# Identity between restricted Cauchy sums for the $q$-Whittaker and skew Schur
polynomials

Takashi Imamura  T. Imamura, Department of Mathematics and Informatics, Chiba
University, Chiba, 263-8522 Japan<EMAIL_ADDRESS>,  Matteo
Mucciconi  M. Mucciconi, Department of Physics, Tokyo Institute of Technology,
Tokyo, 152-8551 Japan<EMAIL_ADDRESS>and  Tomohiro Sasamoto  T.
Sasamoto, Department of Physics, Tokyo Institute of Technology, Tokyo,
152-8551 Japan<EMAIL_ADDRESS>
###### Abstract.

The Cauchy identities play an important role in the theory of symmetric
functions. It is known that Cauchy sums for the $q$-Whittaker and the skew
Schur polynomials produce the same factorized expressions modulo a
$q$-Pochhammer symbol. We consider the sums with restrictions on the length of
the first rows for labels of both polynomials and prove an identity which
relates them. The proof is based on techniques from integrable probability: we
rewrite the identity in terms of two probability measures: the $q$-Whittaker
measure and the periodic Schur measure. The relation follows by comparing
their Fredholm determinant formulas.

###### Contents

  1. 1 Introduction
  2. 2 Fredholm determinants for $q$-Whittaker and shift-mixed periodic Schur measures
  3. 3 Equivalence of two Fredholm determinants by shift of contours
  4. A Notations and formulas
  5. B Properties of the kernels $A(m;r)$ and $B(r;m)$
  6. C Decay estimates of $q$-Pochhammer symbols

##  1\. Introduction

###  1.1. The identity and connections to integrable probability

Symmetric polynomials have been of much interest in various fields of
mathematics such as representation theory, combinatorics etc. Among them the
Schur polynomials $s_{\lambda}(a_{1},\dots,a_{N})$ occupy an important place
because of their simplicity, depth and wide applications. One of their most
well-known properties is the Cauchy identity,

(1.1)
$\displaystyle\sum_{\lambda\in\mathcal{P}}s_{\lambda}(a_{1},\dots,a_{N})s_{\lambda}(b_{1},\dots,b_{M})=\prod_{i=1}^{N}\prod_{j=1}^{M}\frac{1}{1-a_{i}b_{j}},$

where $\mathcal{P}$ is the set of partitions. Along with a number of related
summation identities, (1.1) provides important insights into various field of
mathematics such as combinatorics [Sag00], representation theory [Bum13],
integrable probability [BG16a], etc.

The Cauchy identity for the Schur function (1.1) has been generalized in
various different ways. In this paper, we focus on two generalizations of
(1.1) in which the Schur polynomials in the left hand side are replaced by the
$q$-Whittaker functions $P_{\lambda}(a_{1},\dots,a_{N})$,
$Q_{\lambda}(b_{1},\dots,b_{M})$ and the skew Schur polynomials
$s_{\lambda/\rho}(a_{1},\dots,a_{N})$. These are

(1.2)
$\displaystyle\sum_{\mu\in\mathcal{P}}P_{\mu}(a_{1},\dots,a_{N})Q_{\mu}(b_{1},\dots,b_{M})=\prod_{i=1}^{N}\prod_{j=1}^{M}\frac{1}{(a_{i}b_{j};q)_{\infty}},$
(1.3) $\displaystyle\sum_{\begin{subarray}{c}\lambda,\rho\in\mathcal{P}\\\
\rho\subset\lambda\end{subarray}}q^{|\rho|}s_{\lambda/\rho}(a_{1},\dots,a_{N})s_{\lambda/\rho}(b_{1},\dots,b_{M})=\frac{1}{(q;q)_{\infty}}\prod_{i=1}^{N}\prod_{j=1}^{M}\frac{1}{(a_{i}b_{j};q)_{\infty}},$

where $|\rho|:=\rho_{1}+\rho_{2}+\cdots$, $\rho\subset\lambda$ means
$\rho_{i}\leq\lambda_{i}$ for any $i=1,2,\dots$ and $(x;q)_{\infty}$ is
defined by (A.1). The $q$-Whittaker functions $P_{\lambda}(a_{1},\dots,a_{N})$
are a special case of the Macdonald symmetric polynomials $P_{\lambda}(a;q;t)$
defined in Ch.VI in [Mac98], in which out of two parameters $(q,t)$ of the
latter one puts $t=0$. The dual polynomial $Q_{\mu}(b_{1},\dots,b_{M})$ is
defined as $\prod_{j=1}^{M}(q;q)_{\mu_{j}-\mu_{j+1}}^{-1}\cdot
P_{\mu}(b_{1},\dots,b_{M})$, where $(x;q)_{n}$ is given by (A.2) and we set
$\mu_{M+1}\equiv 0$. When $q=0$, both $P_{\lambda}(a_{1},\dots,a_{N})$ and
$Q_{\lambda}(a_{1},\dots,a_{N})$ reduce to the Schur polynomial and (1.3)
becomes (1.1). The skew Schur polynomials
$s_{\lambda/\rho}(a_{1},\dots,a_{N})$ are labeled by skew partitions
$\lambda/\rho$. For a precise definition, see e.g. Ch.I in [Mac98]. When
$q=0$, since we have $\rho=\phi$ due to the weight $q^{|\rho|}$, they reduce
to (non-skew) Schur polynomial $s_{\lambda}(a_{1},\dots,a_{N})$ and (1.2)
becomes (1.1). Proofs of (1.2) and (1.3) can be found in Sec. VI.4 and Sec.
I.5,28(a) in [Mac98] respectively.

We immediately see that the right-hand sides of (1.2) and (1.3) are exactly
the same up to a factor $1/(q;q)_{\infty}$. Combining this fact with (A.8),
which rewrites $1/(q;q)_{\infty}$ as a sum over partitions $\nu$, we have

(1.4)
$\displaystyle\sum_{\mu,\nu\in\mathcal{P}}q^{|\nu|}P_{\mu}(a_{1},\dots,a_{N})Q_{\mu}(b_{1},\dots,b_{M})=\sum_{\begin{subarray}{c}\lambda,\rho\in\mathcal{P}\\\
\rho\subset\lambda\end{subarray}}q^{|\rho|}s_{\lambda/\rho}(a_{1},\dots,a_{N})s_{\lambda/\rho}(b_{1},\dots,b_{M}).$

This identity suggests that there may exist deeper connections between the
$q$-Whittaker functions and the skew Schur functions resulting in possible
refinements of (1.4). Indeed in this paper we will prove such an identity as
stated in the following theorem.

######  Theorem 1.1.

For any $n\in\mathbb{Z}_{\geq 0}$, we have

(1.5)
$\displaystyle\sum_{\stackrel{{\scriptstyle\mu,\nu\in\mathcal{P}}}{{\mu_{1}+\nu_{1}\leq
n}}}q^{|\nu|}P_{\mu}(a_{1},\dots,a_{N})Q_{\mu}(b_{1},\dots,b_{M})=\sum_{\stackrel{{\scriptstyle\lambda,\rho\in\mathcal{P}}}{{\rho\subset\lambda,\lambda_{1}\leq
n}}}q^{|\rho|}s_{\lambda/\rho}(a_{1},\dots,a_{N})s_{\lambda/\rho}(b_{1},\dots,b_{M}).$

Summations of this type with restrictions on the length of the first row
(mostly for the Schur case) have attracted attention in the theory of
symmetric functions in connection with combinatorics and representation
theory, see for instance [BR01, Ges90, Rai98, RW15]. In this paper, the proof
of Theorem 1.1 will be based on connections of quantities on both hand sides
to integrable probability. In our next paper [IMS21], we will develop a
combinatorial theory which allows a bijective proof of (1.5) and related
identities.

######  Remark 1.2.

In Theorem 1.1 our main identity is stated for complex variables
$a_{1},\dots,a_{N},$ $b_{1},\dots,b_{M}$, but one easily sees that the same
holds also when they are arbitrary specialization of symmetric functions. In
order to establish (1.5) it suffices to prove it for $a_{i},b_{j}$ varying in
an continuous interval, which is in fact what we will prove below using
integrable probabilistic arguments.

Let us rewrite (1.5) in a probabilistic language. The probability measures on
$\mathcal{P}$ related to (1.2) and (1.3) are of much interest in integrable
probability. They are defined as

(1.6) $\displaystyle M_{q\text{W}}(\mu)$
$\displaystyle=P_{\mu}(a_{1},\dots,a_{N})Q_{\mu}(b_{1},\dots,b_{M})/Z_{q\text{W}},$
(1.7) $\displaystyle M_{\text{pS}}(\lambda)$
$\displaystyle=\sum_{\begin{subarray}{c}\rho\in\mathcal{P}\\\
\rho(\subset\lambda)\end{subarray}}q^{|\rho|}s_{\lambda/\rho}(a_{1},\dots,a_{N})s_{\lambda/\rho}(b_{1},\dots,b_{M})/Z_{\text{pS}},$

where we assumed $q\in(0,1)$ $a_{i},b_{j}>0$, with $a_{i}b_{j}<1$ for
$i=1,\dots,N$ and $j=1,\dots,M$ and the normalization constants
$Z_{q\text{W}}$ and $Z_{\text{pS}}$ are given by the right hand sides of the
Cauchy identities (1.2) and (1.3) respectively. $M_{q\text{W}}(\mu)$ is called
the _$q$ -Whittaker measure_ while $M_{\text{pS}}(\lambda)$ is called the
_periodic Schur measure_.

The $q$-Whittaker measure was introduced in [BC14]. It has been playing an
important role in the development of integrable probability in the past
decade. Its marginals are known to describe one point functions of certain
particle systems discretizing the KPZ equation. On the other hand, the
periodic Schur measure was introduced in [Bor07]. It can be regarded as a
model of lozenge tilings in a cylindric domain and free fermions at positive
temperature.

We also introduce two random variables $\chi$ and $S$, which are independent
of other random variables and respectively have distribution,

(1.8)
$\displaystyle\mathbb{P}(\chi=n)=\frac{q^{n}}{(q;q)_{n}}(q;q)_{\infty},\qquad
n=0,1,2,\dots$

and for a fixed parameter $t\in\mathbb{R}_{>0}$,

(1.9)
$\displaystyle\mathbb{P}(S=\ell):=\frac{t^{\ell}q^{\ell^{2}/2}}{(q;q)_{\infty}\theta(-tq^{1/2})},\qquad\ell\in\mathbb{Z}.$

Here the definitions of the $q$-Pochhammer symbols $(x;q)_{\infty}$ and
$(x;q)_{n}$ are given by (A.1) and (A.2) respectively whereas the function
$\theta(x)$ is defined below (A.7).

The random variable $\chi$ is related to the partition $\nu$ in (1.5). Namely,
thanks to the summation identity (A.9), if a partition $\nu$ is sampled with
probability proportional to $q^{|\nu|}$, then its first row $\nu_{1}$ follows
the same law as $\chi$. On the other hand the random variable $S$ is
introduced as it unveils a determinantal structure in the periodic Schur
process. This was observed by Borodin in [Bor07], who noticed that shifting by
$S$ all entries of a random partition $\lambda$ sampled with $M_{\mathrm{pS}}$
law produces a determinantal point process called _shift-mixed periodic Schur
measure_.

With these preparations, Theorem 1.1 can be restated as follows.

######  Theorem 1.3.

Let $\mu_{1},\lambda_{1}$ be the first rows of partitions $\mu,\lambda$,
distributed respectively according to the $q$-Whittaker measure $M_{q{\rm
W}}(\mu)$ (1.6) and to the periodic Schur measure $M_{{\rm pS}}(\lambda)$
(1.7). Let also $\chi$ and $S$ be independent random variables distributed as
(1.8) and (1.9) respectively. The following equalities hold and they are
equivalent. For $n\in\mathbb{Z}_{\geq 0}$, we have

  * (a)

(1.10) $\displaystyle\mathbb{P}\left(\mu_{1}+\chi\leq
n\right)=\mathbb{P}\left(\lambda_{1}\leq n\right),$

  * (b)

(1.11)
$\displaystyle\mathbb{E}\left[\frac{1}{(-tq^{\frac{1}{2}+n-\mu_{1}};q)_{\infty}}\right]=\mathbb{P}(\lambda_{1}+S\leq
n).$

Equivalence between (1.5) and (1.10) is obvious by the definitions of measures
given above. Equivalence between (1.10) and (1.11) will be shown in section
2.3 although it is rather straightforward. The key observation is the formula

(1.12) $\displaystyle\mathbb{P}(\chi+S\leq
n)=\frac{1}{(-tq^{\frac{1}{2}+n};q)_{\infty}}.$

This was already shown in [BB19, Section 2], but we will give a more direct
proof in Lemma 2.4 below. From (1.12) we easily see that (1.11) can be
rewritten as

(1.13) $\displaystyle\mathbb{P}(\mu_{1}+\chi+S\leq
n)=\mathbb{P}(\lambda_{1}+S\leq n),$

which is clearly equivalent to (1.10).

