\
0,&\textnormal{otherwise}.\end{cases}$ (2.11)

###  2.4 Cauchy identity

Along with the branching rule, another fundamental identity for Macdonald
polynomials is the Cauchy identity. Here we present its dual version, see
[Macdonald1995, Chapter VI.4] for the usual version.

######  Proposition 2.3 ([Macdonald1995, (VI.5.4) and Chapter VI.7]).

Let $N,M\geq 1$ be fixed. We have

$\sum_{\lambda\in\mathbb{Y}(N)\colon\lambda_{1}\leq
M}P_{\lambda}(x_{1},\ldots,x_{N}\mid q,t)\hskip
1.0ptP_{\lambda^{\prime}}(y_{1},\ldots,y_{M}\mid
t,q)=\prod_{i=1}^{N}\prod_{j=1}^{M}(1+x_{i}y_{j}).$ (2.12)

Moreover, for any $\mu\in\mathbb{Y}(N)$ we have the following particular case
of the skew Cauchy identity:

$P_{\mu}(x_{1},\ldots,x_{N}\mid
q,t)\prod_{i=1}^{N}(1+x_{i}y)=\sum_{\lambda\in\mathbb{Y}(N)\colon\mu^{\prime}\prec\lambda^{\prime}}P_{\lambda^{\prime}/\mu^{\prime}}(y\mid
t,q)\hskip 1.0ptP_{\lambda}(x_{1},\ldots,x_{N}\mid q,t).$ (2.13)

Identities (2.8) and (2.13) look very similar, but note that in the former we
sum over the smaller partition, while in the latter one we sum over the larger
partition.

###  2.5 Markov kernels from Macdonald polynomials

Let us first recall a general definition from [borodin2016representations,
Chapter 7]. Let $\mathfrak{X},\mathfrak{Y}$ be finite or countable sets. By a
Markov kernel (or a link) from $\mathfrak{X}$ to $\mathfrak{Y}$, we mean a
function $P$ on $\mathfrak{X}\times\mathfrak{Y}$ such that $P(x,y)\in[0,1]$
for all $x\in\mathfrak{X}$, $y\in\mathfrak{Y}$, and

$\sum_{y\in\mathfrak{Y}}P(x,y)=1\qquad\textnormal{for all
$x\in\mathfrak{X}$}.$

We adopt the notation $P\colon\mathfrak{X}\dashrightarrow\mathfrak{Y}$.

Normalizing identities (2.8) and (2.13) with nonnegative variables $x_{i}$ and
$y$ (following [BorFerr2008DF], [Borodin2010Schur] in the Schur case and
[BorodinCorwin2011Macdonald] in the general Macdonald case) leads to the
following two families of links:

$\displaystyle\Lambda^{N}_{N-1}\colon\mathbb{Y}(N)\dashrightarrow\mathbb{Y}(N-1),\qquad\Lambda^{N}_{N-1}(\lambda,\mu)\coloneqq\psi_{\lambda/\mu}(q,t)\hskip
1.0ptx_{N}^{|\lambda|-|\mu|}\hskip 1.0pt\frac{P_{\mu}(x_{1},\ldots,x_{N-1}\mid
q,t)}{P_{\lambda}(x_{1},\ldots,x_{N}\mid q,t)}\,\mathbf{1}_{\mu\prec\lambda};$
(2.14) $\displaystyle
Q_{N}\colon\mathbb{Y}(N)\dashrightarrow\mathbb{Y}(N),\qquad\qquad\quad
Q_{N}(\lambda,\nu)\coloneqq\frac{\psi^{\prime}_{\nu/\lambda}(q,t)\hskip
1.0pty^{|\nu|-|\lambda|}}{\prod_{i=1}^{N}(1+x_{i}y)}\frac{P_{\nu}(x_{1},\ldots,x_{N}\mid
q,t)}{P_{\lambda}(x_{1},\ldots,x_{N}\mid
q,t)}\,\mathbf{1}_{\lambda^{\prime}\prec\nu^{\prime}}.$ (2.15)

Here and throughout the paper by $\mathbf{1}_{A}$ we denote the indicator of
the event (or condition) $A$. For the link (2.15) we also used (2.11).

The links (2.14)–(2.15) satisfy the following intertwining relation:

$Q_{N}\Lambda^{N}_{N-1}=\Lambda^{N}_{N-1}Q_{N-1},\qquad\sum_{\nu\in\mathbb{Y}(N)}Q_{N}(\lambda,\nu)\Lambda^{N}_{N-1}(\nu,\varkappa)=\sum_{\mu\in\mathbb{Y}(N-1)}\Lambda^{N}_{N-1}(\lambda,\mu)Q_{N-1}(\mu,\varkappa),$
(2.16)

where the second identity holds for all $\lambda\in\mathbb{Y}(N)$,
$\varkappa\in\mathbb{Y}(N-1)$, and is simply a more detailed rewriting of the
first one. Intertwining relation (2.16) follows from the skew Cauchy identity,
for example, see [BorodinCorwin2011Macdonald, Proposition 2.3.1].

The Markov chain on $\mathbb{Y}(N)$ defined by the operator $Q_{N}$ is
traditionally viewed as the Macdonald deformation of the Dyson Brownian
motion, see, for example, [BorFerr2008DF], [GorinShkolnikov2012] for the Schur
$q=t$ case, and also [GorinShkolnikov2014] for the general $\beta$ version
based on Jack symmetric polynomials. For the classical $\beta=2$ Dyson
Brownian motion (coming from the Gaussian Unitary Ensemble of random
matrices), intertwining relations were investigated in [warren2005dyson].

In this paper we take a limit of the links $\Lambda^{N}_{N-1}$ as $N\to\infty$
to construct a new Markov process $\Upsilon_{m}$ of $m$ noncolliding particles
depending on the Macdonald parameters $(q,t)$. This process, too, may be
viewed as another Macdonald deformation of the Dyson Brownian motion (in
particular, our process also admits a diffusive scaling to the Dyson Brownian
motion). However, in contrast with $Q_{N}$, the noncolliding particles under
$\Upsilon_{m}$ have negative drift and evolve in the presence of an absorbing
wall at zero.

##  3 Limit transition to noncolliding walks

In this section we perform the limit transition as $N\to+\infty$ in the Markov
kernels $\Lambda^{N}_{N-1}$ (2.14) under the principal specialization (2.3),
and prove Theorem 1.1.

###  3.1 Setup

Denote by $\mathbb{W}_{m}$ the space of $m$-particle configurations in
$\mathbb{Z}_{\geq 0}$, that is,

$\mathbb{W}_{m}\coloneqq\\{\vec{\mathsf{x}}=(\mathsf{x}_{1}>\mathsf{x}_{2}>\ldots>\mathsf{x}_{m}\geq
0)\\}\subset\mathbb{Z}_{\geq 0}^{m}.$ (3.1)

By $\mathbb{W}_{m}(N)$ denote the finite subset of $\mathbb{W}_{m}$ determined
by the condition $\mathsf{x}_{1}\leq N+m-2$. Define the injective maps

$\pi\colon\mathbb{W}_{m}(N)\to\mathbb{Y}(N),\qquad\overline{\pi}\colon\mathbb{W}_{m}(N)\to\mathbb{Y}(N-1),$
(3.2)

as follows. If $\lambda=\pi(\vec{\mathsf{x}})\in\mathbb{Y}(N)$ and
$\mu=\overline{\pi}(\vec{\mathsf{y}})\in\mathbb{Y}(N-1)$, then

$\begin{split}\left\\{\lambda_{1}-1,\lambda_{2}-2,\ldots,\lambda_{N}-N\right\\}&=\left\\{0,1,2,\ldots,N+m-1\right\\}\setminus\\{\mathsf{x}_{1},\ldots,\mathsf{x}_{m}\\};\\\
\left\\{\mu_{1}-1,\mu_{2}-2\ldots,\mu_{N-1}-(N-1)\right\\}&=\left\\{1,2,\ldots,N+m-1\right\\}\setminus\\{\mathsf{y}_{1}+1,\ldots,\mathsf{y}_{m}+1\\}.\end{split}$
(3.3)

For fixed $\vec{\mathsf{x}}$ and growing $N$, almost all parts of
$\lambda=\pi(\vec{\mathsf{x}})$ are equal to $N+m$, and there is a defect in a
few last parts of $\lambda$. The columns of this defect are encoded through
$\vec{\mathsf{x}}$. A similar description holds for
$\mu=\overline{\pi}(\vec{\mathsf{y}})$. In multiplicative notation for
partitions, we have

$\begin{split}\lambda=\pi(\vec{\mathsf{x}})&=(N+m)^{N-\mathsf{x}_{1}+m-1}(N+m-1)^{\mathsf{x}_{1}-\mathsf{x}_{2}-1}\ldots(N+1)^{\mathsf{x}_{m-1}-\mathsf{x}_{m}-1}N^{\mathsf{x}_{m}},\\\
\mu=\overline{\pi}(\vec{\mathsf{y}})&=(N+m)^{N-\mathsf{y}_{1}+m-2}(N+m-1)^{\mathsf{y}_{1}-\mathsf{y}_{2}-1}\ldots(N+1)^{\mathsf{y}_{m-1}-\mathsf{y}_{m}-1}N^{\mathsf{y}_{m}}.\end{split}$
(3.4)

See Figure 2 for an illustration.