Now our goal is to prove (1.11). At this point we remark that, thanks to
results from integrable probability, the expectation and the probability on
both sides are known to be written as Fredholm determinants, see Propositions
2.1 for the expectation on the left hand side and Propositions 2.3 for the
probability on the right hand side. Crucially, the kernels of the two Fredholm
determinants turn out to be very similar: they both possess complex contour
integral expressions with the integrands being the same and they differ from
each other only by a choice of the contours. What we actually prove in this
paper is that this difference of contours does not produce a difference to the
value of the Fredholm determinant, see Theorem 3.1.

###  1.2. Background from integrable probability

In integrable probability, measures on the set of partitions written in terms
of a pair of symmetric functions have attracted much attention. The simplest
case, corresponding to the Cauchy identity (1.1), is the Schur measure,
introduced in [Oko01]. For a partition $\lambda\in\mathcal{P}$, it is defined
by

(1.14) $\displaystyle
M_{\text{S}}(\lambda)=s_{\lambda}(a_{1},\dots,a_{N})s_{\lambda}(b_{1},\dots,b_{M})/Z_{\text{S}},$

where the normalization constant $Z_{\text{S}}$ is the right hand side of
(1.1). From the fact that Schur polynomials can be written as single
determinants due to the Jacobi-Trudi identity, the Schur measure defines a
determinantal point process (DPP) [Sos00].

The Schur measure has connections with a few variants of the totally
asymmetric simple exclusion process (TASEP) or the last passage percolation,
which are typical models in the Kardar-Parisi-Zhang universality class. Such
connection was first discovered by Johansson [Joh00], who showed that the
current of the TASEP has the same law as the marginal $\lambda_{1}$ in the
Schur measure. For our purposes a different variant of the TASEP with a
certain pushing mechanism, called the _PushTASEP_ , is relevant [BF14, WW09].
The precise definition of the dynamics can be found in the references. By
using certain Markov dynamics on the Gelfand-Tsetlin cone, one can show that
the PushTASEP is associated with the Schur measure. If $X_{N}(M)$ is the
position of the $N$-th particle at time $M$ starting from the so called _step
initial condition_ $X_{k}(0)=k$ for $k\in\mathbb{N}$, we have the following
relation,

(1.15) $\displaystyle\mathbb{P}(X_{N}(M)-N\leq n)=\mathbb{P}(\lambda_{1}\leq
n),$

where the right hand side is the marginal of $\lambda_{1}$ from the Schur
measure and can be written as a Fredholm determinant.

The PushTASEP possesses a solvable $q$-deformation called $q$-PushTASEP,
introduced in [BP16, MP17]. This model is of particular interests as it
interpolates between various notable models in the KPZ class including the
Log-Gamma polymer model and the KPZ equation [MP17, BCFV15]. Employing
randomized versions of the RSK correspondence or of the Yang-Baxter equation
one can relate the $q$-PushTASEP with the marginal $\mu_{1}$ of the
$q$-Whittaker measure (1.2) in the same way as in (1.15) [MP17, BMP19, MP20].
Unlike in the $q=0$ case, such matching, although useful, does not lead
immediately to explicit solutions of the model and in order to extract
manageable formulas to study the $q$-PushTASEP additional considerations are
needed. This is because, up to now, relations between the $q$-Whittaker
measure and DPP had remained hidden. Nevertheless, through the use of
Macdonald difference operators, Markov duality, Bethe Ansatz or elliptic
determinants [BC14, BCS14, IS19], Fredholm determinant formulas describing
interesting expectations of marginals of the $q$-Whittaker measure had been
found. In general, following such approaches, one arrives at the following
result

(1.16) $\displaystyle\mathbb{E}\left[\frac{1}{(\zeta
q^{-\mu_{1}};q)_{\infty}}\right]=\det\left(1-K\right)_{\ell^{2}(\mathbb{Z})},$

where $\mu_{1}$ is a marginal of the $q$-Whittaker measure, $\zeta$ is a
parameter and $K$ is some kernel.

In the references listed above several expressions for the right hand side of
(1.16) have been obtained. Techniques involving Macdonald difference operators
or Bethe Ansatz lead to Fredholm determinant formulas where the kernel $K$
does not suggest any immediate relation with natural DPPs. On the other hand
the elliptic determinant approach introduced in [IS19] produces a
substantially simpler formula where $K$ becomes a product between a finite
rank kernel and the Fermi-Dirac distribution. Since such expressions, in
particular the Fermi-Dirac distribution, are well known to appear in the
description of free fermions at finite temperature, findings of [IS19] suggest
a clear connections between discrete KPZ solvable models and DPP.

This observation has led us to a careful comparison between the structure of
the $q$-Whittaker measure and the (shift-mixed) periodic Schur measure
[Bor07], as the latter is a canonical model of free fermions at positive
temperature in one dimension [BB19]. In recent years the same model has
received some attention and its properties and generalizations have been
considered in [BBNV18, Kos21, ARVP21]. The key observation we make is that the
Fredholm determinant appearing in the shift-mixed periodic Schur measure is
almost the same as the one found in [IS19] for the $q$-Whittaker measure, the
only difference being in the contours of the contour integral expression for
the kernels. In fact establishing this equivalence of the two Fredholm
determinants is technically our main result as given in Theorem 3.1.

Before our result (1.10), outside of the TASEP/Schur case described above,
explicit connections between KPZ solvable models and DPPs had remained
concealed. One notable exception to this claim was presented by the KPZ
equation itself. The celebrated solution of the KPZ equation derived in 2010
by [ACQ11, CLDR10, Dot10, SS10a, SS10b] provides, in fact, a clear relation
between the one point function of the random KPZ height with narrow wedge
initial condition and one point edge statistics of a system of trapped free
fermions at finite temperature [DLDMS15]. See also [Joh07, LW20]. The identity
(1.10) extends this fact to a much higher level, widening substantially the
bridge between solvable KPZ models and DPPs or free fermions at finite
temperature.

An additional observation, relating this time the one point function of the
KPZ equation to multiplicative statistics of the Airy point process was made
in [BG16b]. Borodin expanded on this finding in [Bor18] proving a matching of
the expectations between the certain stochastic vertex models and
multiplicative statistics of the Schur measure. Such results had then been
applied to a few models as ASEP, $q$-TASEP, etc. [BO17, OP17]. At the moment
we are not aware of precise connections between results of this kind and the
one given in Theorem 1.3 and further investigation will be needed in the
future.

All the matching discussed in the previous paragraphs, as the one obtained by
us in this paper, consist essentially in comparisons between various formulas.
In our upcoming paper [IMS21] we will further improve on this aspect
developing a bijective theory providing a combinatorial explanation of
identities between models.

###  1.3. Outline

This paper is organized as follows. In section 2 we give Fredholm determinant
formulas for the $q$-Whittaker and shift-mixed Schur measures obtained
respectively in [IS19] and [Bor07]. We also discuss the equivalence between
Theorem 1.1 and Theorem 1.3. The proof of Theorem 1.3 consists in the
equivalence of these Fredholm determinant formulas and it is given in Sec. 3.
A key passage to establish the proof is given by Lemma 3.4 (i)-(iii). In
Appendix A, we summarize some basic notions which we frequently use in this
paper including partitions, $q$-series. In Appendix B, we prove useful bounds
which allow to show Proposition 3.3(i) in Sec. 3.3. In Appendix C, we state
and show estimations of the $q$-Pochhammer symbol used in Sec. 3.4.

##  2\. Fredholm determinants for $q$-Whittaker and shift-mixed periodic Schur
measures

The main purpose of this section is to give Fredholm determinant formulas for
both hand sides of (1.11); see Propositions 2.1 and 2.3. In section 3 we will
show that the two Fredholm determinants are in fact the same, as stated in
Theorem 3.1. In Sec. 2.3 we also give a proof of equivalence between Theorem
1.1 and Theorem 1.3.

Hereafter we consider set of variables $a_{1},\dots,a_{N}$,
$b_{1},\dots,b_{N}$, i.e. we set $M=N$ in the notation used in section 1. The
general case $M\neq N$ of each result is recovered by considerations similar
to the one reported in remark 1.2. We will denote the maximum (resp. minimum)
value of $a_{i}$ and $b_{i}$ $i=1,\dots,N$ by $a_{\max},b_{\max}$ (resp.
$a_{\min}$, $b_{\min}$). We also always omit the factor $1/(2\pi\sqrt{-1})$ in
contour integrals.

###  2.1. Fredholm determinant formula for the $q$-Whittaker measure

We focus on the expectation value $\mathbb{E}\left[1/(\zeta
q^{-\mu_{1}};q)_{\infty}\right]$ with respect to the $q$-Whittaker measure
defined by (1.6). This quantity is the $q$-Laplace transform of the marginal
$\mu_{1}$ and it admits explicit Fredholm determinant expressions [BC14,
BCFV15, MP17, BMP19, IS19]. For our arguments we will make use of the formula
that first appeared in [IS19], which is conceptually different from the ones
of the other references.

The following proposition is obtained by applying a few results in [IS19].

######  Proposition 2.1.

Let $\zeta\in\mathbb{C}\setminus\\{q^{n}\\}_{n\in\mathbb{Z}}$, and
$a_{i},b_{i}\in\mathbb{R}_{>0}$, $i=1,\dots,N$ satisfy $a_{\max}b_{\max}<1$
and $qa_{\max}<a_{\min}$. Let $\mu_{1}$ be the first row of $\mu$ distributed
according to the $q$-Whittaker measure (1.6). We have

(2.1) $\displaystyle\mathbb{E}\left[\frac{1}{(\zeta
q^{-\mu_{1}};q)_{\infty}}\right]=\det\left(1-f_{\zeta}K\right)_{\ell^{2}(\mathbb{Z})},$

where $f_{\zeta}(m)=\frac{-\zeta q^{m}}{1-\zeta q^{m}}$ and

(2.2) $\displaystyle
K(m_{1},m_{2})=\int_{C}\frac{dz}{z}\int_{D}\frac{dw}{w}g_{a,b}(z,w;m_{1},m_{2}),$
(2.3) $\displaystyle
g_{a,b}(z,w;m_{1},m_{2})=\frac{w^{m_{2}}}{z^{m_{1}}}\frac{w}{z-w}\prod_{i=1}^{N}\frac{(a_{i}z;q)_{\infty}}{(a_{i}w;q)_{\infty}}\cdot\prod_{j=1}^{N}\frac{(b_{j}/w;q)_{\infty}}{(b_{j}/z;q)_{\infty}},$

where the contour $C$ encloses anticlockwise the poles at $z=0,b_{i}q^{j}$,
$i=1,\dots,N$, $j\in\mathbb{Z}_{\geq 0}$ whereas $D$ encloses anticlockwise
the poles at $w=1/a_{i}$, $i=1,\dots,N$ but not the ones at
$w=0,1/a_{i}q^{j}$, $i=1,\dots,N$, $j\in\mathbb{Z}_{\geq 1}$. They are
illustrated in Fig. 1.

Figure 1. The contours $C$, $D$, and $\bar{D}$ appearing in (2.2) and (2.17).
“$\bullet$”s and “”s represent the poles of $w$ and $z$ respectively.

To prove Proposition 2.1, we recall the discussions in Sec. 3 in [IS19]. There
we consider a two-sided $q$-Whittaker measure, which is a probability measure
on the set of signatures,

(2.4)
$\displaystyle\mathcal{S}_{N}=\\{\lambda=(\lambda_{1},\dots,\lambda_{N})\in\mathbb{Z}^{N}|\lambda_{1}\geq\cdots\geq\lambda_{N}\\}.$

Note that $\lambda_{i}$, $i=1,\dots,N$ can be negative. Such probability
measures on the extended space can be realized by replacing the function
$Q_{\mu}(x)$ in (1.6) with $Q_{\mu}(\rho)$, where $\rho$ is a certain
specialization of the algebra of symmetric functions. Here we state only a
particular case which corresponds to Def. 3.4 in [IS19], setting
$a_{i}=x_{i}$, $i=1,\dots,N$, $t=0$ and $\alpha_{j}=b_{j}$, $j=1,\dots,N$.
With this choice $M_{q\text{W}}(\mu)$ in (1.6) becomes

(2.5) $\displaystyle
P_{\lambda}(x_{1},\dots,x_{N})Q_{\lambda}(\rho^{-}_{b})/\Pi^{(-)}(x;b),$

where $\lambda\in\mathcal{S}_{N}$, the parameters $x_{i},~{}b_{i}$, satisfy
$b_{\max}/x_{\min}<1$ and

(2.6) $\displaystyle
Q_{\lambda}\left(\rho^{-}_{b}\right)=\prod_{i=1}^{N-1}(q^{\lambda_{i}-\lambda_{i+1}+1};q)_{\infty}\int_{C^{N}}\prod_{i=1}^{N}\frac{dz_{i}}{z_{i}}\cdot
P_{\lambda}\left(1/z\right)\Pi^{(-)}\left(z;b\right)m_{N}^{q}\left(z\right).$

Here $C$ is the same contour as the one in (2.2) i.e. it encloses positively
the poles at $z=0,b_{i}q^{j}$, $i=1,\dots,N$ and $j\in\mathbb{Z}_{\geq 0}$,
$1/z$ in $P_{\lambda}$ is a shorthand notation for $(1/z_{1},\dots,1/z_{N})$
and

(2.7) $\displaystyle m_{N}^{q}(z)=\frac{1}{N!}\prod_{1\leq i<j\leq
N}(z_{i}/z_{j};q)_{\infty}(z_{j}/z_{i};q)_{\infty},$ (2.8)
$\displaystyle\Pi^{(-)}\left(z;b\right)=\prod_{i,j=1}^{N}\frac{1}{(b_{i}/z_{j};q)_{\infty}}.$

For more detail see Appendix B in [IS19].