Figure 2: Left: Young diagram of $\pi(\vec{\mathsf{x}})$ defined by
(3.2)–(3.3). The column lengths of the defect are $\mathsf{x}_{j}-(m-j)$.
Right: Young diagram of $\overline{\pi}(\vec{\mathsf{y}})$.

######  Lemma 3.1.

For any $\vec{\mathsf{x}},\vec{\mathsf{y}}\in\mathbb{W}_{m}(N)$, we have
$\overline{\pi}(\vec{\mathsf{y}})\prec\pi(\vec{\mathsf{x}})$ if and only if
$\mathsf{y}_{i}=\mathsf{x}_{i}$ or $\mathsf{y}_{i}=\mathsf{x}_{i}-1$ for all
$i=1,\ldots,m$.

###### Proof.

Straightforward verification. ∎

In the rest of this section we fix arbitrary
$\vec{\mathsf{x}},\vec{\mathsf{y}}\in\mathbb{W}_{m}$ and compute the limit of
$\Lambda^{N}_{N-1}(\pi(\vec{\mathsf{x}}),\overline{\pi}(\vec{\mathsf{y}}))$
(2.14) under principal specialization $x_{j}=t^{j-1}$ as $N\to+\infty$.
Clearly, for sufficiently large $N$ the partitions
$\pi(\vec{\mathsf{x}}),\overline{\pi}(\vec{\mathsf{y}})$ are well-defined. We
will show that this limit is equal to the Markov kernel
$\Upsilon_{m}(\vec{\mathsf{x}},\vec{\mathsf{y}})$ defined by (1.3).

Throughout the computation we adopt the convention that
$\lambda=\pi(\vec{\mathsf{x}})$, $\mu=\overline{\pi}(\vec{\mathsf{y}})$.

###  3.2 Initial Markov kernel

Our starting point is the formula for
$\Lambda^{N}_{N-1}(\pi(\vec{\mathsf{x}}),\overline{\pi}(\vec{\mathsf{y}}))$
(2.14) under the principal specialization $x_{j}=t^{j-1}$, which takes the
form

$\displaystyle\Lambda^{N}_{N-1}(\lambda,\mu)$
$\displaystyle=\psi_{\lambda/\mu}(q,t)\hskip
1.0ptx_{N}^{|\lambda|-|\mu|}\hskip 1.0pt\frac{P_{\mu}(x_{1},\ldots,x_{N-1}\mid
q,t)}{P_{\lambda}(x_{1},\ldots,x_{N}\mid q,t)}$
$\displaystyle=\psi_{\lambda/\mu}(q,t)\hskip
1.0ptt^{(N-1)(|\lambda|-|\mu|)}\hskip
1.0pt\frac{P_{\mu}(1,t,\ldots,t^{N-2}\mid
q,t)}{P_{\lambda}(1,t,\ldots,t^{N-2},t^{N-1}\mid q,t)}$
$\displaystyle=t^{(N-1)(|\lambda|-|\mu|)}\prod_{1\leq i<j\leq
N-1}\frac{\mathsf{f}(q^{\mu_{i}-\mu_{j}}t^{j-i})\hskip
1.0pt\mathsf{f}(q^{\lambda_{i}-\lambda_{j+1}}t^{j-i})}{\mathsf{f}(q^{\lambda_{i}-\mu_{j}}t^{j-i})\hskip
1.0pt\mathsf{f}(q^{\mu_{i}-\lambda_{j+1}}t^{j-i})}$ $\displaystyle\qquad\times
t^{n(\mu)}\prod_{\square\in\mu}\frac{1-q^{a^{\prime}(\square)}t^{N-1-l^{\prime}(\square)}}{1-q^{a(\square)}t^{l(\square)+1}}t^{-n(\lambda)}\prod_{\square\in\lambda}\frac{1-q^{a(\square)}t^{l(\square)+1}}{1-q^{a^{\prime}(\square)}t^{N-l^{\prime}(\square)}}.$
(3.5)

Here we used (2.3) and (2.7). Our next steps are devoted to taking the limit
as $N\to+\infty$ in various parts of the product (3.5).

###  3.3 Power of $t$

Let us first consider the overall power of $t$ in (3.5) which is equal to
$t^{(N-1)(|\lambda|-|\mu|)+n(\mu)-n(\lambda)}$. Our aim is to express
quantities depending on $\lambda,\mu$ through
$\vec{\mathsf{x}},\vec{\mathsf{y}}$. Adopt the convention
$\mathsf{x}_{0}=N+m$, $\mathsf{y}_{0}=N+m-1$,
$\mathsf{x}_{m+1}=\mathsf{y}_{m+1}=-1$, so that

$N+m-\mathsf{x}_{1}-1=\mathsf{x}_{0}-\mathsf{x}_{1}-1,\qquad
N+m-\mathsf{y}_{1}-2=\mathsf{y}_{0}-\mathsf{y}_{1}-1.$

We have from (3.4):

$\displaystyle|\lambda|-|\mu|=\sum_{i=0}^{m}(N+i)\left(\left(\mathsf{x}_{m-i}-\mathsf{x}_{m-i+1}-1\right)-\left(\mathsf{y}_{m-i}-\mathsf{y}_{m-i+1}-1\right)\right)$
$\displaystyle\hskip
10.0pt=\sum_{i=0}^{m}(N+i)\left(\left(\mathsf{x}_{m-i}-\mathsf{x}_{m-i+1}\right)-\left(\mathsf{y}_{m-i}-\mathsf{y}_{m-i+1}\right)\right)$
$\displaystyle\hskip
10.0pt=N\sum_{i=0}^{m}\left(\mathsf{x}_{m-i}-\mathsf{x}_{m-i+1}\right)-\left(\mathsf{y}_{m-i}-\mathsf{y}_{m-i+1}\right)+\sum_{i=0}^{m}i\left(\left(\mathsf{x}_{m-i}-\mathsf{x}_{m-i+1}\right)-\left(\mathsf{y}_{m-i}-\mathsf{y}_{m-i+1}\right)\right)$
$\displaystyle\hskip 10.0pt=N+m+|\vec{\mathsf{y}}|-|\vec{\mathsf{x}}|.$

Moreover, from (2.4) and (3.4) we have (using our conventions about
$\mathsf{x}_{0},\mathsf{y}_{0},\mathsf{x}_{m+1},\mathsf{y}_{m+1}$):

$\displaystyle n(\mu)-n(\lambda)$
$\displaystyle=\sum_{j=0}^{m}(N+m-j)\left[\binom{\mathsf{y}_{0}-\mathsf{y}_{j+1}-(j+1)}{2}-\binom{\mathsf{y}_{0}-\mathsf{y}_{j}-j}{2}\right]$
$\displaystyle\hskip
50.0pt-\sum_{j=0}^{m}(N+m-j)\left[\binom{\mathsf{x}_{0}-\mathsf{x}_{j+1}-(j+1)}{2}-\binom{\mathsf{x}_{0}-\mathsf{x}_{j}-j}{2}\right]$
$\displaystyle=-(N-1)(N+m)-\binom{m}{2}+|\vec{\mathsf{x}}|+\frac{1}{2}\sum_{i=1}^{m}(\mathsf{y}_{i}-\mathsf{x}_{i})(\mathsf{x}_{i}+\mathsf{y}_{i}+3+2i-2m-2N).$

The $i$-th term in the last sum is rewritten as

$\frac{1}{2}(\mathsf{y}_{i}-\mathsf{x}_{i})(\mathsf{x}_{i}+\mathsf{y}_{i}+3+2i-2m-2N)=-(N-1)(\mathsf{y}_{i}-\mathsf{x}_{i})+\frac{1}{2}(\mathsf{y}_{i}-\mathsf{x}_{i})(-\mathsf{x}_{i}+\mathsf{y}_{i}+1)\\\
+(\mathsf{y}_{i}-\mathsf{x}_{i})(\mathsf{x}_{i}-m+i).$

Since $\mathsf{y}_{i}-\mathsf{x}_{i}$ is equal to $0$ or $-1$, the quantity
$(\mathsf{y}_{i}-\mathsf{x}_{i})(-\mathsf{x}_{i}+\mathsf{y}_{i}+1)$ is
identically zero. Thus, we have

$\displaystyle
n(\mu)-n(\lambda)=-(N-1)(N+m+|\vec{\mathsf{y}}|-|\vec{\mathsf{x}}|)+|\vec{\mathsf{x}}|-\binom{m}{2}+\sum_{i=1}^{m}(\mathsf{x}_{i}-m+i)(\mathsf{y}_{i}-\mathsf{x}_{i}).$