From (3.15) in Lemma 3.3 in [IS19], one sees that
$Q_{\lambda}(\rho_{b}^{-})=0$ unless
$0\geq\lambda_{1}\geq\cdots\geq\lambda_{N}$ and thus the measure (2.5) has
support only on the negative region. Although the measures (1.6) and (2.5)
have different support, they enjoy the following relation:

######  Lemma 2.2.

Set $a_{i}=1/x_{i},~{}i=1,\dots,N$ and define the signature $\mu$ by the
negation of $\lambda$,
$\mu:=(-\lambda_{N},-\lambda_{N-1},\dots,-\lambda_{1})$. Then we have

(2.9) $\displaystyle
P_{\lambda}(x_{1},\dots,x_{N})Q_{\lambda}(\rho^{-}_{b})/\Pi^{(-)}(x;b)=M_{q{\rm
W}}(\mu)$

and in particular

(2.10) $\displaystyle\mathbb{E}_{q{\rm
W}(x;\rho_{b}^{-})}\left[\frac{1}{(\zeta
q^{\lambda_{N}};q)_{\infty}}\right]=\mathbb{E}_{q{\rm
W}(a;b)}\left[\frac{1}{(\zeta q^{-\mu_{1}};q)_{\infty}}\right],$

where $\mathbb{E}_{q{\rm W}(x;\rho_{b}^{-})}[\cdot]$ and $\mathbb{E}_{q{\rm
W}(a;b)}[\cdot]$ are averages over the measures (2.5) and (1.6) respectively.

###### Proof.

Equation (2.9) follows straightforwardly from a combination of definition
(2.5) and relations (3.14) and (3.15) in [IS19],

(2.11)
$\begin{split}P_{\lambda}(x_{1},\dots,x_{N})=P_{\mu}(a_{1},\dots,a_{N}),\\\
Q_{\lambda}(\rho^{-}_{b})=Q_{\mu}(b_{1},\dots,b_{N}),\end{split}$

once we recall hypothesis relating $a_{i}$ (resp. $\mu$) to $x_{i}$ (resp.
$\lambda$) . ∎

###### Proof of Proposition 2.1.

It follows immediately from Lemma 2.2 and the Fredholm determinant formula for
the left-hand side of (2.10). Applying the change of notations
$a_{i}\rightarrow 1/x_{i}$, $v\rightarrow w$, $n\rightarrow-m_{1}$,
$m\rightarrow-m_{2}$ to (4.37) in [IS19], we obtain the Fredholm determinant
formula

(2.12)
$\displaystyle\mathbb{E}_{q\text{W}(x;\rho_{b}^{-})}\left[\frac{1}{(\zeta
q^{\lambda_{N}};q)_{\infty}}\right]=\det\left(1-f_{\zeta}M\right)_{\ell^{2}(\mathbb{Z})}.$

Here $f_{\zeta}(m)$ is defined above (2.2) and

(2.13) $\displaystyle
M(m_{1},m_{2})=\int_{C}\frac{dz}{z}\int_{D}\frac{dw}{w}\frac{w^{m_{2}}}{z^{m_{1}}}\frac{w}{z-w}\prod_{i=1}^{N}\frac{(z/x_{i};q)_{\infty}}{(w/x_{i};q)_{\infty}}\cdot\prod_{j=1}^{N}\frac{(b_{j}/w;q)_{\infty}}{(b_{j}/z;q)_{\infty}},$

where $C$ is the same contour as (2.6) and $D$ encloses positively
$x_{i},~{}i=1,\dots,N$. Combining (2.10) with (2.12), we get (2.1).

∎

###  2.2. Fredholm determinant formula for the shift-mixed periodic Schur
measure

We have discussed in section 1 how the periodic Schur measure
$M_{\text{pS}}(\lambda)$ (1.7) becomes a determinantal point process once all
the rows of $\lambda$ get shifted by an independent random quantity $S$
distributed as (1.9). This was originally observed by Borodin in [Bor07] and
the measure describing the process

(2.14) $\displaystyle(\lambda_{1}+S,\lambda_{2}+S,\dots)$

takes the name of _shift-mixed periodic Schur measure_. The probability
distribution of rightmost element $\lambda_{1}+S$, can be represented as a
single Fredholm determinant and this is reported in the next proposition.

######  Proposition 2.3.

Let $\lambda_{1}+S$ be the first element of (2.14) distributed according to
the shift-mixed periodic Schur measure. For $k\in\mathbb{Z}_{\geq 0}$, we have

(2.15) $\displaystyle\mathbb{P}\left(\lambda_{1}+S\leq
k\right)=\det\left(1+fL\right)_{\ell^{2}(\mathbb{Z})},$

where $f(m)$ is given in terms of $f_{\zeta}(m)$ defined above (2.2),

(2.16) $\displaystyle
f(m)=f_{\zeta}(m)|_{\zeta=-tq^{1/2+k}}=\frac{tq^{1/2+k+m}}{1+tq^{1/2+k+m}}$

and by using $g_{a,b}(z,w,m_{1},m_{2})$ (2.3), the kernel $L(m_{1},m_{2})$ is
defined by

(2.17) $\displaystyle
L(m_{1},m_{2})=\int_{C}\frac{dz}{z}\int_{\bar{D}}\frac{dw}{w}g_{a,b}(z,w,m_{1},m_{2}).$

Here contour $C$ is oriented counterclockwise and it encloses the origin with
its radius $r$ satisfying $b_{\max}<r<1/a_{\max}$ whereas $\bar{D}$ encloses
$C$ anticlockwise. $C$ and $\bar{D}$ are depicted in Fig.1 given below
Proposition 2.1.

###### Proof.

From Theorem 2.2 in [Bor07], we have, for $k\in\mathbb{Z}_{\geq 0}$,

(2.18) $\displaystyle\mathbb{P}\left(\lambda_{1}+S\leq
k\right)=\det\left(1-\tilde{L}\right)_{\ell^{2}(k+1,k+2,\cdots)},$

where the kernel $\tilde{L}$ is defined as

(2.19)
$\displaystyle\tilde{L}(\ell_{1},\ell_{2})=\int_{|w|=r}\frac{dw}{w}\int_{|z|=r^{\prime}}\frac{dz}{z}\frac{z^{\ell_{2}-1}}{w^{\ell_{1}-1}}\prod_{i=1}^{N}\frac{(a_{i}z;q)_{\infty}}{(a_{i}w;q)_{\infty}}\cdot\prod_{j=1}^{N}\frac{(b_{j}/w;q)_{\infty}}{(b_{j}/z;q)_{\infty}}\cdot\sum_{m\in\mathbb{Z}}\frac{tq^{\frac{1}{2}+m}}{1+tq^{\frac{1}{2}+m}}\left(\frac{w}{z}\right)^{m},$

and $r,r^{\prime}$ satisfy 111In [Bor07, BB19], the conditions for
$r,r^{\prime}$ are given by using the analyticity of the integrand for the
periodic Schur process with positive specializations. This condition can be
translated into the explicit one (2.20) in our case (1.7).

(2.20) $\displaystyle
b_{\max}<r,r^{\prime}<\frac{1}{a_{\max}},~{}~{}1<\frac{r}{r^{\prime}}<q^{-1}.$

We will now show that
$\det(1-\tilde{L})_{\ell^{2}(k+1,k+2,\dots)}=\det(1+L)_{\ell^{2}(\mathbb{Z})}$.
Note that the kernel $\tilde{L}(\ell_{1},\ell_{2})$ (2.19) can be written as

(2.21)
$\displaystyle\tilde{L}(\ell_{1},\ell_{2})=\sum_{m\in\mathbb{Z}}A(\ell_{1},m)B(m,\ell_{2}),$

where

(2.22) $\displaystyle A(\ell,m)$
$\displaystyle=\int_{|w|=r}\frac{dw}{w}w^{m-\ell+1}\prod_{i=1}^{N}\frac{1}{(a_{i}w;q)_{\infty}}\cdot\prod_{j=1}^{N}(b_{j}/w;q)_{\infty},$
(2.23) $\displaystyle B(m,\ell)$
$\displaystyle=\frac{tq^{1/2+m}}{1+tq^{1/2+m}}\int_{|z|=r^{\prime}}\frac{dz}{z}z^{\ell-1-m}\prod_{i=1}^{N}(a_{i}z;q)_{\infty}\cdot\prod_{j=1}^{N}\frac{1}{(b_{j}/z;q)_{\infty}}.$

By a basic property of the determinant, we have

(2.24)
$\displaystyle\det(1-AB)_{\ell^{2}(k+1,k+2,\dots)}=\det(1-BA)_{\ell^{2}(\mathbb{Z})},$

and evaluating the kernel in the right hand side we find

(2.25) $\displaystyle~{}(BA)(n_{1},n_{2})$
$\displaystyle=\frac{tq^{n_{1}+\frac{1}{2}}}{1+tq^{n_{1}+\frac{1}{2}}}\int_{|z|=r^{\prime}}\frac{dz}{z}\int_{|w|=r}\frac{dw}{w}\frac{w^{n_{2}}}{z^{n_{1}}}\prod_{i=1}^{N}\frac{(a_{i}z;q)_{\infty}}{(a_{i}w;q)_{\infty}}\cdot\prod_{j=1}^{N}\frac{(b_{i}/w;q)_{\infty}}{(b_{j}/z;q)_{\infty}}\cdot\sum_{\ell=k+1}^{\infty}\left(\frac{z}{w}\right)^{\ell-1}$
$\displaystyle=\frac{-tq^{\frac{1}{2}+k+m_{1}}}{1+tq^{\frac{1}{2}+k+m_{1}}}\int_{|z|=r^{\prime}}\frac{dz}{z}\int_{|w|=r}\frac{dw}{w}\frac{w^{m_{2}}}{z^{m_{1}}}\prod_{i=1}^{N}\frac{(a_{i}z;q)_{\infty}}{(a_{i}w;q)_{\infty}}\cdot\prod_{j=1}^{N}\frac{(b_{i}/w;q)_{\infty}}{(b_{j}/z;q)_{\infty}}\cdot\frac{w}{z-w}$
$\displaystyle=-f(m_{1})L(m_{1},m_{2}).$

Setting $m_{i}=n_{i}-k$ for $i=1,2$ completes the proof. ∎

###  2.3. Equivalence of Theorem 1.1 and Theorem 1.3

This subsection is devoted prove that Theorem 1.1 is equivalent to Theorem
1.3. Since equivalence between Theorem 1.1 and Theorem 1.3 (a) is
straightforward, the only nontrivial part is to show the equivalence between
Theorem 1.3 (a) and (b) i.e., (1.10) and (1.11). This is a consequence of the
computation reported in the following lemma.

######  Lemma 2.4.

For $n\in\mathbb{Z}$, $0<q<1$ and $t>0$, we have

(2.26) $\displaystyle\mathbb{P}(\chi+S\leq
n)=\frac{1}{(-tq^{\frac{1}{2}+n};q)_{\infty}},$

where $\chi$ and $S$ are independent random variables with distributions (1.8)
and (1.9) respectively.

###### Proof.

We write the left-hand side of (2.26) as

(2.27) $\displaystyle\mathbb{P}(\chi+S\leq
n)=\sum_{\ell\in\mathbb{Z}}\mathbb{P}(S=\ell)\mathbb{P}(\chi\leq n-\ell).$

Applying (1.9) and

(2.28) $\displaystyle\mathbb{P}(\chi\leq
m)=\begin{cases}(q;q)_{\infty}/(q;q)_{m},&m\in\mathbb{Z}_{\geq 0},\\\
0,&m\in\mathbb{Z}_{<0},\end{cases}$

which follows from (1.8), to the right-hand side of (2.27), we find

(2.29) $\displaystyle\mathbb{P}(\chi+S\leq
n)=\frac{1}{\theta(-q^{1/2}t)}\sum_{\ell=-\infty}^{n}\frac{t^{\ell}q^{\frac{\ell^{2}}{2}}}{(q;q)_{n-\ell}}=\frac{t^{n}q^{\frac{n^{2}}{2}}(-q^{\frac{1}{2}-n}t^{-1};q)_{\infty}}{\theta(-q^{1/2}t)}.$

where, in the second equality we used a version of the $q$-binomial theorem
(A.5). Noting from the definitions of the $q$-Pochhammer symbol (A.1) and
(A.2), the numerator of the last expression becomes

(2.30) $\displaystyle
t^{n}q^{n^{2}/2}\left(-q^{1/2}/tq^{n};q\right)_{\infty}=(-tq^{1/2};q)_{n}(-q^{1/2}/t;q)_{\infty},$

we arrive at the right-hand side of (2.26). ∎

By using (2.26), we see that the left-hand side of (1.11) is expressed as

(2.31)
$\displaystyle\mathbb{E}\left[\frac{1}{(-tq^{\frac{1}{2}+k-\mu_{1}})}\right]=\mathbb{P}\left(\mu_{1}+\chi+S\leq
k\right).$

Thus (1.11) can be rewritten as

(2.32) $\displaystyle\mathbb{P}\left(\mu_{1}+\chi+S\leq
k\right)=\mathbb{P}(\lambda_{1}+S\leq k),$

which is clearly equivalent to (1.10).

##  3\. Equivalence of two Fredholm determinants by shift of contours

In this section we will force some conditions on parameters $a_{i},b_{i}$,
$i=1,\dots,N$. Namely we assume that $a_{i},b_{i}\in(0;1)$, $i=1,\dots,N$ are
distinct and ordered as $a_{1}>a_{2}>\cdots>a_{N}$ (hence $a_{\min}=a_{1}$ and
$a_{\max}=a_{N}$) and moreover

(3.1) $a_{1}/a_{N}<q^{-\frac{1}{2}+\epsilon},$

for a fixed $\epsilon\in(0,1/2)$. Recall that for the sake of our main result
Theorem 1.1 these restrictions do not constitute a problem, see remark 1.2.

###  3.1. Restating the main theorem as a determinant identity

In the previous section, precisely in Proposition 2.1 and Proposition 2.3, we
have seen that both hand sides of (1.11) can be written as Fredholm
determinants. By this equality (1.11) is equivalent to the following identity
for the two Fredholm determinants.

######  Theorem 3.1.

Let $K(m_{1},m_{2})$ be the kernel given in (2.2) and $L(m_{1},m_{2})$ in
(2.17). Also let $f$ be given by (2.16). Then we have

(3.2)
$\displaystyle\det\left(1-fK\right)_{\ell^{2}(\mathbb{Z})}=\det\left(1+fL\right)_{\ell^{2}(\mathbb{Z})}.$

In this section we will elaborate the proof of Theorem 3.1. Comparing the
expressions of the two kernels in (2.2) and in (2.17), we notice that both of
them are written in the double integral forms with the common integrand
$g_{a,b}(z,w;m_{1},m_{2})$, given by (2.3). The only difference between them
is the integration contour of $w$: the contour $D$ in $K(m_{1},m_{2})$
encloses $1/a_{j},~{}j=1,\dots,N$, while $\bar{D}$ in $L(m_{1},m_{2})$ the
origin and the contour of $z$. See Fig.1.

Result of Theorem 3.1 is not trivial outside of the case $q=0$, when the
kernel themselves are equal, i.e. $K(m_{1},m_{2})=-L(m_{1},m_{2})$ for all
$m_{1},m_{2}\in\mathbb{Z}_{\leq 0}$. This can be shown as follows. When $q=0$,
we notice that $f(m)$ becomes the indicator function of $\mathbb{Z}_{\leq 0}$
and the integrand $g_{a,b}(z,w;m_{1},m_{2})$ in (2.3) becomes

(3.3) $\displaystyle
g_{a,b}(z,w;m_{1},m_{2})=\frac{w^{m_{2}}}{z^{m_{1}}}\frac{w}{z-w}\prod_{i=1}^{N}\frac{(1-a_{i}z)}{(1-a_{i}w)}\frac{(1-b_{j}/w)}{(1-b_{j}/z)},$

and we see that the poles in the $w$ variable include only $w=0,z,1/a_{i}$,
$i=1,\dots,N$ and in particular $\infty$ is not a pole since $m_{2}\leq 0$.
Thus using Cauchy’s integral theorem we see that the contour $D$ can be
deformed to $\bar{D}$ without changing the value of the kernel up to sign.

For general $0<q<1$, this is no longer the case. In this case equality does
not hold for the kernels, but it only does at the level of the Fredholm
determinants (3.2). To prove this we consider a sequence of changes of the
contours of $w$ starting from $D$, and including, sequentially, poles of the
form $1/a_{j}q^{k},j=1,\dots,N$ starting from $k=0,1,\dots$ and finally
including the singularity at infinity. We find that such modifications of the
kernel do not affect its Fredholm determinant, so that, once we the all the
singularities have been added in the contour one can switch such contour to
$\bar{D}$ by using the Cauchy theorem.

Notations for the series of contours just described, their corresponding
kernels and Fredholm determinants are provided next.

######  Definition 3.2.

For $\ell\in\mathbb{Z}_{\geq 0}$ we define the kernels

(3.4) $\displaystyle
K_{\ell}(m_{1},m_{2})=\int_{C}\frac{dz}{z}\int_{D_{\ell}}\frac{dw}{w}g_{a,b}(z,w;m_{1},m_{2}),$
(3.5) $\displaystyle
K_{\infty}(m_{1},m_{2})=\int_{C}\frac{dz}{z}\int_{D_{\infty}}\frac{dw}{w}g_{a,b}(z,w;m_{1},m_{2}).$

The function $g_{a,b}$ is given by (2.3) and the contour $C$ encloses the
origin and it has radius larger than $b_{\max}$ as in (2.2), (2.17). The
integration contours $D_{\ell}$ are positively oriented and encircle poles
$1/q^{i}a_{j}$, $i=0,\dots,\ell$, $j=1,\dots,N$ and no other singularity.
Analogously, the contour $D_{\infty}$ encircles all poles encircle poles
$1/q^{i}a_{j}$, $i\in\mathbb{Z}_{\geq 0}$, $j=1,\dots,N$ and no other
singularity. Typical choices for $D_{\ell},D_{\infty}$ are reported in fig. 2.
Recalling the function $f(m)$ defined by (2.16) we further define the Fredholm
determinants

(3.6) $\displaystyle
F_{\ell}:=\det(1-fK_{\ell})_{\ell^{2}(\mathbb{Z})}~{}\text{for
$\ell=0,1,2,\dots$},$ (3.7) $\displaystyle
F_{\infty}:=\det(1-fK_{\infty})_{\ell^{2}(\mathbb{Z})}.$

Figure 2. Contours $D_{\ell}$, $\tilde{D}_{\ell}$, and $D_{\infty}$ are
illustrated. We set $c_{\ell}$ to be the center of the interval
$[\frac{1}{a_{N}q^{\ell}},\frac{1}{a_{1}q^{\ell+1}}]$, i.e.
$c_{\ell}=\frac{1}{2q^{\ell}}\left(\frac{1}{a_{1}q}+\frac{1}{a_{N}}\right)$.

Note that by definition, we see $K_{0}(m_{1},m_{2})=K(m_{1},m_{2})$. Thus by
Proposition 2.2, we immediately see $F_{0}=\det(1-fK)_{\ell^{2}(\mathbb{Z})}$.
In Proposition 4.7 in [IS19], we showed that the defined by the kernel
$K(m_{1},m_{2})$ is trace-class, thus $\det(1-fK)_{\ell^{2}(\mathbb{Z})}$ is
well-defined. This fact can be generalized to $F_{\ell}$ for any
$\ell\in\mathbb{Z}_{\geq 0}$ and also $F_{\infty}$.

For this purpose and also for later use, we will give different
representations for the kernels (3.4) and (3.5). To simplify notation, we
introduce a relabelling of all the inverse of poles $a_{i}q^{j}$ in decreasing
order and call them $\tilde{a}_{r},r\in\mathbb{Z}_{>0}$. Namely, we write

(3.8) $\displaystyle\tilde{a}_{r}=a_{k}q^{u}$

where $r=uN+k$ with $r\in\mathbb{Z}_{>0}$, $r\in\\{1,\dots,N\\}$ and
$u\in\mathbb{Z}_{\geq 0}$. Since we assumed $a_{1}>a_{2}>\cdots>a_{N}$ without
loss of generality, $\tilde{a}_{1}>\tilde{a}_{2}>\cdots$ holds.

For $m\in\mathbb{Z}$ and $r\in\mathbb{Z}_{\geq 0}$, we define two infinite
dimensional matrices, $A(m;r)$, $B(r;m)$, as

(3.9) $\displaystyle A(m;r)$
$\displaystyle=f(m)\int_{C}\frac{dz}{z^{1+m}}\frac{1}{z-\tilde{a}_{r}^{-1}}\prod_{i=1}^{N}\frac{(a_{i}z;q)_{\infty}}{(b_{i}/z;q)_{\infty}},$
(3.10) $\displaystyle B(r;m)$
$\displaystyle=\tilde{a}_{r}^{-m}\underset{w=\tilde{a}_{r}^{-1}}{\text{Res}}\prod_{i=1}^{N}\frac{(b_{i}/w;q)_{\infty}}{(a_{i}w;q)_{\infty}}.$

For $\ell\in\mathbb{Z}_{\geq 0}$, we also introduce their truncated version,
$A_{\ell}(m;r)$, $B_{\ell}(r;m)$, defined by

(3.11) $\displaystyle
A_{\ell}(m;r):=A(m;r)\mathbf{1}_{\\{1,\dots,N(\ell+1)\\}}(s),~{}B_{\ell}(r;m):=\mathbf{1}_{\\{1,\dots,N(\ell+1)\\}}(r)B(r;m),$

where for a set $A\subset\mathbb{Z}_{\geq 0}$, $\mathbf{1}_{A}$ is the
indicator function of $A$. By evaluating the poles inside $D_{\ell}$ of (3.4),
we easily find

(3.12) $\displaystyle
f(m_{1})K_{\ell}(m_{1},m_{2})=\sum_{r=1}^{N(\ell+1)}A_{\ell}(m_{1};r)B_{\ell}(r;m_{2}).$

Below in the proof of Proposition 3.3(iii) we will also see that

(3.13) $\displaystyle
f(m_{1})K_{\infty}(m_{1},m_{2})=\sum_{r=1}^{\infty}A(m_{1};r)B(r;m_{2}).$

In the following discussions it is also useful to consider the conjugated
matrices

(3.14) $\displaystyle\tilde{A}(m;r)$
$\displaystyle=\tau(m)A(m;r)\sigma(r),\qquad\tilde{B}(m;r)=\sigma^{-1}(r)B(r;m)\tau^{-1}(m),$

where

(3.15)
$\displaystyle\tau(m)=\begin{cases}a_{1}^{-m}q^{-\frac{m^{2}}{2N}-\frac{m}{2}+\epsilon
m},&m\geq 0,\\\
1,&m<0\end{cases},\qquad\sigma(r)=\begin{cases}q^{-(1-\omega)u},&m\geq 0,\\\
1,&m<0,\end{cases}$

for $\epsilon\in(0,1/2)$ and $\omega\in(0,1/2-\epsilon)$, and also their
truncated versions $\tilde{A}_{\ell}(m;r)$, and $\tilde{B}_{\ell}(r;m)$,

(3.16)
$\displaystyle\tilde{A}_{\ell}(m;r):=\tilde{A}(m;r)\mathbf{1}_{\\{1,\dots,N(\ell+1)\\}}(r),~{}\tilde{B}_{\ell}(r;m):=\mathbf{1}_{\\{1,\dots,N(\ell+1)\\}}(r)\tilde{B}(r;m).$

Note that the value of the Fredholm determinant would not change when we
replace $A_{\ell},B_{\ell},A,B$ by their conjugated versions,
$\tilde{A}_{\ell},\tilde{B}_{\ell},\tilde{A},\tilde{B}$ in the kernel
(3.12),(3.13).

######  Proposition 3.3.

Assume $a_{i},b_{i}\in(0,1)$, $i=1,\dots,N$, $a_{1}>\cdots>a_{N}$, and
$a_{1}/a_{N}<q^{-\frac{1}{2}+\epsilon}$ with $\epsilon\in(0,1/2)$.

  * (i)

The operators with the kernels, $\tilde{A}(m;r)$, $\tilde{B}(r;m)$,
$\tilde{A}_{\ell}(m;r)$, $\tilde{B}_{\ell}(r;m)$ for $m\in\mathbb{Z}$ and
$r,\ell\in\mathbb{Z}_{\geq 0}$ are of Hilbert-Schmidt class.

  * (ii)

$K_{\infty}(m_{1},m_{2})$ can be expressed as the limit
$\ell\rightarrow\infty$ of $K_{\ell}(m_{1},m_{2})$,

(3.17)
$K_{\infty}(m_{1},m_{2})=\lim_{\ell\rightarrow\infty}K_{\ell}(m_{1},m_{2}).$

  * (iii)

The Fredholm determinants $F_{\ell}$ and $F_{\infty}$, defined by (3.6) and
(3.7) respectively, are well-defined.