We see that the $N$-dependent terms in
$(N-1)(|\lambda|-|\mu|)+n(\mu)-n(\lambda)$ cancel out, and the overall factor
containing the power of $t$ in
$\Lambda^{N}_{N-1}(\pi(\vec{\mathsf{x}}),\overline{\pi}(\vec{\mathsf{y}}))$
has the form

$t^{-\binom{m}{2}+|\vec{\mathsf{x}}|+\sum_{i=1}^{m}(\mathsf{x}_{i}-m+i)(\mathsf{y}_{i}-\mathsf{x}_{i})}.$
(3.6)

###  3.4 Coarms and colegs

Addressing the factors in (3.5) containing coarms and colegs of $\lambda$ and
$\mu$, we obtain using (2.5):

$\prod_{\square\in\lambda}(1-q^{a^{\prime}(\square)}t^{N-l^{\prime}(\square)})=\prod_{i=1}^{N}\prod_{j=1}^{\lambda_{i}}(1-q^{j-1}t^{N-i+1})=\prod_{i=1}^{N}(t^{N+1-i};q)_{\lambda_{i}}.$

This product is in the denominator, and a similar factor
$\prod_{i=1}^{N-1}(t^{N-i};q)_{\mu_{i}}$ appears in the numerator. Let us show
that the contribution coming from coarms and colegs goes to one, that is,

$\lim_{N\to+\infty}\frac{\prod_{i=1}^{N-1}(t^{N-i};q)_{\mu_{i}}}{\prod_{i=1}^{N}(t^{N+1-i};q)_{\lambda_{i}}}=1.$
(3.7)

To see this, observe that $\lambda_{i+1}-\mu_{i}$ for all $i=1,\ldots,N-1$ is
a nonnegative integer independent of $N$, see (3.4). Therefore,

$\frac{(t^{N-i};q)_{\mu_{i}}}{(t^{N+1-(i+1)},q)_{\lambda_{i+1}}}=\frac{(t^{N-i};q)_{\mu_{i}}}{(t^{N-i},q)_{\lambda_{i+1}}}=\frac{1}{(1-t^{N-i}q^{\mu_{i}})(1-t^{N-i}q^{\mu_{i}+1})\ldots(1-t^{N-i}q^{\lambda_{i+1}-1})},$

where the product in the right-hand side is finite. Since $\mu_{i}$ goes to
infinity as $N\to\infty$, this product converges to $1$. There is one more
factor in the denominator of (3.7), namely,

$\frac{1}{(t^{N};q)_{\lambda_{1}}}=\frac{(t^{N}q^{N+m};q)_{\infty}}{(t^{N};q)_{\infty}}.$

This factor also goes to $1$ as $N\to+\infty$, and so the limit (3.7) is
established.

###  3.5 Arms and legs

We now consider the product in (3.5) involving arms and legs:

$\prod_{\square\in\mu}\frac{1}{1-q^{a(\square)}t^{l(\square)+1}}\prod_{\square\in\lambda}\bigl{(}1-q^{a(\square)}t^{l(\square)+1}\bigr{)}.$
(3.8)

Recall the notation $\lambda=\pi(\vec{\mathsf{x}})$,
$\mu=\overline{\pi}(\vec{\mathsf{y}})$, see (3.4). Observe that the Young
diagrams $\lambda=\pi(\vec{\mathsf{x}})$ and
$\overline{\pi}(\vec{\mathsf{x}})$ differ only by adding the first row (cf.
Figure 1). For each box $\square$ in the first row, the quantity $l(\square)$
is of order $N$, and so $1-q^{a(\square)}t^{l(\square)+1}$ is close to $1$ for
large $N$. Therefore, the product (3.8) has the same limit as

$\prod_{\square\in\overline{\pi}(\vec{\mathsf{y}})}\frac{1}{1-q^{a(\square)}t^{l(\square)+1}}\prod_{\square\in\overline{\pi}(\vec{\mathsf{x}})}\bigl{(}1-q^{a(\square)}t^{l(\square)+1}\bigr{)}.$
(3.9)

######  Proposition 3.2.

As $N\to+\infty$, the product (3.9) converges to

$\frac{V_{t,q}(\vec{\mathsf{y}})}{V_{t,q}(\vec{\mathsf{x}})}\prod_{i=1}^{m}\left(1-q^{m-i}t^{\mathsf{x}_{i}-m+i}\mathbf{1}_{\mathsf{y}_{i}=\mathsf{x}_{i}-1}\right),$
(3.10)

where $V_{t,q}$ is the $(t,q)$-Vandermonde given by (1.1).

###### Proof.

Recall the notation for the $(q,t)$-deformed hook polynomials [Macdonald1995,
VI.(8.1), VI.(8.1’)]:

$c_{\nu}(q,t)=\prod_{\square\in\nu}\bigl{(}1-q^{a(\square)}t^{l(\square)+1}\bigr{)},\qquad
c^{\prime}_{\nu}(q,t)=\prod_{\square\in\nu}\bigl{(}1-q^{a(\square)+1}t^{l(\square)}\bigr{)},$

and observe that $c_{\nu}(q,t)=c^{\prime}_{\nu^{\prime}}(t,q)$, where
$\nu^{\prime}$ is the transposed Young diagram. Note that, in multiplicative
notation,

$\overline{\pi}(\vec{\mathsf{x}})=(N-1)^{N}(N-1-\mathsf{x}_{m})(N-1-\mathsf{x}_{m-1}+1)(N-1-\mathsf{x}_{m-2}+2)\ldots(N-1-\mathsf{x}_{1}+m-1)$

and similarly for $\overline{\pi}(\vec{y})$, see Figure 2. Adopt the
convention $\mathsf{x}_{m+j}=\mathsf{y}_{m+j}=-j$ for $j=1,2,\ldots$. Then we
may shift the indices to encode
$\overline{\pi}(\vec{\mathsf{x}})^{\prime}_{j}=N+j-1-\mathsf{x}_{m-j}$, where
$-N\leq j\leq m-1$, and similarly for $\vec{\mathsf{y}}$. We have using
[kaneko1996q, Proposition 3.2]:

$\displaystyle c_{\overline{\pi}(\vec{\mathsf{x}})}(q,t)$
$\displaystyle=c^{\prime}_{\overline{\pi}(\vec{\mathsf{x}})^{\prime}}(t,q)$
$\displaystyle=\frac{(t;t)_{\infty}^{N+m}}{\prod_{j=-N}^{m-1}(t^{N+j-\mathsf{x}_{m-j}}q^{m-1-j};t)_{\infty}}\prod_{-N\leq
i<j\leq
m-1}\frac{(q^{j-i}t^{i-j-\mathsf{x}_{m-i}+\mathsf{x}_{m-j}+1};t)_{\infty}}{(q^{j-i-1}t^{i-j-\mathsf{x}_{m-i}+\mathsf{x}_{m-j}+1};t)_{\infty}}$
$\displaystyle=\frac{(t;t)_{\infty}^{N+m}}{\prod_{j=-N}^{m-1}(t^{N+j-\mathsf{x}_{m-j}}q^{m-1-j};t)_{\infty}}\prod_{-N\leq
i<j\leq-1}\frac{(q^{j-i}t;t)_{\infty}}{(q^{j-i-1}t;t)_{\infty}}$
$\displaystyle\hskip 50.0pt\times\prod_{\begin{subarray}{c}-N\leq i\leq-1\\\
0\leq j\leq
m-1\end{subarray}}\frac{(q^{j-i}t^{-j+\mathsf{x}_{m-j}+1};t)_{\infty}}{(q^{j-i-1}t^{-j+\mathsf{x}_{m-j}+1};t)_{\infty}}\underbrace{\prod_{1\leq
i<j\leq
m}\frac{(q^{j-i}t^{i-j-\mathsf{x}_{j}+\mathsf{x}_{i}+1};t)_{\infty}}{(q^{j-i-1}t^{i-j-\mathsf{x}_{j}+\mathsf{x}_{i}+1};t)_{\infty}}}_{1/V_{t,q}(\vec{\mathsf{x}})}.$

Therefore, the product (3.9) becomes

$\frac{c_{\overline{\pi}(\vec{\mathsf{x}})}(q,t)}{c_{\overline{\pi}(\vec{\mathsf{y}})}(q,t)}=\frac{V_{t,q}(\vec{\mathsf{y}})}{V_{t,q}(\vec{\mathsf{x}})}\prod_{j=1}^{m}\frac{(t^{N+m-j-\mathsf{y}_{j}}q^{j-1};t)_{\infty}}{(t^{N+m-j-\mathsf{x}_{j}}q^{j-1};t)_{\infty}}\\\
\times\prod_{j=1}^{m}\prod_{i=-N}^{-1}\frac{(q^{m-j-i}t^{j-m+\mathsf{x}_{j}+1};t)_{\infty}}{(q^{m-j-i-1}t^{j-m+\mathsf{x}_{j}+1};t)_{\infty}}\frac{(q^{m-j-i-1}t^{j-m+\mathsf{y}_{j}+1};t)_{\infty}}{(q^{m-j-i}t^{j-m+\mathsf{y}_{j}+1};t)_{\infty}}.$