###### Proof.

(i) By the estimates of $\tilde{A}(m;r),\tilde{B}(r;m)$ in Corollary B.2, it
is easy to check the conditions of the Hilbert-Schmidt operators

(3.18)
$\displaystyle\sum_{m\in\mathbb{Z}}\sum_{r=1}^{\infty}|\tilde{A}(m;r)|^{2}<\infty,\qquad\sum_{m\in\mathbb{Z}}\sum_{r=1}^{\infty}|\tilde{B}(r;m)|^{2}<\infty.$

Obviously the same relations also hold for
$\tilde{A}_{\ell},\tilde{B}_{\ell}$.

For (ii) let us introduce a kernel,

(3.19) $\displaystyle
K(m_{1},m_{2};\sharp)=\int_{C}\frac{dz}{z}\int_{\sharp}\frac{dw}{w}g_{a,b}(z,w,m_{1},m_{2}),\quad
m_{1},m_{2}\in\mathbb{Z},$

where $g_{a,b}(z,w,m_{1},m_{2})$ is defined by (2.3) and
$\sharp\in\\{D_{\ell},\tilde{D}_{\ell}\\}_{\ell=0,1,\dots}\cup\\{D_{\infty}\\}$
represents a contour for $w$. They are illustrated in Fig.2. By definition,
one sees

(3.20) $\displaystyle
K_{\ell}(m_{1},m_{2})=K(m_{1},m_{2};D_{\ell}),~{}K_{\infty}(m_{1},m_{2})=K(m_{1},m_{2};D_{\infty}),~{}$

Using (3.19), we show (3.17). Noting
$D_{\infty}=D_{\ell}\cup\tilde{D}_{\ell}$, we see that

$\displaystyle K(m_{1},m_{2};D_{\infty})$
$\displaystyle=K(m_{1},m_{2};D_{\ell})+K(m_{1},m_{2};\tilde{D}_{\ell})$ (3.21)
$\displaystyle=\lim_{\ell\rightarrow\infty}K(m_{1},m_{2};D_{\ell})+\lim_{\ell\rightarrow\infty}K(m_{1},m_{2};\tilde{D}_{\ell})=\lim_{\ell\rightarrow\infty}K(m_{1},m_{2};D_{\ell})$

Here in the second equality we used the fact that
$D_{\ell}\cup\tilde{D}_{\ell}(=D_{\infty})$ does not depend on $\ell$ while in
the third equality we used the fact
$\lim_{\ell\rightarrow\infty}K(m_{1},m_{2};\tilde{D}_{\ell})=0$, which follows
from estimates of Lemma C.1 (iii).

(iii) By (3.12), (3.16) and (i) we see that the operators with the kernel
$\tau(m_{1})K_{\ell}(m_{1},m_{2})\tau^{-1}(m_{2})$, $\ell=0,1,2,\dots$ are
trace-class. Since the factor $\tau(m_{1})/\tau(m_{2})$ does not affect the
value of the determinant, then $F_{\ell}$ in (3.6) is well-defined. Next
combining (3.12) and (3.17), we have (3.13). Then by the same arguments used
above, the operator with the kernel
$\tau(m_{1})K_{\infty}(m_{1},m_{2})\tau^{-1}(m_{2})$ is trace-class and
$F_{\infty}$ in (3.7) is well-defined. ∎

Theorem 3.1 can be immediately proved by the following lemma.

######  Lemma 3.4.

Denote the left-hand side of (3.2) by $F$, i.e.,
$F:=\det\left(1-fK\right)_{\ell^{2}(\mathbb{Z})}$. The following statements
hold.

  * (i)

$F_{\ell}$ does not depend on $\ell=0,1,2,\dots$ and is equal to $F$, i.e. we
have

(3.22) $\displaystyle F_{\ell}=F,\quad\forall\ell\in\mathbb{Z}_{\geq 0}.$

  * (ii)

We have

(3.23) $\displaystyle\lim_{\ell\rightarrow\infty}F_{\ell}=F_{\infty}.$

  * (iii)

We have $K_{\infty}(m_{1},m_{2})=-L(m_{1},m_{2})$ and thus

(3.24) $\displaystyle
F_{\infty}=\det\left(1+fL\right)_{\ell^{2}(\mathbb{Z})}.$

The proofs of the above (i)-(iii) will be given in Sec. 3.2-3.4 respectively.

###### Proof of Theorem 3.1.

This is a straightforward consequence of Lemma 3.4 which implies

(3.25)
$\displaystyle\det(1-fK)_{\ell^{2}(\mathbb{Z})}=F_{\ell}=F_{\infty}=\det(1+fL)_{\ell^{2}(\mathbb{Z})},$

under the assumptions given at the beginning of the section. ∎

Notice that one can significantly relax conditions on parameters set at the
beginning of the section for Theorem 3.1. For this one could relate matching
of determinants to identity (1.10). Since the latter is an equality between
polynomials of bounded degree in variables $a_{1},\dots,a_{N}$,
$b_{1},\dots,b_{N}$ whose coefficients are rational functions in $q$, it can
be extended to all choices of $a_{i},b_{j}$ and in particular to all choices
for which determinants in (3.2) make sense.

###  3.2. Proof of Lemma 3.4 (i)

The case of $\ell=0$ was already mentioned before Proposition 3.3. It is then
sufficient to show $F_{\ell}=F_{\ell-1}$ for any $\ell=1,2,\dots$. For this
purpose, We rewrite $F_{\ell}$ (3.6) as a determinant with rank $N(\ell+1)$.

######  Lemma 3.5.

We have

(3.26) $\displaystyle
F_{\ell}=\det\left(W_{n,n^{\prime}}\right)_{n,n^{\prime}=1,2,\dots,N(\ell+1)}.$

Here $W_{n,n^{\prime}}$ is defined by

(3.27) $\displaystyle
W_{n,n^{\prime}}=\underset{w=\tilde{a}_{n}^{-1}}{\mathrm{Res}}\prod_{i=1}^{N}\frac{(b_{i}/w;q)_{\infty}}{(a_{i}w;q)_{\infty}}\times\int_{C_{n}}\frac{dz}{z}\frac{-1}{z-\tilde{a}_{n^{\prime}}^{-1}}\prod_{i=1}^{N}\frac{(a_{i}z;q)_{\infty}}{(b_{i}/z;q)_{\infty}}\cdot\frac{\theta(-\tilde{a}_{n}z/tq^{\frac{1}{2}+k})(q;q)_{\infty}^{2}}{\theta(-1/tq^{\frac{1}{2}+k})\theta(\tilde{a}_{n}z)},$

where $C_{n}$ is a positively oriented circle with the condition
$\tilde{a}_{n}^{-1}<|z|<(q\tilde{a}_{n})^{-1}$ and $\theta(x)$ is defined
below (A.7).

###### Proof.

Let us write (3.12) as $fK_{\ell}=A_{\ell}B_{\ell}$, so that
$F_{\ell}=\det(1-fK_{\ell})_{\ell^{2}(\mathbb{Z})}=\det(1-A_{\ell}B_{\ell})_{\ell^{2}(\mathbb{Z})}$.
By Proposition 3.3(i), $A_{\ell},B_{\ell}$ are Hilbert-Schmidt operators and
hence $B_{\ell}A_{\ell}$ is trace class. Thus we can use the basic property of
the determinant
$\det(1-A_{\ell}B_{\ell})_{\ell^{2}(\mathbb{Z})}=\det(\delta_{i,j}-(B_{\ell}A_{\ell})_{i,j})_{i,j=1}^{N(\ell+1)}$
and rewrite $F_{\ell}$ as

(3.28) $\displaystyle
F_{\ell}=\det(1-A_{\ell}B_{\ell})_{\ell^{2}(\mathbb{Z})}=\det\left(\delta_{n,n^{\prime}}-(B_{\ell}A_{\ell})_{n,n^{\prime}}\right)_{n,n^{\prime}=1}^{N(\ell+1)}.$

Here

$\displaystyle(B_{\ell}A_{\ell})_{n,n^{\prime}}$
$\displaystyle=\underset{w=\tilde{a}_{n}^{-1}}{\text{Res}}\prod_{i=1}^{N}\frac{(b_{i}/w;q)_{\infty}}{(a_{i}w;q)_{\infty}}\times\int_{C}\frac{dz}{z}\frac{1}{z-\tilde{a}_{n^{\prime}}^{-1}}\prod_{i=1}^{N}\frac{(a_{i}z;q)_{\infty}}{(b_{i}/z;q)_{\infty}}\sum_{m\in\mathbb{Z}}\frac{f(m)}{(z\tilde{a}_{n})^{m}}$
(3.29)
$\displaystyle=\underset{w=\tilde{a}_{n}^{-1}}{\text{Res}}\prod_{i=1}^{N}\frac{(b_{i}/w;q)_{\infty}}{(a_{i}w;q)_{\infty}}\times\int_{C_{n}^{\prime}}\frac{dz}{z}\frac{1}{z-\tilde{a}_{n^{\prime}}^{-1}}\prod_{i=1}^{N}\frac{(a_{i}z;q)_{\infty}}{(b_{i}/z;q)_{\infty}}\cdot\frac{\theta(-\tilde{a}_{n}z/tq^{\frac{1}{2}+k})(q;q)_{\infty}^{2}}{\theta(-1/tq^{\frac{1}{2}+k})\theta(\tilde{a}_{n}z)},$

where in the second equality we applied a version of the Ramanujan summation
formula (A.7) and $\theta(x)$ is defined below (A.7). Note that to apply
(A.7), we extend the contour $C$ to $C^{\prime}_{n}$, which represents the
circle centered at the origin with radius
$R\in(q/\tilde{a}_{n},1/\tilde{a}_{n})$.

Now we show that the entry $\delta_{n,n^{\prime}}-(BA)_{n,n^{\prime}}$ is
equal to $W_{n,n^{\prime}}$ (3.27). For this we further extend in (3.29) the
radius $R$ of the contour such that $R\in(1/\tilde{a}_{n},1/q\tilde{a}_{n})$.
The integrand in (3.29) does not have a pole at $z=1/\tilde{a}_{n}$ except for
$n=n^{\prime}$, in which case the residue can be easily found to be 1.
Therefore this extension of the contour cancels the term
$\delta_{n,n^{\prime}}$ in the last expression in (3.28) and we arrive at the
expression (3.27). ∎

Now we prove Lemma 3.4 (i). We focus on the representation (3.26) and
especially on the last $N(\ell+1)$-th row of the matrix $W$. We decompose the
elements into two parts as

(3.30) $\displaystyle W_{N(\ell+1),n^{\prime}}$
$\displaystyle=W^{(1)}_{N(\ell+1),n^{\prime}}+W^{(2)}_{N(\ell+1),n^{\prime}}$

for $n^{\prime}=1,\dots,N(\ell+1)$. Here $W^{(1)}_{N(\ell+1),n^{\prime}}$
corresponds to the residue at $z=\tilde{a}_{N(\ell+1)}^{-1}$ while
$W^{(2)}_{N(\ell+1),n^{\prime}}$ corresponds to the element with the shrunk
contour satisfying $q\tilde{a}_{n}^{-1}<|z|<\tilde{a}_{n}^{-1}$. As we stated
in the last paragraph in the proof of Lemma 3.5, one easily sees
$W^{(1)}_{N(\ell+1),n^{\prime}}=\delta_{N(\ell+1),n^{\prime}}$. On the other
hand from the computations

(3.31)
$\displaystyle\underset{w=\tilde{a}_{n}^{-1}}{\text{Res}}\prod_{i=1}^{N}\frac{(b_{i}/w;q)_{\infty}}{(a_{i}w;q)_{\infty}}=\prod_{i=1}^{N}\frac{\tilde{a}_{n}}{\tilde{a}_{n}-a_{i}}\cdot\underset{w=\tilde{a}_{n-N}^{-1}}{\text{Res}}\prod_{i=1}^{N}\frac{(b_{i}/w;q)_{\infty}}{(a_{i}w;q)_{\infty}},$
(3.32)
$\displaystyle\theta(\tilde{a}_{n}z)=\frac{-q}{\tilde{a}_{n}z}\theta(\tilde{a}_{n-N}z),$

we have

(3.33) $\displaystyle
W^{(2)}_{N(\ell+1),n^{\prime}}=-tq^{\frac{1}{2}+k}\prod_{i=1}^{N}\frac{\tilde{a}_{N(\ell+1)}}{\tilde{a}_{N(\ell+1)}-a_{i}}\cdot
W_{N\ell,n^{\prime}}.$

Combining them with the multi-linearity of determinants, we get

(3.34) $\displaystyle
F_{\ell}=\det[W_{n,n^{\prime}}]_{n,n^{\prime}=1}^{N(\ell+1)}=\det[W_{n,n^{\prime}}]_{n,n^{\prime}=1}^{N(\ell+1)-1}.$

Repeating this procedure $N-1$ times we have $F_{\ell}=F_{\ell-1}$. ∎

###  3.3. Proof of Lemma 3.4 (ii)

From (3.12) and Corollary B.2, we find a uniform bound for the kernel
$\tau(m_{1})f(m_{1})K_{\ell}(m_{1},m_{2})\tau^{-1}(m_{2})$, where
$K_{\ell}(m_{1},m_{2})$, $\tau(m)$ and $f(m)$ are defined by (3.4), (3.15) and
(2.16) respectively. We have

(3.35)
$\displaystyle\left|\tau(m_{1})f(m_{1})K_{\ell}(m_{1},m_{2})\tau^{-1}(m_{2})\right|\leq
Dd^{|m_{1}|+|m_{2}|}$

for some constants $D>0$ and $d\in(0,1)$ which do not depend on $\ell$,
$m_{1}$ or $m_{2}$.