The first product over $1\leq j\leq m$ converges to $1$ as $N\to+\infty$
thanks to the presence of $t^{N}$. In the second product over $1\leq j\leq m$,
the terms where $\mathsf{y}_{j}=\mathsf{x}_{j}$ are simply equal to $1$. When
$\mathsf{y}_{j}=\mathsf{x}_{j}-1$, we have

$\displaystyle\prod_{i=-N}^{-1}\frac{(q^{m-j-i}t^{j-m+\mathsf{x}_{j}+1};t)_{\infty}}{(q^{m-j-i-1}t^{j-m+\mathsf{x}_{j}+1};t)_{\infty}}\frac{(q^{m-j-i-1}t^{j-m+\mathsf{x}_{j}};t)_{\infty}}{(q^{m-j-i}t^{j-m+\mathsf{x}_{j}};t)_{\infty}}$
$\displaystyle=\prod_{i=1}^{N}\frac{1-q^{m-j+i-1}t^{j-m+\mathsf{x}_{j}}}{1-q^{m-j+i}t^{j-m+\mathsf{x}_{j}}}$
$\displaystyle=\frac{1-q^{m-j}t^{j-m+\mathsf{x}_{j}}}{1-q^{m-j+N}t^{j-m+\mathsf{x}_{j}}},$

and the $N\to+\infty$ limit of this expression is
$1-q^{m-j}t^{\mathsf{x}_{j}-m+j}$. This completes the proof. ∎

###  3.6 Pieri coefficient

It remains to consider the $N\to+\infty$ limit of the Pieri coefficient
$\psi_{\lambda/\mu}(q,t)$ entering (3.5). Recall the convention
$\mathsf{x}_{m+j}=\mathsf{y}_{m+j}=-j$ for $j=1,2,\ldots$, and encode
(shifting the indices)

$\lambda^{\prime}_{i}=\pi(\vec{\mathsf{x}})^{\prime}_{i}=N+i-\mathsf{x}_{m-i},\qquad\mu^{\prime}_{j}=\overline{\pi}(\vec{\mathsf{y}})^{\prime}_{j}=N-1+j-\mathsf{y}_{m-j}.$

We can now use the well-known duality [Macdonald1995, Proof of (6.24)]

$\psi_{\lambda/\mu}(q,t)=\psi^{\prime}_{\lambda^{\prime}/\mu^{\prime}}(t,q),$

and obtain from (2.10):

$\displaystyle\psi^{\prime}_{\lambda^{\prime}/\mu^{\prime}}(t,q)$
$\displaystyle=\prod_{\begin{subarray}{c}-N\leq i<j\leq m-1\\\
\lambda^{\prime}_{i}=\mu^{\prime}_{i},\,\lambda^{\prime}_{j}=\mu^{\prime}_{j}+1\end{subarray}}\frac{(1-t^{\mu^{\prime}_{i}-\mu^{\prime}_{j}}q^{j-i-1})(1-t^{\lambda^{\prime}_{i}-\lambda^{\prime}_{j}}q^{j-i+1})}{(1-t^{\mu^{\prime}_{i}-\mu^{\prime}_{j}}q^{j-i})(1-t^{\lambda^{\prime}_{i}-\lambda^{\prime}_{j}}q^{j-i})}$
$\displaystyle=\prod_{\begin{subarray}{c}-N\leq i<j\leq m-1\\\
\mathsf{y}_{m-i}=\mathsf{x}_{m-i}-1,\,\mathsf{y}_{m-j}=\mathsf{x}_{m-j}\end{subarray}}\frac{(1-t^{i-j+\mathsf{y}_{m-j}-\mathsf{y}_{m-i}}q^{j-i-1})(1-t^{i-j+\mathsf{x}_{m-j}-\mathsf{x}_{m-i}}q^{j-i+1})}{(1-t^{i-j+\mathsf{y}_{m-j}-\mathsf{y}_{m-i}}q^{j-i})(1-t^{i-j+\mathsf{x}_{m-j}-\mathsf{x}_{m-i}}q^{j-i})}.$

Split the product over $i<j$ into three parts. The first part with $-N\leq
i<j\leq-1$ cancels out and is equal to $1$. The second part with $-N\leq
i\leq-1$ and $0\leq j\leq m-1$ is also equal to $1$ since we have
$\mathsf{y}_{m-i}=\mathsf{x}_{m-i}$ for all $-N\leq i\leq-1$. The third part
with $0\leq i<j\leq m-1$ equals

$\prod_{\begin{subarray}{c}1\leq i<j\leq m\\\
\mathsf{y}_{i}=\mathsf{x}_{i},\,\mathsf{y}_{j}=\mathsf{x}_{j}-1\end{subarray}}\frac{(1-t^{i-j+\mathsf{y}_{i}-\mathsf{y}_{j}}q^{j-i-1})(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}}q^{j-i+1})}{(1-t^{i-j+\mathsf{y}_{i}-\mathsf{y}_{j}}q^{j-i})(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}}q^{j-i})},$
(3.11)

which is independent of $N$. Note that (3.11) is equal to a Pieri coefficient
$\psi^{\prime}_{(\vec{\mathsf{x}}-\delta_{m})/(\vec{\mathsf{y}}-\delta_{m})}(t,q)$,
where $\delta_{m}=(m-1,m-2,\ldots,2,1,0)$ is the staircase partition of length
$m$.

###  3.7 Final result

Putting together all the computations from this section, we see that the
$N\to+\infty$ limit of the Markov kernel
$\Lambda^{N}_{N-1}(\pi(\vec{\mathsf{x}}),\overline{\pi}(\vec{\mathsf{y}}))$ is
the Markov kernel (on $\mathbb{W}_{m}$) of the Macdonald noncolliding walks
with the absorbing wall at zero:

$\displaystyle\Upsilon_{m}(\vec{\mathsf{x}},\vec{\mathsf{y}})$
$\displaystyle=t^{-\binom{m}{2}+|\vec{\mathsf{x}}|+\sum_{i=1}^{m}(\mathsf{x}_{i}-m+i)(\mathsf{y}_{i}-\mathsf{x}_{i})}\hskip
1.0pt\frac{V_{t,q}(\vec{\mathsf{y}})}{V_{t,q}(\vec{\mathsf{x}})}\hskip
1.0pt\psi^{\prime}_{(\vec{\mathsf{x}}-\delta_{m})/(\vec{\mathsf{y}}-\delta_{m})}(t,q)\prod_{i=1}^{m}\left(1-q^{m-i}t^{\mathsf{x}_{i}-m+i}\mathbf{1}_{\mathsf{y}_{i}=\mathsf{x}_{i}-1}\right)$
$\displaystyle=t^{-\binom{m}{2}}\hskip 1.0pt\prod_{1\leq i<j\leq
m}\frac{(q^{j-i}t^{\mathsf{x}_{i}-\mathsf{x}_{j}-j+i+1};t)_{\infty}(q^{j-i-1}t^{\mathsf{y}_{i}-\mathsf{y}_{j}-j+i+1};t)_{\infty}}{(q^{j-i-1}t^{\mathsf{x}_{i}-\mathsf{x}_{j}-j+i+1};t)_{\infty}(q^{j-i}t^{\mathsf{y}_{i}-\mathsf{y}_{j}-j+i+1};t)_{\infty}}$
$\displaystyle\hskip 70.0pt\times\prod_{\begin{subarray}{c}1\leq i<j\leq m\\\
\mathsf{y}_{i}=\mathsf{x}_{i},\,\mathsf{y}_{j}=\mathsf{x}_{j}-1\end{subarray}}\frac{(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}+1}q^{j-i-1})(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}}q^{j-i+1})}{(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}+1}q^{j-i})(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}}q^{j-i})}$
(3.12) $\displaystyle\hskip
70.0pt\times\prod_{i\colon\mathsf{y}_{i}=\mathsf{x}_{i}}t^{\mathsf{x}_{i}}\prod_{i\colon\mathsf{y}_{i}=\mathsf{x}_{i}-1}(t^{m-i}-q^{m-i}t^{\mathsf{x}_{i}}).$

The second expression is obtained by rewriting the powers of $t$, and
expanding the notation of $V_{t,q}$ and $\psi^{\prime}$. This completes the
proof of Theorem 1.1.

###  3.8 Absorbing wall

Let us show that $\Upsilon_{m}$ (3.12) indeed possesses an absorbing wall at
zero:

######  Proposition 3.3.

Started from any initial configuration, the process $\Upsilon_{m}$ eventually
reaches the absorbing state $\delta_{m}=(m-1,m-2,\ldots,1,0)$.