Lemma 3.4 (ii) is then immediately proven combining (3.35) with the Hadamard
inequality

(3.36) $\displaystyle|\det
A|\leq\prod_{i=1}^{n}\left(\sum_{j=1}^{n}|a_{i,j}|^{2}\right)^{1/2}$

which holds for any for $n\times n$ matrix $A=(a_{ij})_{i,j=1,\dots,n}$. Note
that $F_{\ell}$ (3.6) can be written as

(3.37) $\displaystyle F_{\ell}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\sum_{m_{1}=-\infty}^{\infty}\cdots\sum_{m_{n}=-\infty}^{\infty}\det\left(\tau(m_{i})\tau^{-1}(m_{j})f(m_{i})K_{\ell}(m_{i},m_{j})\right)_{i,j=1}^{n}.$

By (3.35) and (3.36), the determinant in (3.37) can be bounded as

$\displaystyle~{}\left|\det\left(\tau(m_{i})\tau^{-1}(m_{j})f(m_{i})K_{\ell}(m_{i},m_{j})\right)_{i,j=1}^{n}\right|$
$\displaystyle\leq\prod_{i=1}^{n}\left(\sum_{j=1}^{n}\left|\tau(m_{i})\tau^{-1}(m_{j})f(m_{i})K_{\ell}(m_{i},m_{j})\right|^{2}\right)^{1/2}$
(3.38)
$\displaystyle\leq\prod_{i=1}^{n}\left(\sum_{j=1}^{n}\left|Dd^{|m_{i}|+|m_{j}|}\right|^{2}\right)^{1/2}=D^{n}\prod_{i=1}^{n}d^{|m_{i}|}\left(\sum_{j=1}^{n}d^{2|m_{j}|}\right)^{1/2}\leq
D^{n}n^{\frac{n}{2}}\prod_{i=1}^{n}d^{|m_{i}|}.$

Note that bound (3.3) holds for any $\ell\in\mathbb{Z}_{\geq 0}$. Moreover, we
have

(3.39)
$\displaystyle\sum_{m_{1}=-\infty}^{\infty}\cdots\sum_{m_{n}=-\infty}^{\infty}D^{n}n^{\frac{n}{2}}\prod_{i=1}^{n}d^{|m_{i}|}=D^{\prime
n}n^{\frac{n}{2}}<\infty,$ (3.40)
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}D^{\prime
n}n^{\frac{n}{2}}<\infty$

with some constant $D^{\prime}>0$. Thus by dominated convergence theorem,
Lemma 3.4 (ii) holds. ∎

###  3.4. Proof of Lemma 3.4 (iii)

In this subsection, we will prove $K_{\infty}(m_{1},m_{2})=-L(m_{1},m_{2})$ by
deforming the contour of $w$ from $D_{\infty}$ to $\tilde{C}$ going through
$\Gamma_{c}$, where all contours are depicted in fig. 3. We have to care about
the behavior of the integrand $g_{a,b}(z,w;m_{1},m_{2})$ (2.3) when $|w|$ is
large since the poles $w=1/a_{i}q^{j}$, $i=1,\dots,N$, $j\in\mathbb{Z}_{\geq
0}$ accumulate at the infinite point of $w$ producing an essential
singularity. We estimate the behavior for large $|w|$ using appendix C.

Here for convenience we also use the kernel (3.19) where we set $\sharp$ to be
some contours depicted in Fig.3. Note that by definition, one sees
$K_{\infty}(m_{1},m_{2})=K(m_{1},m_{2};D_{\infty})$, and
$L(m_{1},m_{2})=-K(m_{1},m_{2};\tilde{C})$. Note that $\tilde{C}$ goes around
the origin in negative direction. We now show that, from the properties of the
$q$-Pochhammer symbol discussed in Appendix C, the following holds,

(3.41) $\displaystyle K(m_{1},m_{2};\sharp)=0,~{}\text{when
$\sharp=A_{d},A_{-d},C_{d},C_{-d}$ and $\Gamma_{c_{-}}$.}$

In Fig. 3, these contours are displayed in grey. These facts can be seen in
the following way. In (3.19), we focus on the integration of $w$,

(3.42)
$\displaystyle\int_{\sharp}dw\frac{w^{m}}{z-w}\prod_{i=1}^{N}\frac{{(b_{i}/w;q)_{\infty}}}{(a_{i}w;q)_{\infty}},$

with $m\in\mathbb{Z}$ and $z\in C$ fixed. Here the contour $C$ is illustrated
in fig. 1. We explain only the case $\sharp=\Gamma_{c_{-}}$ since the other
cases follow in a similar way.

First we note that (3.42) does not depend on the value of $c_{-}$, as can be
seen as follows. We consider the contour integral (3.42) with
$\sharp=R_{1}\cup R_{2}\cup R_{3}\cup R_{4}$ where $R_{i},i=1,2,3,4$ are
depicted in Fig. 3. Since there are no singularities inside the rectangle, the
value of this integral is zero by the Cauchy’s theorem. On the other hand from
Lemma C.1 (ii) we find the contributions from both $R_{2}$ and $R_{4}$ are
bounded by $c\exp[-Nc^{\prime}\log^{2}d]$ when $d>1$ for some constants $c$
and $c^{\prime}$ thus they vanish when $d\rightarrow\infty$. Combining these
facts, we find the contribution from $R_{1}$ is equal to that from $R_{3}$ as
$d\rightarrow\infty$ up to sign.

Next, we will show that (3.42) with $\sharp=\Gamma_{c_{-}}$ vanishes when
$c_{-}\rightarrow-\infty$. Combining Lemma C.1 (i) and (ii), we see that for
$z=a+b\sqrt{-1}$ with $a<-1$ and $b\in\mathbb{R}$,

(3.43) $\displaystyle|(z;q)_{\infty}|\geq
c_{1}\exp[c_{2}\log^{2}(|a|\vee|b|)]$

where $x\vee y=\max(x,y)$. Applying (3.43) to (3.42) with
$\sharp=\Gamma_{c_{-}}$, we have

$\displaystyle\left|\int_{\Gamma_{c_{-}}}dwh(w)\right|\leq\int_{\Gamma^{(1)}_{c_{-}}}dw\left|h(w)\right|+\int_{\Gamma^{(2)}_{c_{-}}}dw\left|h(w)\right|$
(3.44) $\displaystyle\leq
c_{3}|c_{-}|^{m_{2}}\exp(-Nc_{2}\log^{2}(|c_{-}|))+c_{4}\int_{\Gamma_{c-}^{(2)}}dw|w|^{m_{2}}\exp(-Nc_{2}\log^{2}|w|)$

where $h(w)$ is the integrand of (3.42), $\Gamma_{c_{-}}^{(1)}$ is the
vertical line from $c_{-}+c_{-}\sqrt{-1}$ to $c_{-}-c_{-}\sqrt{-1}$ and
$\Gamma_{c_{-}}^{(2)}$ is its complement
$\Gamma_{c_{-}}\setminus\Gamma_{c_{-}}^{(1)}$. Thus we find that in the last
expression in (3.44), both two terms vanishes as $|c_{-}|\rightarrow\infty$
and we arrive at the conclusion.

Now we will prove Lemma 3.4 (iii) by showing the following two relations

(3.45a) $\displaystyle K(m_{1},m_{2};\Gamma_{c})=K_{\infty}(m_{1},m_{2}),$
(3.45b) $\displaystyle K(m_{1},m_{2};\Gamma_{c})=-L(m_{1},m_{2}).$

First we prove (3.45a). Noting $\Gamma_{c}=A_{d}\cup\tilde{D}\cup A_{-d}$, and
(3.41), we have

(3.46) $\displaystyle
K(m_{1},m_{2};\Gamma_{c})=K(m_{1},m_{2};\tilde{D})+K(m_{1},m_{2};A_{d})+K(m_{1},m_{2};A_{-d})=K(m_{1},m_{2};\tilde{D}).$

Next we show the relation (3.45b). Noting the decomposition
$\tilde{C}=\Gamma_{c}\cup C_{d}\cup C_{-d}\cup\Gamma_{c_{-}}$, and (3.41), we
have

(3.47) $\displaystyle K(m_{1},m_{2},\tilde{C})=K(m_{1},m_{2},\Gamma_{c}).$

Combining this with the fact $L(m_{1},m_{2})=-K(m_{1},m_{2};\tilde{C})$, we
arrive at (3.45b). ∎

Figure 3. All of the contours appearing in the proof of Lemma 3.4. We set
$c>0$, $c_{-}<0$.

#### Acknowledgments

This work was initiated during the MATRIX program ”Non-equilibrium systems and
special functions” in 2018. The authors are grateful to Dan Betea and Jérémie
Bouttier for introduction to their recent work [BB19] and for discussions
about the periodic Schur measure and the free Fermion at positive temperature
during the program. In particular TS thanks DB for pointing out I.5,28(a) of
Macdonald’s book. We greatly appreciate creative atmosphere and warm
hospitality of the MATRIX institute. The work of TI has been supported by JSPS
KAKENHI Grant Nos. 16K05192, 19H01793, and 20K03626. The work of TS has been
supported by JSPS KAKENHI Grants No.JP15K05203, No. JP16H06338, No.
JP18H01141, No. JP18H03672, No. JP19L03665, No. JP21H04432.

##  Appendix A Notations and formulas

In this sections we summarize the notations and formulas which we frequently
use in this paper.

###  A.1. Partitions

A partition $\lambda$ is a sequence of positive integers
$\lambda=(\lambda_{1},\lambda_{2},\dots),~{}\lambda_{j}\in\mathbb{Z}_{>0}$
with non decreasing orders $\lambda_{1}\geq\lambda_{2}\geq\cdots$. The length
of partition $\ell(\lambda)$ is the number of nonzero elements in $\lambda$
and the size of $\lambda$ is defined as $|\lambda|=\sum_{j}\lambda_{j}$. Let
$\mathcal{P}$ be the set of the partitions whose lengths are finite. The
partition $\lambda\in\mathcal{P}$ can be expressed in another way
$\lambda=1^{m_{1}}2^{m_{2}}\cdots$, where $m_{j}$ represents the number of the
elements $\lambda_{k}$s such that $\lambda_{k}=j$. Note that $\sum_{j}m_{j}=n$
and $|\lambda|=\sum_{j}jm_{j}$. For example $\lambda=(3,3,2,2,1,1,1,0,\cdots)$
is expressed as $1^{3}\,2^{2}\,3^{2}$. For $\lambda,\mu\in\mathcal{P}$, we
denote $\mu\subset\lambda$ if $\mu_{i}\leq\lambda_{i},~{}i=1,2,\dots$.

###  A.2. $q$-Series and sum over partitions

Here we summarize a few formulas for $q$-series and related summations over
partitions. For more detail, we refer the reader to Secs.10 and 11 in [AAR00].
Define

(A.1) $\displaystyle(a;q)_{\infty}=\prod_{j=0}^{\infty}(1-aq^{j}),$ (A.2)
$\displaystyle(a;q)_{n}=\frac{(a;q)_{\infty}}{(aq^{n};q)_{\infty}}=\begin{cases}(1-a)(1-aq)\cdots(1-aq^{n-1}),&n\geq
0,\\\ (1-a/q)(1-a/q^{2})\cdots(1-a/q^{m}),&n=-m,~{}m>0.\end{cases}$

Note that $(q;q)_{-m}^{-1}=0$ for $m\in\mathbb{Z}_{>0}$.

Next we give some formulas for the $q$-series. First we state the $q$-binomial
theorem: For $|z|<1,|q|<1$ and $a\in\mathbb{C}$,

(A.3)
$\displaystyle\sum_{n=0}^{\infty}\frac{(a;q)_{n}}{(q;q)_{n}}z^{n}=\frac{(az;q)_{\infty}}{(z;q)_{\infty}}.$

For the proof, see e.g. Theorem 10.2.1 in [AAR00]. In this paper we use the
two special cases,

(A.4)
$\displaystyle\sum_{n=0}^{\infty}\frac{z^{n}}{(q;q)_{n}}=\frac{1}{(z;q)_{\infty}},$
(A.5)
$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\frac{n(n-1)}{2}}}{(q;q)_{n}}z^{n}=(z;q)_{\infty}.$

The first equality (A.4) holds for $|z|<1$ while the second one (A.5) holds
for arbitrary $z\in\mathbb{C}$. They are written in Corollary 10.2.2 (a) and
(b) respectively in [AAR00]. One immediately obtains (A.4) (resp. (A.5)) by
setting $a=0$ (resp. replacing $a\rightarrow 1/a$ $x\rightarrow ax$, then
taking the limit $a\rightarrow 0$) in (A.3).