###### Proof.

First, observe that if $\mathsf{x}_{m}=0$, then the term
$t^{m-i}-q^{m-i}t^{\mathsf{x}_{m}}$ (present in (3.12) if $\mathsf{y}_{m}=-1$)
vanishes. This means that once the leftmost particle $\mathsf{x}_{m}$ reaches
$0$, it stays there forever. Moreover, this implies that $\delta_{m}$ is
indeed an absorbing state.

Now, if the process does not eventually reach $\delta_{m}$, then by the
monotonicity it will stay an infinite amount of time in some configuration
$\vec{\mathsf{x}}$ with $|\vec{\mathsf{x}}|>|\delta_{m}|=\binom{m}{2}$.
However, this cannot happen with positive probability because $0<t<1$ and

$\Upsilon_{m}(\vec{\mathsf{x}},\vec{\mathsf{x}})=t^{|\vec{\mathsf{x}}|-\binom{m}{2}}<1.$

This completes the proof. ∎

##  4 Degenerations and limits of Macdonald noncolliding walks

In this section we discuss various degenerations of the Macdonald noncolliding
walks $\Upsilon_{m}$.

###  4.1 Schur degeneration

When $q=t$, recall that the Macdonald polynomials turn into the Schur
polynomials (2.2). This degeneration simplifies our Markov chain, too:

######  Proposition 4.1.

When $t=q$, the Macdonald noncolliding walks $\Upsilon_{m}$ (3.12) on
$\mathbb{W}_{m}$ become

$\Upsilon_{m}^{\mathrm{Schur}}(\vec{\mathsf{x}},\vec{\mathsf{y}})=q^{-\binom{m}{2}+(m-1)(|\vec{\mathsf{x}}|-|\vec{\mathsf{y}}|)}\hskip
1.0pt\prod_{1\leq i<j\leq
m}\frac{q^{\mathsf{y}_{j}}-q^{\mathsf{y}_{i}}}{q^{\mathsf{x}_{j}}-q^{\mathsf{x}_{i}}}\prod_{i=1}^{m}\left(q^{\mathsf{x}_{i}}\mathbf{1}_{\mathsf{y}_{i}=\mathsf{x}_{i}}+(1-q^{\mathsf{x}_{i}})\mathbf{1}_{\mathsf{y}_{i}=\mathsf{x}_{i}-1}\right).$
(4.1)

###### Proof.

The right-hand side of (4.1) is a straightforward simplification of (3.12) at
$q=t$. ∎

This Markov chain may be viewed as the Doob $h$-transform (cf. [konig2002non],
[Konig2005]) of $m$ independent random walks on $\mathbb{Z}_{\geq 0}$ with
transition probabilities

$\mathrm{Prob}(k\to k)=q^{k},\qquad\mathrm{Prob}(k\to k-1)=1-q^{k-1},\qquad
k\in\mathbb{Z}_{\geq 0},$

and absorbing wall at zero. The corresponding positive harmonic function for
the Markov transition matrix of $m$ independent walks is

$h(\vec{\mathsf{x}})=q^{-(m-1)|\vec{\mathsf{x}}|}\prod_{1\leq i<j\leq
m}(q^{\mathsf{x}_{i}}-q^{\mathsf{x}_{j}}),\qquad\vec{\mathsf{x}}\in\mathbb{W}_{m},$

and it has the eigenvalue $q^{\binom{m}{2}}$. The statement about the
eigenvalue is equivalent to

$\sum_{\vec{\mathsf{y}}\in\mathbb{W}_{m}}\Upsilon_{m}^{\mathrm{Schur}}(\vec{\mathsf{x}},\vec{\mathsf{y}})=1,$

which follows as a $t=q$ specialization of Theorem 1.1.

The noncolliding walks $\Upsilon_{m}^{\mathrm{Schur}}$ are somewhat similar to
the ones studied in [BG2011non]. Indeed, the latter are obtained from $m$
independent random walks on the whole line $\mathbb{Z}$ by means of the Doob
$h$-transform with the $q$-Vandermonde $\prod_{1\leq i<j\leq
m}(q^{\mathsf{x}_{i}}-q^{\mathsf{x}_{j}})$, and this harmonic function has
eigenvalue $1$. The resulting process studied in [BG2011non] is invariant with
respect to space translations $\vec{\mathsf{x}}\mapsto\vec{\mathsf{x}}+L$
(where $L\in\mathbb{Z}$ is arbitrary). In contrast, our walks are not
translation invariant and live on $\mathbb{W}_{m}\subset\mathbb{Z}_{\geq
0}^{m}$. As time goes to infinity, our process $\Upsilon_{m}^{\mathrm{Schur}}$
is eventually absorbed at $\delta=(m-1,m-2,\ldots,1,0)\in\mathbb{W}_{m}$. It
also seems that our noncolliding walks $\Upsilon_{m}^{\mathrm{Schur}}$ do not
admit a limit (while keeping $q$ fixed) to the process of [BG2011non].

In [BG2011non] it was shown that the space-time distribution of the
noncolliding $q$-dependent random walks on the whole line $\mathbb{Z}$ is a
determinantal point process. The determinantal structure of the process
$\Upsilon_{m}^{\mathrm{Schur}}$ (4.1) will be explored in a forthcoming work.

###  4.2 Jack limit

Fix $\alpha>0$. When $q=t^{\alpha}$ and $t\to 1$, it is known [Macdonald1995,
Chapter VI.10] that Macdonald polynomials turn into Jack symmetric
polynomials. The parameter $\alpha$ is sometimes denoted by $1/\theta$ (in
literature on asymptotic representation theory, for example, [Kerov1998]), and
is related to the parameter $\beta$ in Random Matrix Theory as
$\alpha=2/\beta$.

We aim to take the Jack limit of the Markov chain $\Upsilon_{m}$ (3.12). As
the factors $t^{m-i}-q^{m-i}t^{\mathsf{x}_{i}}$ in $\Upsilon_{m}$ tend to $0$
for fixed $\vec{\mathsf{x}}$, we need to scale $\vec{\mathsf{x}}$ with $t$.
This scaling necessarily moves the process away from the absorbing wall. More
precisely, we take the limit as

$t\to 1,\qquad L\to+\infty,\qquad q=t^{\alpha},\qquad t^{L}\to b\in(0,1),$
(4.2)

and shift $\vec{\mathsf{x}},\vec{\mathsf{y}}$ by $L$ as follows:

$\mathsf{x}_{i}=\mathsf{x}_{i}(L)=\mathsf{X}_{i}+i(\alpha-1)+L,\qquad\mathsf{y}_{i}=\mathsf{y}_{i}(L)=\mathsf{Y}_{i}+i(\alpha-1)+L,$
(4.3)

where

$\mathsf{X}_{i},\mathsf{Y}_{i}\in\mathbb{Z}+i(\alpha-1),\quad
i=1,\ldots,m,\qquad\mathsf{X}_{i}-\mathsf{X}_{i+1}\geq\alpha,\quad\mathsf{Y}_{i}-\mathsf{Y}_{i+1}\geq\alpha.$
(4.4)

The latter inequalities come from the strict ordering of
$\vec{\mathsf{x}},\vec{\mathsf{y}}\in\mathbb{W}_{m}$ (3.1). We also have
$\mathsf{Y}_{i}=\mathsf{X}_{i}$ or $\mathsf{Y}_{i}=\mathsf{X}_{i}-1$ for all
$i=1,\ldots,m$.

######  Proposition 4.2.

Under (4.2)–(4.3), the transition probabilities
$\Upsilon_{m}(\vec{\mathsf{x}}(L),\vec{\mathsf{y}}(L))$ converge to

$\Upsilon_{m}^{\mathrm{Jack}}(\vec{\mathsf{X}},\vec{\mathsf{Y}})\coloneqq
b^{m-|\vec{\mathsf{X}}|+|\vec{\mathsf{Y}}|}(1-b)^{|\vec{\mathsf{X}}|-|\vec{\mathsf{Y}}|}\prod_{1\leq
i<j\leq m}\frac{(\mathsf{X}_{i}-\alpha\hskip
1.0pt\mathbf{1}_{\mathsf{Y}_{i}=\mathsf{X}_{i}-1})-(\mathsf{X}_{j}-\alpha\hskip
1.0pt\mathbf{1}_{\mathsf{Y}_{j}=\mathsf{X}_{j}-1})}{\mathsf{X}_{i}-\mathsf{X}_{j}}.$
(4.5)

###### Proof.

We have from (3.12) for any fixed
$\vec{\mathsf{x}},\vec{\mathsf{y}}\in\mathbb{W}_{m}$:

$\displaystyle\Upsilon_{m}(\vec{\mathsf{x}}+L,\vec{\mathsf{y}}+L)$
$\displaystyle=t^{-\binom{m}{2}}\hskip 1.0pt\prod_{1\leq i<j\leq
m}\frac{(q^{j-i}t^{\mathsf{x}_{i}-\mathsf{x}_{j}-j+i+1};t)_{\infty}(q^{j-i-1}t^{\mathsf{y}_{i}-\mathsf{y}_{j}-j+i+1};t)_{\infty}}{(q^{j-i-1}t^{\mathsf{x}_{i}-\mathsf{x}_{j}-j+i+1};t)_{\infty}(q^{j-i}t^{\mathsf{y}_{i}-\mathsf{y}_{j}-j+i+1};t)_{\infty}}$
$\displaystyle\hskip 70.0pt\times\prod_{\begin{subarray}{c}1\leq i<j\leq m\\\
\mathsf{y}_{i}=\mathsf{x}_{i},\,\mathsf{y}_{j}=\mathsf{x}_{j}-1\end{subarray}}\frac{(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}+1}q^{j-i-1})(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}}q^{j-i+1})}{(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}+1}q^{j-i})(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}}q^{j-i})}$
$\displaystyle\hskip
70.0pt\times\prod_{i\colon\mathsf{y}_{i}=\mathsf{x}_{i}}t^{\mathsf{x}_{i}+L}\prod_{i\colon\mathsf{y}_{i}=\mathsf{x}_{i}-1}(t^{m-i}-q^{m-i}t^{\mathsf{x}_{i}+L}).$

The third line is the only part involving $L$, and it turns into
$b^{m-|\vec{\mathsf{x}}|+|\vec{\mathsf{y}}|}(1-b)^{|\vec{\mathsf{x}}|-|\vec{\mathsf{y}}|}$.
The limits of the other factors are obtained in a standard way, for example,
see [Macdonald1995, Chapter VI.10]. We have for
$\vec{\mathsf{x}},\vec{\mathsf{y}}$ as in (4.3):

$\displaystyle\frac{(q^{j-i}t^{\mathsf{x}_{i}-\mathsf{x}_{j}-j+i+1};t)_{\infty}(q^{j-i-1}t^{\mathsf{y}_{i}-\mathsf{y}_{j}-j+i+1};t)_{\infty}}{(q^{j-i-1}t^{\mathsf{x}_{i}-\mathsf{x}_{j}-j+i+1};t)_{\infty}(q^{j-i}t^{\mathsf{y}_{i}-\mathsf{y}_{j}-j+i+1};t)_{\infty}}$
$\displaystyle\hskip
20.0pt\to\frac{\Gamma(\alpha(j-i-1)+\mathsf{x}_{i}-\mathsf{x}_{j}-j+i+1)\Gamma(\alpha(j-i)+\mathsf{y}_{i}-\mathsf{y}_{j}-j+i+1)}{\Gamma(\alpha(j-i)+\mathsf{x}_{i}-\mathsf{x}_{j}-j+i+1)\Gamma(\alpha(j-i-1)+\mathsf{y}_{i}-\mathsf{y}_{j}-j+i+1)}$
$\displaystyle\hskip
40.0pt=\frac{\Gamma(\mathsf{X}_{i}-\mathsf{X}_{j}+1-\alpha)\Gamma(\mathsf{Y}_{i}-\mathsf{Y}_{j}+1)}{\Gamma(\mathsf{X}_{i}-\mathsf{X}_{j}+1)\Gamma(\mathsf{Y}_{i}-\mathsf{Y}_{j}+1-\alpha)},$

and

$\displaystyle\frac{(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}+1}q^{j-i-1})(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}}q^{j-i+1})}{(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}+1}q^{j-i})(1-t^{i-j+\mathsf{x}_{i}-\mathsf{x}_{j}}q^{j-i})}$
$\displaystyle\hskip
20.0pt\to\frac{(\alpha(j-i-1)+i-j+\mathsf{x}_{i}-\mathsf{x}_{j}+1)(\alpha(j-i+1)+i-j+\mathsf{x}_{i}-\mathsf{x}_{j})}{(\alpha(j-i)+i-j+\mathsf{x}_{i}-\mathsf{x}_{j}+1)(\alpha(j-i)+i-j+\mathsf{x}_{i}-\mathsf{x}_{j}}$
$\displaystyle\hskip
40.0pt=\frac{(\mathsf{X}_{i}-\mathsf{X}_{j}+1-\alpha)(\mathsf{X}_{i}-\mathsf{X}_{j}+\alpha)}{(\mathsf{X}_{i}-\mathsf{X}_{j}+1)(\mathsf{X}_{i}-\mathsf{X}_{j})}.$

One can check by considering four cases depending on
$\mathsf{Y}_{i}-\mathsf{X}_{i}\in\left\\{-1,0\right\\}$,
$\mathsf{Y}_{j}-\mathsf{X}_{j}\in\left\\{-1,0\right\\}$ that the ratio of the
gamma functions simplifies as

$\prod_{1\leq i<j\leq
m}\frac{\Gamma(\mathsf{X}_{i}-\mathsf{X}_{j}+1-\alpha)\Gamma(\mathsf{Y}_{i}-\mathsf{Y}_{j}+1)}{\Gamma(\mathsf{X}_{i}-\mathsf{X}_{j}+1)\Gamma(\mathsf{Y}_{i}-\mathsf{Y}_{j}+1-\alpha)}\prod_{\begin{subarray}{c}1\leq
i<j\leq m\\\
\mathsf{Y}_{i}=\mathsf{X}_{i},\,\mathsf{Y}_{j}=\mathsf{X}_{j}-1\end{subarray}}\frac{(\mathsf{X}_{i}-\mathsf{X}_{j}+1-\alpha)(\mathsf{X}_{i}-\mathsf{X}_{j}+\alpha)}{(\mathsf{X}_{i}-\mathsf{X}_{j}+1)(\mathsf{X}_{i}-\mathsf{X}_{j})}\\\
=\prod_{1\leq i<j\leq m}\frac{(\mathsf{X}_{i}-\alpha\hskip
1.0pt\mathbf{1}_{\mathsf{Y}_{i}=\mathsf{X}_{i}-1})-(\mathsf{X}_{j}-\alpha\hskip
1.0pt\mathbf{1}_{\mathsf{Y}_{j}=\mathsf{X}_{j}-1})}{\mathsf{X}_{i}-\mathsf{X}_{j}},$

which completes the proof. ∎

The $m$-particle Markov chain $\Upsilon_{m}^{\mathrm{Jack}}$ (4.5), where each
particle leaves on its own shifted copy of $\mathbb{Z}$ (see (4.4)), is a
discrete time “Bernoulli” analogue of the $\beta$-nonintersecting Poisson
random walks considered in [huang2021beta]. Indeed, sending $b\to 1$,
rescaling time from discrete to continuous, and reversing the direction of
jumps from left to right turns $\Upsilon_{m}^{\mathrm{Jack}}$ into the
$\beta$-nonintersecting Poisson walks.

When $\alpha=1$ (so the random matrix parameter is $\beta=2$), the process
$\Upsilon_{m}^{\mathrm{Jack}}$ turns into the process of noncolliding
Bernoulli walks conditioned to never collide. The trajectory of this Markov
process started from an arbitrary fixed initial configuration is a
determinantal point process. This structure was utilized in
[GorinPetrov2016universality] to establish local universality.

###  4.3 Hall–Littlewood degeneration and a continuous time limit

Let us now take $q=0$. Under this degeneration, the Macdonald polynomials
become the Hall–Littlewood polynomials [Macdonald1995, Chapter III].

######  Proposition 4.3.

When $q=0$, the Macdonald noncolliding walks $\Upsilon_{m}$ (3.12) on
$\mathbb{W}_{m}$ become

$\begin{split}\Upsilon_{m}^{\mathrm{HL}}(\vec{\mathsf{x}},\vec{\mathsf{y}})&=t^{-\binom{m}{2}}(t^{\mathsf{x}_{m}}\mathbf{1}_{\mathsf{y}_{m}=\mathsf{x}_{m}}+(1-t^{\mathsf{x}_{m}})\mathbf{1}_{\mathsf{y}_{m}=\mathsf{x}_{m}-1})\\\
&\hskip
30.0pt\times\prod_{i=1}^{m-1}(t^{\mathsf{x}_{i}}\mathbf{1}_{\mathsf{y}_{i}=\mathsf{x}_{i}}+t^{m-i}\mathbf{1}_{\mathsf{y}_{i}=\mathsf{x}_{i}-1})(1-t^{\mathsf{x}_{i}-\mathsf{x}_{i+1}-1}\mathbf{1}_{\mathsf{y}_{i}=\mathsf{x}_{i}-1}\mathbf{1}_{\mathsf{y}_{i+1}=\mathsf{x}_{i+1}}).\end{split}$
(4.6)

###### Proof.

A straightforward simplification of (3.12) at $q=0$. ∎

In (4.6), let us send $t=(1-\varepsilon)\nearrow 1$ and scale discrete time by
$\varepsilon$ to continuous time. This would amount to a Poisson-type limit
transition in our Markov chain $\Upsilon_{m}^{\mathrm{HL}}$.