The Ramanujan’s summation formula is a generalization of the $q$-Binomial
theorem (A.3) to the sum over $\mathbb{Z}$: For $|q|<1,|b/a|<|z|<1$,

(A.6)
$\sum_{n\in\mathbb{Z}}\frac{(a;q)_{n}}{(b;q)_{n}}z^{n}=\frac{(az;q)_{\infty}(\frac{q}{az};q)_{\infty}(q;q)_{\infty}(\frac{b}{a};q)_{\infty}}{(z;q)_{\infty}(\frac{q}{a};q)_{\infty}(b;q)_{\infty}(\frac{b}{az};q)_{\infty}}.$

For a proof see, e.g. Theorem 10.5.1 in [AAR00]. In this paper we use the
special case $a=-t$, $b=-qt$, $z=q/w$ with $t\in\mathbb{C}\setminus\\{0\\}$,

(A.7)
$\displaystyle\sum_{n\in\mathbb{Z}}\frac{tq^{n}w^{-n}}{1+tq^{n}}=\frac{\theta(-w/t)(q;q)_{\infty}^{2}}{\theta(-1/t)\theta(w)},$

where $\theta(x)$ is defined by using (A.1) as
$\theta(x)=(x;q)_{\infty}(q/x;q)_{\infty}$ and according to the condition on
$z$ in (A.6), $w$ should satisfy $|q|<|w|<1$.

Finally we present a well-known formula for the generating function of
$\lambda$:

(A.8) $\sum_{\lambda\in\mathcal{P}}q^{|\lambda|}=\frac{1}{(q;q)_{\infty}}$

and its refinement with a fixed $\lambda_{1}$:

(A.9) $\displaystyle\sum_{\begin{subarray}{c}\lambda\in\mathcal{P}\\\
\lambda_{1}=n\end{subarray}}q^{|\lambda|}=\frac{q^{n}}{(q;q)_{n}}.$

(A.9) is obtained as follows. Switching to the notation of partition as
$\lambda=1^{m_{1}}2^{m_{2}}\cdots$, and noting $|\lambda|=\sum_{j}jm_{j}$, we
have

(A.10) $\displaystyle\sum_{\begin{subarray}{c}\lambda\in\mathcal{P}\\\
\lambda_{1}=n\end{subarray}}q^{|\lambda|}=\prod_{j=1}^{n-1}\sum_{m_{j}=0}^{\infty}q^{jm_{j}}\cdot\sum_{m_{n}=1}^{\infty}q^{nm_{n}}=\prod_{j=1}^{n-1}\frac{1}{1-q^{j}}\cdot\frac{q^{n}}{1-q^{n}}.$

(A.8) follows immediately from (A.9) by taking the sum over $n$ and noting
(A.4).

##  Appendix B Properties of the kernels $A(m;r)$ and $B(r;m)$

In this appendix we give upper bounds for matrix elements of $A(m;r)$ (3.9)
and $B(r;m)$ (3.10), which are useful to prove the well-definedness of the
Fredholm determinants $F_{\ell}$(3.6) and $F_{\infty}$(3.7) in Proposition 3.3
(ii) and Lemma 3.4 (ii) in section 3.3.

######  Lemma B.1.

Recall the assumption on $a_{i},b_{i}$, $i=1,\dots,N$ stated at the begining
of section 3. Recall also the notation (3.8), $\tilde{a}_{r}=a_{k}q^{u}$. Let
$C_{+},~{}C_{-},~{}D_{+},$ and $D_{-}$ be constants which do not depend on $m$
or $r$.

  * (i)

Assume $b$ satisfies $b_{\max}<b<1$. We have

(B.1)
$\displaystyle|A(m;r)|\begin{cases}<C_{+}a_{1}^{m}q^{\frac{m^{2}}{2N}+\frac{m}{2}+u},&m\geq
0,\\\ <C_{-}b^{-m}q^{u},&m<0.\end{cases}$

  * (ii)

Assume $a_{1}/a_{N}<q^{-\frac{1}{2}+\epsilon}$ with $\epsilon\in(0,1/2)$. Then
we have

(B.2)
$\displaystyle|B(r;m)|\begin{cases}<D_{+}a_{N}^{-m}q^{-\frac{m^{2}}{2N}+\epsilon
m-(\frac{1}{2}+\epsilon)u},&m\geq 0,\\\
<D_{-}a_{1}^{-m}q^{u},&m<0.\end{cases}$

From (B.1), (B.2), we immediately obtain the bounds for $\tilde{A}(m;s)$ and
$\tilde{B}(s;m)$ defined by (3.14).

######  Corollary B.2.

With the same notation and assumptions as in Lemma B.1, we have

(B.3) $\displaystyle|\tilde{A}(m;r)|\begin{cases}<C_{+}q^{\epsilon m+\omega
u},&m\geq 0,\\\
<C_{-}b^{-m}q^{u},&m<0,\end{cases}~{}~{}|\tilde{B}(r;m)|\begin{cases}<D_{+}q^{\epsilon
m+(\frac{1}{2}-\epsilon-\omega)u},&m\geq 0,\\\
<D_{-}a_{1}^{-m}q^{u},&m<0.\end{cases}$

Note that both $\tilde{A}(m;r)$ and $\tilde{B}(r;m)$ are bounded exponentially
for $m$ and $r$.

###### Proof of Lemma B.1.

In this proof $C_{i}$, $i=1,\dots,4$ and $D_{j}$, $j=1,2,3$ will denote
constants which do not depend on $m$ or $r$ (i.e. $k,u$).

(i) First we consider the case $m\geq 0$. We note that the singularities of
the integrand of (3.9) are $0$ and $b_{i}q^{j},$ $i=1,\dots,N$,
$j\in\mathbb{Z}_{\geq 0}$ and do not include $\tilde{a}_{r}^{-1}$ because of
the precence of the factor $\prod_{i=1}^{N}(a_{i}z;q)_{\infty}$. Thus we can
extend the contour $C$ arbitrarily without changing the value of $A(m;r)$.
First we give an estimate setting the radius to be $q^{-n}$ with
$n\in\mathbb{Z}_{>0}$ and then specify $n$ giving an optimal estimate. With
this choice of the radius $|z|=q^{-n}$, $f(m)$ in (2.16) and some factors in
the integrand in $A(m;r)$ can be estimated as

(B.4) $\displaystyle|f(m)|<q^{m},\hskip 14.22636pt\frac{2\pi
q^{-n}}{|z^{1+m}|}=2\pi q^{nm},\hskip
14.22636pt\left|\prod_{j=1}^{N}\frac{1}{(b_{j}/z;q)_{\infty}}\right|<\prod_{j=1}^{N}\frac{1}{(b_{j}q^{n};q)_{\infty}}<\prod_{j=1}^{N}\frac{1}{(b_{j};q)_{\infty}},$

where the factor $2\pi q^{-n}$ represents the circumference with radius
$q^{-n}$. We rewrite the remaining factor as

(B.5)
$\displaystyle\frac{1}{z-\tilde{a}_{r}^{-1}}\prod_{i=1}^{N}(a_{i}z;q)_{\infty}=-a_{k}q^{u}\prod_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}^{N}(a_{i}z;q)_{\infty}\cdot(a_{k}z;q)_{u}(a_{k}q^{u+1}z;q)_{\infty}.$

The factor $-a_{k}q^{u}\prod_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{N}(a_{i}z;q)_{\infty}$ in (B.5) can be estimated as

$\displaystyle\left|-a_{k}q^{u}\prod_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{N}(a_{i}z;q)_{\infty}\right|$
$\displaystyle<a_{k}q^{u}\prod_{\begin{subarray}{c}i=1\\\ i\neq
k\end{subarray}}^{N}(-a_{i}/q^{n};q)_{\infty}=a_{k}q^{u-(N-1)\frac{n(n+1)}{2}}\prod_{\begin{subarray}{c}i=1\\\
i\neq k\end{subarray}}^{N}a_{i}^{n}(-q/a_{i};q)_{n}(-a_{i};q)_{\infty}$ (B.6)
$\displaystyle<C_{1}q^{u-(N-1)\frac{n(n+1)}{2}}a_{1}^{(N-1)n},$

where in the second equality we used the relation

(B.7)
$\displaystyle(x/q^{n};q)_{\infty}=(-x)^{n}q^{-\frac{n(n+1)}{2}}(q/x;q)_{n}(x;q)_{\infty}.$

For the remaining factors in (B.5), we find

$\displaystyle\left|(a_{k}z;q)_{u}(a_{k}q^{u+1}z;q)_{\infty}\right|$
$\displaystyle<(-a_{k}/q^{n};q)_{u}(-a_{k}q^{u-n+1};q)_{\infty}$ (B.8)
$\displaystyle<\begin{cases}C_{2}a_{k}^{n}q^{-\frac{n(n+1)}{2}},&u\geq n,\\\
C_{3}a_{k}^{n}q^{-\frac{n(n+1)}{2}+n-u},&n>u.\end{cases}$

Here in the second inequality we used (B.7). Thus we have a bound regardless
of the order of $n$ and $u$

(B.9)
$\displaystyle\left|(a_{k}z;q)_{u}(a_{k}q^{u+1}z;q)_{\infty}\right|<C_{4}a_{1}^{n}q^{-\frac{n(n+1)}{2}}.$

Putting together (B.4), (B), and (B.9), we have

$\displaystyle|A(m;r)|$
$\displaystyle<C_{+}a_{1}^{Nn}q^{nm-N\frac{n(n+1)}{2}+u+m}$ (B.10)
$\displaystyle=C_{+}a_{1}^{Nn}q^{-\frac{N}{2}\left(n-\frac{m}{N}+\frac{1}{2}\right)^{2}+\frac{N}{2}\left(\frac{m}{N}-\frac{1}{2}\right)^{2}+u+m}$

Now we specify $n$ which gives an optimal estimate. From the last expression
in (B), a simple computation shows that such optimal bound is given by the
nonnegative integer which is closest to $m/N-1/2$ and this proves the first
inequality in (B.1).

Next we consider the case $m<0$. In this case, we set the radius of contour
$C$ to be $b$ appearing in the assumption of (i). We find

$\displaystyle|f(m)|<C_{4}\qquad\frac{2\pi b}{|z|^{1+m}}=2\pi
b^{-m},\qquad\frac{1}{|z-\tilde{a}^{-1}_{r}|}<\frac{a_{k}q^{u}}{1-a_{k}bq^{u}}<\frac{a_{k}}{1-a_{k}}q^{\ell},$
(B.11)
$\displaystyle\left|\prod_{j=1}^{M}\frac{(a_{i}z;q)_{\infty}}{(b_{j}/z;q)_{\infty}}\right|<\prod_{i=1}^{N}\frac{(-a_{i}b;q)_{\infty}}{(b_{i}/b;q)_{\infty}}.$

Combining these estimates, we arrive at the bottom estimate of (B.1).

(ii) Writing $\tilde{a}_{r}=a_{k}q^{u}$ and calculating explicitly the residue
in (3.10) , we have

$\displaystyle B(r;m)$
$\displaystyle=-a_{k}^{-(m+1)}q^{-u(m+1)}\prod_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\frac{1}{(a_{i}/a_{k}q^{u};q)_{\infty}}\cdot\frac{1}{(1/q^{u};q)_{\infty}(q;q)_{\infty}}\cdot\prod_{j=1}^{N}\frac{1}{(a_{k}b_{j}q^{u};q)_{\infty}}$
(B.12)
$\displaystyle=-a_{k}^{-(m+1)}q^{-u(m+1)+N\frac{u(u+1)}{2}}\prod_{\begin{subarray}{c}i=1\\\
i\neq
k\end{subarray}}\frac{(-a_{k}/a_{i})^{u}}{(a_{i}/a_{k}q^{u};q)_{\infty}}\cdot\frac{(-1)^{u}}{(q;q)_{u}(q;q)_{\infty}}\cdot\prod_{j=1}^{N}\frac{1}{(a_{k}b_{j}q^{u};q)_{\infty}},$

where in the second equality we used (B.7). Using the condition
$a_{1}/a_{N}<q^{-1/2+\epsilon}$ we have the following estimate,

(B.13) $\displaystyle|B(r;m)|$
$\displaystyle<D_{1}a_{k}^{-(m+1)}q^{-um+N\frac{u(u+1)}{2}-N\left(\frac{1}{2}-\epsilon\right)u-\left(\frac{1}{2}+\epsilon\right)u}.$

For $m\geq 0$, combining (B.13) with the following bound

(B.14) $\displaystyle
q^{-um+N\frac{u(u+1)}{2}-N\left(\frac{1}{2}-\epsilon\right)u}=q^{\frac{N}{2}\left(u-\frac{m}{N}+\epsilon\right)^{2}-\frac{N}{2}\left(\frac{m}{N}-\epsilon\right)^{2}}<D_{2}q^{-\frac{m^{2}}{2N}+\epsilon
m},$

we have the desired estimate. On the other hand for $m<0$, we use

(B.15) $\displaystyle q^{-um}<q^{\ell},\qquad
q^{N\frac{u(u+1)}{2}-N\left(\frac{1}{2}-\epsilon\right)u-(\frac{1}{2}+\epsilon)\ell}<D_{3}.$

From (B.13) and (B.15), we obtain the bottom relation in (B.2). ∎

##  Appendix C Decay estimates of $q$-Pochhammer symbols

In this section, we provide simple bounds for the $q$-Pochhammer symbol
$(z;q)_{\infty}$, where $z\in\mathbb{C}$ and $0<q<1$. For our purpose, it is
sufficient to have bounds in the following three cases though they are not
exclusive nor exhaustive.