We have

$\Upsilon_{m}^{\mathrm{HL}}(\vec{\mathsf{x}},\vec{\mathsf{x}})=t^{-\binom{m}{2}+|\vec{\mathsf{x}}|}\prod_{i=1}^{m-1}\sim
1-\left(|\vec{\mathsf{x}}|-\tbinom{m}{2}\right)\varepsilon+O(\varepsilon^{2}).$
(4.7)

This implies that as $\varepsilon\to 0$, a single step of the Markov chain
$\Upsilon_{m}^{\mathrm{HL}}$ typically does not change $\vec{\mathsf{x}}$.
However, occasionally, with probability proportional to $\varepsilon$, a
change in $\vec{\mathsf{x}}$ may occur. All probabilities of order
$O(\varepsilon^{2})$ vanish in the scaling limit.

Therefore, a jump in continuous time can happen in the presence of only one
factor in (4.6) proportional to $(1-t^{k})$ for some $k>0$. Such a factor is
associated to a particle $\mathsf{x}_{i}$ which has jumped to the left by $1$
while the particle $\mathsf{x}_{i+1}$ has stayed (if $i=m$, the latter
condition is replaced by $\mathsf{x}_{m}>0$). This leads to the jump rate
$\mathsf{x}_{i}-\mathsf{x}_{i+1}-1$, $i=1,\ldots,m$, where, by agreement,
$\mathsf{x}_{m+1}=-1$. Moreover, if one particle $\mathsf{x}_{i}$,
$i=1,\ldots,m$, jumps to the left by $1$, then any adjacent block of particles
$\mathsf{x}_{i-1},\mathsf{x}_{i-2},\ldots,\mathsf{x}_{i-r}$ can also jump to
the left by $1$, at the same rate. This leads to the following definition of a
continuous time process.

######  Definition 4.4.

Let $\Upsilon_{m}^{\mathrm{cont}}$ be a continuous time Markov process on
$\mathbb{W}_{m}$ (3.1) with jump rates defined as follows. Attach to each
particle $\mathsf{x}_{i}\in\mathbb{Z}_{\geq 0}$, $i=1,\ldots,m$, an
independent exponential clock of rate $i\hskip
1.0pt(\mathsf{x}_{i}-\mathsf{x}_{i+1}-1)$, where, by agreement,
$\mathsf{x}_{m+1}=-1$. When the clock of $\mathsf{x}_{i}$ rings, we
additionally select an index $j\in\left\\{1,\ldots,i\right\\}$ uniformly at
random, and all the particles
$\mathsf{x}_{i},\mathsf{x}_{i-1},\ldots,\mathsf{x}_{j}$ simultaneously jump to
the left by $1$.

The total jump rate from $\vec{\mathsf{x}}$ under the process from Definition
4.4 is equal to

$m\mathsf{x}_{m}+(m-1)(\mathsf{x}_{m-1}-\mathsf{x}_{m}-1)+\ldots+(\mathsf{x}_{1}-\mathsf{x}_{2}-1)=|\vec{\mathsf{x}}|-\tbinom{m}{2},$

which agrees with (4.7).

Therefore, we have established the following Poisson-type limit transition:

######  Proposition 4.5.

Let the Hall–Littlewood parameter $t$ be scaled as $t=1-\varepsilon$, where
$\varepsilon\searrow 0$. Let us scale the discrete time as
$\lfloor\varepsilon^{-1}\tau\rfloor$, where $\tau\in\mathbb{R}_{\geq 0}$ is
the new continuous time parameter. Under this scaling, the Hall–Littlewood
noncolliding walks $\Upsilon_{m}^{\mathrm{HL}}$ (4.6) on $\mathbb{W}_{m}$ turn
into the continuous time Markov process $\Upsilon_{m}^{\mathrm{cont}}$ on
$\mathbb{W}_{m}$ with jump rates given by Definition 4.4 above.

The dynamics $\Upsilon_{m}^{\mathrm{cont}}$ is somewhat similar to the
backwards, inhomogeneous version of the Hammersley process introduced in
[PetrovSaenz2019backTASEP] in that the jump rate attached to each particle
$\mathsf{x}_{i}$ is equal to $i$ times the size of the gap behind
$\mathsf{x}_{i}$. However, the jumping mechanism in
$\Upsilon_{m}^{\mathrm{cont}}$ is very different from the one in the backwards
Hammersley process.

Another observation about $\Upsilon_{m}^{\mathrm{cont}}$ is that it is not
clear how to define the “bulk” version of the dynamics living on the full line
$\mathbb{Z}$ and preserving a translation invariant probability distribution
on $\left\\{0,1\right\\}^{\mathbb{Z}}$. Indeed, the uniformly random selection
of the number of particles which simultaneously jump at each clock ring is not
readily extendable to infinitely many particles on the full line. This
presents an obstacle to hydrodynamic analysis of
$\Upsilon_{m}^{\mathrm{cont}}$, even at an heuristic level.

Figure 3: (a) A trajectory of the process $\Upsilon_{m}$ with $m=4$ with
initial configuration $\vec{\mathsf{x}}=(7,6,3,1)$; (b) A bijective
interpretation of the trajectory as a lozenge tiling (the shaded triangles
correspond to $\mathbb{Z}_{\geq 0}\setminus\vec{\mathsf{x}}$ and are removed
from the strip); (c) The corresponding plane partition of shape
$(4,4,2,1)=(7,6,3,1)-(3,2,1,0)$; (d) Encoding the elements of the plane
partition. It must satisfy $\pi_{41}\geq 0$, $\pi_{32}\geq 1$, $\pi_{24}\geq
3$, and $\pi_{14}\geq 3.$

##  5 Plane partitions

Here we present an interpretation of trajectories of our noncolliding walks as
a certain ensemble of plane partitions with an arbitrary cascade-like front
wall.

###  5.1 Bijection to lozenge tilings and plane partitions

Let $t\in\mathbb{Z}_{\geq 0}$ denote the discrete time in $\Upsilon_{m}$, and
let $\vec{\mathsf{x}}(0)=\vec{\mathsf{x}}\in\mathbb{W}_{m}$ be the initial
configuration of the process. Recall from Proposition 3.3 that the final
configuration of the process $\Upsilon_{m}$ is
$\delta_{m}=(m-1,m-2,\ldots,1,0)$. Via a suitable affine transform, let us
bijectively map the trajectory of $\Upsilon_{m}$ into a lozenge tiling of the
vertical strip of width $\mathsf{x}_{1}+1$, see Figure 3, (a) and (b). The
bottom boundary of the vertical strip is encoded by $\vec{\mathsf{x}}$ in the
following way. Viewing $\vec{\mathsf{x}}$ as a particle configuration in
$\mathbb{Z}_{\geq 0}$, each particle $\mathsf{x}_{i}$ corresponds to a
straight piece in the boundary of slope $(-1/\sqrt{3})$, and each hole
corresponds to cutting a small triangle out of the strip.

Due to the absorption (Proposition 3.3), the lozenge tiling is “frozen” far at
the top, with $\mathsf{x}_{1}+1-m$ tiles of one type on the left followed by
$m$ tiles of the other type. Thus, the lozenge tiling contains only finitely
many horizontal lozenges. Therefore, we may view the tiling as a graph of a
function $\pi_{i,j}$ defined on cells of a Young diagram of size

$\lambda=\vec{\mathsf{x}}-\delta_{m}=(\mathsf{x}_{1}-m+1,\mathsf{x}_{2}-m+2,\ldots,\mathsf{x}_{m-1}-1,\mathsf{x}_{m}),$

see Figure 3, (c) and (d). That is, in $\pi_{i,j}$ we have $1\leq i\leq m$ and
$1\leq j\leq\lambda_{i}$. Since the lozenge tiling cannot have holes inside,
this function must satisfy $\pi_{i,j}\geq\pi_{i+1,j}$ and
$\pi_{i,j}\geq\pi_{i,j+1}$. Such functions are often called plane partitions,
e.g., see, [Stanley1999, Chapter 7].

Moreover, due to the sloped bottom boundary of the lozenge tiling in Figure 3,
(b), the plane partition must also satisfy

$\pi_{i,\lambda_{i}}\geq\lambda_{i}-\mathsf{x}_{m}=(\mathsf{x}_{i}+i)-(\mathsf{x}_{m}+m)\quad\textnormal{for
all $1\leq i\leq m$}.$ (5.1)

This last condition means that the plane partition (or rather the
corresponding lozenge tiling) has a cascade-like front wall. This front wall
is encoded by the $\Upsilon_{m}$’s initial configuration $\vec{\mathsf{x}}$
which may be an arbitrary element of the space $\mathbb{W}_{m}$ (3.1).

###  5.2 Boltzmann factors

Fix $m\geq 1$. For a given fixed initial configuration, the space of possible
trajectories of the Markov process $\Upsilon_{m}$ is countable. Here we give a
different characterization of the probability measure on this space induced by
the transition probabilities $\Upsilon_{m}$ (3.12). Namely, we compute the so-
called Boltzmann factors, that is, the ratios of the probability weights
coming from trajectories related by an elementary transformation. In this way,
the probability of each given trajectory of $\Upsilon_{m}$ is proportional to
the product of the Boltzmann factors associated with this trajectory. Note
that such a product does not depend on the order of elementary transformations
since the result is proportional to the probability (the Gibbs weight) of a
trajectory which we started with.