######  Lemma C.1.

Let us write $z=a+\sqrt{-1}b$ with $a,b\in\mathbb{R}$. For $0<q<1$ fixed,
there exist positive constants $c_{1}=c_{1}(q)$ and $c_{2}=c_{2}(q)$ such that

  * (i)

for $a\leq-1$,

(C.1) $\displaystyle|(z;q)_{\infty}|\geq
c_{1}\exp\left[c_{2}\log^{2}|a|\right]$

  * (ii)

for $a<1$ and $|b|>1$,

(C.2) $\displaystyle|(z;q)_{\infty}|\geq
c_{1}\exp\left[c_{2}\log^{2}|b|\right]$

  * (iii)

for $a>1$,

(C.3) $\displaystyle|(z;q)_{\infty}|\geq
c_{1}[b^{2}\vee(q^{\alpha-1}-1)(1-q^{\alpha})]\exp\left[c_{2}\log^{2}|a|\right],$

where $x\vee y:=max(x,y)$ and
$\alpha=\left\lfloor\log_{q^{-1}}a\right\rfloor-\log_{q^{-1}}a$ with $\lfloor
x\rfloor$ being the minimum integer greater than $x$.

###### Proof.

We repeatedly use the simple estimate

(C.4) $\displaystyle|1-w|=\sqrt{(1-\Re w)^{2}+(\Im
w)^{2}}\geq\begin{cases}|1-\Re w|,\\\ |\Im w|,\end{cases}$

which hold for any $w\in\mathbb{C}$. We use the top or bottom estimate
depending on the situations.

(i) In terms of $\alpha\in[0,1)$ and $J\in\mathbb{Z}_{>0}$, we express
$a(<-1)$ as $a=-q^{\alpha-J}$. Applying the top estimate in (C.4) to the
factor $|1-zq^{k}|$, we have

(C.5) $\displaystyle|1-zq^{k}|=\sqrt{(1+aq^{k})^{2}+(bq^{k})^{2}}\geq
1+q^{\alpha-J+k}\geq\begin{cases}q^{\alpha-J+k},&{\text{for
$k=0,1,\dots,J-1$}},\\\ 1-q^{k+1-J},&{\text{for $k=J,J+1,\dots$}}.\end{cases}$

Thus we have

(C.6) $\displaystyle|(z;q)_{\infty}|\geq q^{\alpha
J-\frac{J(J+1)}{2}}(q;q)_{\infty}=q^{-\frac{1}{2}(\alpha-J)^{2}+\frac{1}{2}(\alpha-J)+\frac{\alpha^{2}-\alpha}{2}}(q;q)_{\infty}\geq
q^{-\frac{1}{2}(\alpha-J)^{2}}(q;q)_{\infty},$

where in the last inequality we used the fact $0\leq\alpha<1$ and $1\leq J$.
Recalling $\alpha-J=\log_{q}|a|$, one easily sees

(C.7) $\displaystyle
q^{-\frac{(\alpha-J)^{2}}{2}}=e^{\frac{-(\log|a|)^{2}}{2\log q}}.$

From (C.6) and (C.7), we obtain (C.1).

(ii) We adopt the same strategy as above. In this case we write
$|b|=q^{\alpha-J}$, where $0\leq\alpha<1$ and $J\in\mathbb{Z}_{>0}$. Using
(C.4), we have

(C.8)
$\displaystyle|1-zq^{k}|\geq\begin{cases}|b|q^{k}=q^{\alpha-J+k},&{\text{for
$k=0,1,\dots,J-1$}},\\\ 1-aq^{k}\geq 1-q^{k+1-J},&{\text{for
$k=J,J+1,\dots$}},\end{cases}$

where we apply the top (resp. bottom) estimate in (C.4) to the case $k<J$
(resp $k\geq J$) and in the inequailty in the bottom row we used the
assumption $a<1$. Note that the estimate (C.8) is exactly the same as (C.5)
except that $\alpha-J$ represents $\log_{q}|b|$. Thus similarly to the case
(i) we get (C.2).

(iii) As in the case (i), we express $a$ as $a=q^{\alpha-J}$ with $a\in[0,1)$
and $J\in\\{1,2,\dots\\}$. First we consider the case where
$b\geq(q^{\alpha-1}-1)(1-q^{\alpha})$. We use the estimate as follows. For
$k\in\mathbb{Z}_{\geq 0}$, we have

(C.9) $\displaystyle~{}~{}|1-zq^{k}|$
$\displaystyle\geq\begin{cases}|1-aq^{k}|=q^{\alpha-J+k}(1-q^{J-\alpha-k})\geq
q^{\alpha-J+k}({1-q}),&{\text{for $0\leq k\leq J-2$}},\\\ bq^{k},&{\text{for
$k=J-1,J$}},\\\ |1-aq^{k}|=1-q^{\alpha-J+k}\geq 1-q^{-J+k},&{\text{for
$J+1\leq k$}}.\end{cases}$

Thus we have

$\displaystyle|(z;q)_{\infty}|$
$\displaystyle\geq\prod_{k=0}^{J-2}q^{\alpha-J+k}(1-q)\cdot\prod_{k=J-1}^{J}bq^{k}\cdot\prod_{k=J+1}^{\infty}(1-q^{-J+k})$
(C.10)
$\displaystyle=q^{\alpha(J-1)-\frac{J(J+1)}{2}+2J}(1-q)^{J}b^{2}(q;q)_{\infty}.$

The dominant factor in the right-hand side of (C) is $q^{-(\alpha-J)^{2}/2}$
again and thus from (C.7), we obtain the desired form (C.3).

The case $b^{2}<(q^{\alpha-1}-1)(1-q^{\alpha})$ of (C.3) can be readily
obtained by replacing the second inequality in (C.9) (corresponding to the
case $k=J-1,~{}J$) with $|1-zq^{k}|\geq|1-aq^{k}|=|1-q^{\alpha-J+k}|$. ∎

## References

  * [AAR00] G.E. Andrews, R. Askey, and R. Roy.  Special Functions.  Encyclopedia of Mathematics and its Applications 71. Cambridge University Press, 2000.
  * [ACQ11] G. Amir, I. Corwin, and J Quastel.  Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions.  Comm. Pure Appl. Math., 64:466–537, 2011.
  * [ARVP21] A. Ahn, M. Russkikh, and R. Van Peski.  Lozenge tilings and the Gaussian free field on a cylinder.  arXiv:2105.00551, 2021.
  * [BB19] D. Betea and J. Bouttier.  The periodic Schur process and free Fermions at finite temperature.  Math. Phys. Anal. Geom., 22:3, 2019.
  * [BBNV18] D. Betea, J. Bouttier, P. Nejjar, and M. Vuletić.  The free boundary schur process and applications I.  Ann. Henri Poincare, 19:3663–3742, 2018.
  * [BC14] A. Borodin and I. Corwin.  Macdonald processes.  Probab. Theory Relat. Fields, 158:225–400, 2014.
  * [BCFV15] A. Borodin, I. Corwin, P. Ferrari, and B. Vető.  Height fluctuations for the stationary KPZ equation.  Math. Phys. Anal. Geom., 18:20, 2015.
  * [BCS14] A. Borodin, I. Corwin, and T. Sasamoto.  From duality to determinants for $q$-TASEP and ASEP.  Ann. Probab., 42:2314–2382, 2014.
  * [BF14] A. Borodin and P. L. Ferrari.  Anisotropic growth of random surfaces in 2 + 1 dimensions.  Commun. Math. Phys., 325:603–684, 2014.
  * [BG16a] A. Borodin and V. Gorin.  Lectures on integrable probability.  In Probability and Statistical Physics in St. Petersburg, volume 91 of Proceedings of Symposia in Pure Mathematics, pages 155–214. AMS, 2016.  arXiv:1212.3351 [math.PR].
  * [BG16b] A. Borodin and V. Gorin.  Moments match between the KPZ equation and the Airy point process.  SIGMA, 12:102, 2016.
  * [BMP19] A. Bufetov, M. Mucciconi, and L. Petrov.  Yang-Baxter random fields and stochastic vertex models.  to appear in Adv. Math., arXiv:1905.06815, 2019.
  * [BO17] A. Borodin and G. Olshanski.  The ASEP and determinantal point processes.  Commun. Math. Phys., 353:853–903, 2017.
  * [Bor07] A. Borodin.  Periodic Schur process and cylindric partitions.  Duke Math. J., 140:391–468, 2007.
  * [Bor18] A. Borodin.  Stochastic higher spin six vertex model and Macdonald measures.  J. Math. Phys., 59:023301, 2018.
  * [BP16] A. Borodin and L. Petrov.  Nearest neighbor Markov dynamics on Macdonald processes.  Adv. Math., 300:71–155, 2016.  arXiv:1305.5501 [math.PR].
  * [BR01] J. Baik and E.M. Rains.  Algebraic aspects of increasing subsequences.  Duke Math. J., 109:1–65, 2001.
  * [Bum13] D. Bump.  Lie Groups.  Graduate texts in mathematics. Springer, 2013.
  * [CLDR10] P. Calabrese, P. Le Doussal, and A. Rosso.  Free-energy distribution of the directed polymer at high temperature.  EPL, 90:20002, 2010.
  * [DLDMS15] D. Dean, P. Le Doussal, S. Majumdar, and G. Schehr.  Finite-temperature free fermions and the Kardar-Parisi-Zhang equation at finite time.  Phys. Rev. Lett., 114:110402, 2015.
  * [Dot10] V. Dotsenko.  Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers.  EPL, 90:20003, 2010.
  * [Ges90] I.M. Gessel.  Symmetric functions and $P$-recursiveness.  J. Comb. Theory Ser. A., 53:257–285, 1990.
  * [IMS21] T. Imamura, M. Mucciconi, and T. Sasamoto.  Skew RSK dynamics: Greene invariants, affine crystals and applications to $q$-Whittaker polynomials.  in preparation, 2021.
  * [IS19] T. Imamura and T. Sasamoto.  Fluctuations for stationary $q$-TASEP.  Probab. Theory Relat. Fields, 174:647–730, 2019.
  * [Joh00] K. Johansson.  Shape fluctuations and random matrices.  Commun. Math. Phys., 209:437–476, 2000.
  * [Joh07] K. Johansson.  From Gumbel to Tracy–Widom.  Probab. Theory Relat. Fields, 138:75–112, 2007.
  * [Kos21] S. Koshida.  Free field theory and observables of periodic Macdonald processes.  J. Comb. Theory Ser. A., 182:105473, 2021.
  * [LW20] K. Liechty and D. Wang.  Asymptotics of free fermions in a quadratic well at finite temperature and the Moshe-Neuberger-Shapiro random matrix model.  Ann. Inst. H. Poincaré Probab. Statist., 56:1072–1098, 2020.
  * [Mac98] I.G. Macdonald.  Symmetric Functions and Hall Polynomials.  Oxford classic texts in the physical sciences. Clarendon Press, 1998.
  * [MP17] K. Matveev and L. Petrov.  $q$-randomized Robinson-Schensted-Knuth correspondences and random polymers.  Annales de l’IHP D, 4(1):1–123, 2017.
  * [MP20] M. Mucciconi and L. Petrov.  Spin q-whittaker polynomials and deformed quantum toda.  arXiv:2003.14260, 2020.
  * [Oko01] A. Okounkov.  Infinite wedge and random partitions.  Sel. Math. New Ser., 7(1):57–81, 2001.
  * [OP17] D. Orr and L. Petrov.  Stochastic higher spin six vertex model and $q$-taseps.  Adv. Math., 317:473–525, 2017.
  * [Rai98] E.M. Rains.  Increasing subsequences and the classical groups.  Electron. J. Comb., 5:R12, 1998.
  * [RW15] E.M. Rains and S.O. Warnaar.  Bounded Littlewood identities.  arXiv:1506.02755, 2015.
  * [Sag00] B. E. Sagan.  The Symmetric Group.  Graduate texts in mathematics. Springer, 2000.
  * [Sos00] A. Soshnikov.  Determinantal random point fields.  Russ. Math. Surv., 55:923, 2000.
  * [SS10a] T. Sasamoto and H. Spohn.  Exact height distributions for the KPZ equation with narrow wedge initial condition.  Nucl. Phys. B, 834:523–542, 2010.
  * [SS10b] T. Sasamoto and H. Spohn.  One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality.  Phys. Rev. Lett., 104:230602, 2010.
  * [WW09] J. Warren and P. Windridge.  Some examples of dynamics for gelfand-tsetlin patterns.  Electron. J. Probab., 14:1745–1769, 2009.