We need some notation. Fix time $t\in\mathbb{Z}_{\geq 1}$, and let
$\vec{\mathsf{x}}=\vec{\mathsf{x}}(t-1)$,
$\vec{\mathsf{y}}=\vec{\mathsf{x}}(t)$,
$\vec{\mathsf{z}}=\vec{\mathsf{x}}(t+1)$ be three consecutive states of our
Markov chain $\Upsilon_{m}$. Let us also change $\vec{\mathsf{y}}$ in an
elementary way to $\vec{\mathsf{w}}$ such that for some fixed $1\leq k\leq m$:

$\mathsf{w}_{i}=\begin{cases}\mathsf{y}_{i},&i\neq k;\\\
\mathsf{y}_{k}-1,&i=k.\end{cases}$

In the lozenge tiling interpretation of Figure 3, (b), the piece of the
trajectory $\vec{\mathsf{x}}\to\vec{\mathsf{y}}\to\vec{\mathsf{z}}$ differs
from $\vec{\mathsf{x}}\to\vec{\mathsf{w}}\to\vec{\mathsf{z}}$ by moving a
horizontal lozenge down by $1$. In the 3-dimensional interpretation of the
lozenge tiling, this means removing a $1\times 1\times 1$ box from the stack
of boxes. Let us also denote

$\omega(a,b)\coloneqq\frac{(1-t^{a-b+1}q^{b-1})(1-t^{a-b}q^{b+1})}{(1-t^{a-b+1}q^{b})(1-t^{a-b}q^{b})}$

to shorten some of the formulas below.

######  Proposition 5.1.

With the above notation, we have

$\begin{split}\frac{\Upsilon_{m}(\vec{\mathsf{x}},\vec{\mathsf{y}})\Upsilon_{m}(\vec{\mathsf{y}},\vec{\mathsf{z}})}{\Upsilon_{m}(\vec{\mathsf{x}},\vec{\mathsf{w}})\Upsilon_{m}(\vec{\mathsf{w}},\vec{\mathsf{z}})}&=t\hskip
1.0pt\prod_{i=1}^{k-1}\frac{\mathbf{1}_{\mathsf{z}_{i}=\mathsf{y}_{i}-1}+\omega(\mathsf{y}_{i}-\mathsf{y}_{k},k-i)\mathbf{1}_{\mathsf{z}_{i}=\mathsf{y}_{i}}}{\mathbf{1}_{\mathsf{y}_{i}=\mathsf{x}_{i}-1}+\omega(\mathsf{x}_{i}-\mathsf{x}_{k},k-i)\mathbf{1}_{\mathsf{y}_{i}=\mathsf{x}_{i}}}\\\
&\hskip
50.0pt\times\prod_{j=k+1}^{m}\frac{\mathbf{1}_{\mathsf{y}_{j}=\mathsf{x}_{j}}+\omega(\mathsf{x}_{k}-\mathsf{x}_{j},j-k)\mathbf{1}_{\mathsf{y}_{j}=\mathsf{x}_{j}-1}}{\mathbf{1}_{\mathsf{z}_{j}=\mathsf{y}_{j}}+\omega(\mathsf{y}_{k}-\mathsf{y}_{j}-1,j-k)\mathbf{1}_{\mathsf{z}_{j}=\mathsf{y}_{j}-1}}.\end{split}$
(5.2)

###### Proof.

We use the formula

$\Upsilon_{m}(\vec{\mathsf{x}},\vec{\mathsf{y}})=t^{-\binom{m}{2}+|\vec{\mathsf{x}}|+\sum_{i=1}^{m}(\mathsf{x}_{i}-m+i)(\mathsf{y}_{i}-\mathsf{x}_{i})}\hskip
1.0pt\frac{V_{t,q}(\vec{\mathsf{y}})}{V_{t,q}(\vec{\mathsf{x}})}\hskip
1.0pt\psi^{\prime}_{(\vec{\mathsf{x}}-\delta_{m})/(\vec{\mathsf{y}}-\delta_{m})}(t,q)\prod_{i=1}^{m}\left(1-q^{m-i}t^{\mathsf{x}_{i}-m+i}\mathbf{1}_{\mathsf{y}_{i}=\mathsf{x}_{i}-1}\right)$

for the transition probability. Clearly, in their combination in the left-hand
side of (5.2) all factors $V_{t,q}$ cancel out. Next, recall that
$|\vec{\mathsf{y}}|=|\vec{\mathsf{w}}|+1$, and this gives rise to the factor
$t$ in front of the right-hand side of (5.2). Next, one can readily check that
all the factors coming from

$t^{-\binom{m}{2}+\sum_{i=1}^{m}(\mathsf{x}_{i}-m+i)(\mathsf{y}_{i}-\mathsf{x}_{i})}\prod_{i=1}^{m}\left(1-q^{m-i}t^{\mathsf{x}_{i}-m+i}\mathbf{1}_{\mathsf{y}_{i}=\mathsf{x}_{i}-1}\right)$

in the left-hand side of (5.2) cancel out, too. Finally, we are left with the
ratio

$\displaystyle\frac{\psi^{\prime}_{(\vec{\mathsf{x}}-\delta_{m})/(\vec{\mathsf{y}}-\delta_{m})}(t,q)\psi^{\prime}_{(\vec{\mathsf{y}}-\delta_{m})/(\vec{\mathsf{z}}-\delta_{m})}(t,q)}{\psi^{\prime}_{(\vec{\mathsf{x}}-\delta_{m})/(\vec{\mathsf{w}}-\delta_{m})}(t,q)\psi^{\prime}_{(\vec{\mathsf{w}}-\delta_{m})/(\vec{\mathsf{z}}-\delta_{m})}(t,q)}.$

Recalling the definition of $\psi^{\prime}$ (2.10), we may rewrite this ratio
as

$\displaystyle\prod_{\begin{subarray}{c}1\leq i<j\leq m\\\
\mathsf{y}_{i}=\mathsf{x}_{i},\,\mathsf{y}_{j}=\mathsf{x}_{j}-1\end{subarray}}\omega(\mathsf{x}_{i}-\mathsf{x}_{j},j-i)\prod_{\begin{subarray}{c}1\leq
i<j\leq m\\\
\mathsf{z}_{i}=\mathsf{y}_{i},\,\mathsf{z}_{j}=\mathsf{y}_{j}-1\end{subarray}}\omega(\mathsf{y}_{i}-\mathsf{y}_{j},j-i)$
$\displaystyle\hskip 40.0pt\times\prod_{\begin{subarray}{c}1\leq i<j\leq m\\\
\mathsf{w}_{i}=\mathsf{x}_{i},\,\mathsf{w}_{j}=\mathsf{x}_{j}-1\end{subarray}}\frac{1}{\omega(\mathsf{x}_{i}-\mathsf{x}_{j},j-i)}\prod_{\begin{subarray}{c}1\leq
i<j\leq m\\\
\mathsf{z}_{i}=\mathsf{w}_{i},\,\mathsf{z}_{j}=\mathsf{w}_{j}-1\end{subarray}}\frac{1}{\omega(\mathsf{w}_{i}-\mathsf{w_{j}},j-i)}.$

Here all terms where neither $i$ nor $j$ is equal to $k$ cancel out. When
$j=k$, we may only get nontrivial contributions from the second and the third
products, and when $i=k$, nontrivial contributions may only come from the
first and the fourth products. In the fourth product, we use
$\mathsf{w}_{k}-\mathsf{w}_{j}=\mathsf{y}_{k}-\mathsf{y}_{j}-1$. This
completes the proof. ∎

In the Schur case $t=q$, we have $\omega(a,b)=1$, so the expression (5.2)
reduces simply to $q$. In this way, adding a $1\times 1\times 1$ box
multiplies the probability weight of a lozenge tiling by $q$, so the whole
probability of a tiling is proportional to $q^{\mathsf{vol}}$, where
$\mathsf{vol}$ is the volume under the corresponding 3-dimensional surface.
Note that this volume is the same as the sum of the entries of the plane
partition as in Figure 3, (c) and (d).

Gibbs measures on lozenge tilings of various infinite regions in which the
probability weight of a tiling is proportional to $q^{\mathsf{vol}}$ have been
widely studied, see, for example, [okounkov2003correlation], [Okounkov2005],
[BMRT2009BackWall]. Most well-known ensembles of such lozenge tilings are
solvable by means of Schur processes, and feature an arbitrary back wall. Our
ensemble of $q^{\mathsf{vol}}$ weighted lozenge tilings possesses a different
kind of boundary conditions, namely, an arbitrary cascade front wall, as seen
in Figure 3, (b).

## References

L. Petrov, University of Virginia, Department of Mathematics, 141 Cabell
Drive, Kerchof Hall, P.O. Box 400137, Charlottesville, VA 22904, USA

E-mail<EMAIL_ADDRESS>