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# Biorthogonal measures, polymer partition functions, and random matrices

Mattia Cafasso  Univ Angers, SFR MATHSTIC, F-49000 Angers,
<EMAIL_ADDRESS>Tom Claeys  Institut de Recherche en
Mathématique et Physique, UCLouvain, Chemin du Cyclotron 2, 1348 Louvain-la-
Neuve<EMAIL_ADDRESS>
###### Abstract

We develop the study of a particular class of biorthogonal measures,
encompassing at the same time several random matrix models and partition
functions of polymers. This general framework allows us to characterize the
partition functions of the Log Gamma polymer and the mixed polymer in terms of
explicit biorthogonal measures, as it was previously done for the homogeneous
O’Connell-Yor polymer by Imamura and Sasamoto. In addition, we show that the
biorthogonal measures associated to these three polymer models (Log Gamma,
O’Connell-Yor, and mixed polymer) converge to random matrix eigenvalue
distributions in small temperature limits. We also clarify the connection
between different Fredholm determinant representations and explain how our
results might be useful for asymptotic analysis and large deviation estimates.

##  1 Introduction

We study a class of biorthogonal measures on $\mathbb{R}^{N}$, characterized
by kernels admitting double contour integral representations. Specific
measures of this form are known to be eigenvalue distributions of random
matrices, for instance in the Gaussian, Laguerre, and Jacobi Unitary Ensembles
(GUE, LUE, and JUE) [30, 48], as well as in generalizations of these like the
GUE with external source [19, 20], the LUE with external source [4, 29], sums
of LUE and GUE matrices [24], product random matrix ensembles [1, 2, 31, 32,
41, 45, 46], and Muttalib-Borodin ensembles [11]. Another measure of this form
characterizes the partition function of the O’Connell-Yor polymer partition
function [36]. The first purpose of this paper is to study general properties
of this class of biorthogonal measures. This will allow us to interpret all of
the above models as instances of a unified framework, and to identify other
models which fit in our scheme. More precisely, we will prove that the Log
Gamma polymer [50] and the mixed polymer [13] partition functions can also be
characterized in terms of explicit biorthogonal measures. These
characterizations are direct analogues of a result by Imamura and Sasamoto
[36] for the homogeneous O’Connell-Yor polymer. Moreover, they can be seen as
finite temperature generalizations of the connection between last passage
percolation with exponential weights and the LUE [39, 17, 27], and of the
connection between Brownian queues and the GUE [7, 35] (see also [5] for a
different last passage percolation model). From another perspective, they are
finite $N$ variants of the connection between the Kardar-Parisi-Zhang (KPZ)
equation and the Airy point process [3]. In addition, the biorthogonal
measures enable us to clarify the connection between various Fredholm
determinant expressions which appeared in the literature before, and to derive
a novel representation of these Fredholm determinants, which is possibly more
convenient in view of asymptotic analysis and large deviation estimates.

A probability measure on $\mathbb{R}^{N}$ of the form

$\frac{1}{Z_{N}}\det\left(f_{m}(x_{k})\right)_{k,m=1}^{N}\det\left(g_{m}({x_{k}})\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm
d}x_{k},$ (1.1)

for certain sets of functions $f_{1},\ldots,f_{N}$ and $g_{1},\ldots,g_{N}$,
is called a biorthogonal ensemble [11]. Observe that there is a lot of freedom
in the choices of $f_{1},\ldots,f_{N}$ and $g_{1},\ldots,g_{N}$ for a given
biorthogonal ensemble, since one can take linear combinations of columns and
rows of the matrices $\left(f_{m}(x_{k})\right)_{k,m=1}^{N}$ and
$\left(g_{m}(x_{k})\right)_{m,k=1}^{N}$ without affecting the distribution.
Biorthogonal ensembles are special cases of determinantal point processes: for
$n=1,\ldots,N$, the $n$-point correlation function

$\rho_{n,N}(x_{1},\ldots,x_{n}):=\frac{N!}{(N-n)!Z_{N}}\int_{\mathbb{R}^{N-n}}\det\left(f_{m}(x_{k})\right)_{k,m=1}^{N}\det\left(g_{m}({x_{k}})\right)_{m,k=1}^{N}\prod_{k=n+1}^{N}{\rm
d}x_{k}$ (1.2)

can be expressed as a determinant:

$\rho_{n,N}(x_{1},\ldots,x_{n})=\det\left(L_{N}(x_{m},x_{k})\right)_{m,k=1}^{n}.$
(1.3)

The correlation kernel $L_{N}$ can be taken of the form

$L_{N}(x,x^{\prime})=\sum_{k=1}^{N}F_{k}(x)G_{k}(x^{\prime}),$ (1.4)

where $F_{1},\ldots,F_{N}$ have the same linear span as $f_{1},\ldots,f_{N}$,
$G_{1},\ldots,G_{N}$ have the same linear span as $g_{1},\ldots,g_{N}$, and
they satisfy the biorthogonality relations

$\int_{\mathbb{R}}F_{k}(x)G_{m}(x){\rm d}x=\delta_{km},\qquad k,m=1,\ldots,N.$
(1.5)

Average multiplicative statistics in biorthogonal ensembles can be expressed
as Fredholm series: for any bounded measurable
$\sigma:\mathbb{R}\to\mathbb{C}$, we have

$\mathbb{E}\left[\prod_{k=1}^{N}(1-\sigma(x_{k}))\right]=\sum_{k=0}^{N}\frac{(-1)^{k}}{k!}\int_{\mathbb{R}^{k}}\det\left(L_{N}(x_{j},x_{m})\right)_{j,m=1}^{k}\prod_{j=1}^{k}\sigma(x_{k}){\rm
d}x_{k}.$ (1.6)

The right hand side is the (truncated) Fredholm series associated to the
kernel $\sigma(x)L_{N}(x,x^{\prime})$, which we will denote henceforth as
$\det(1-\sigma L_{N})_{L^{2}(\mathbb{R})}$. The subscript $L^{2}(\mathbb{R})$
indicates that we are integrating over $\mathbb{R}$, however at this point, we
do not interpret $\sigma L_{N}$ yet as a trace-class operator acting on
$L^{2}(\mathbb{R})$, and we use $\det(1-\sigma L_{N})_{L^{2}(\mathbb{R})}$
simply as a short-hand notation for the right hand side of (1.6).

Orthogonal polynomial ensembles [42] form an important subclass of
biorthogonal ensembles, corresponding to $f_{m}(x)=x^{m-1}$ and
$g_{m}(x)=x^{m-1}{w(x)}$ for some weight function $w$. Then we can take
$F_{k}=P_{k-1}$ to be the degree $k-1$ normalized orthogonal polynomial on the
real line with respect to the weight $w$, and $G_{k}=P_{k-1}w$. Orthogonal
polynomial ensembles arise for instance as eigenvalue distributions of
unitarily invariant Hermitian random matrices, see e.g. [25]. Polynomial
ensembles [44] are more general and correspond to $f_{m}(x)=x^{m-1}$ with
arbitrary $g_{m}$. Examples of them arise as distributions of singular values
of products of random matrices and in Muttalib-Borodin ensembles [11]. Pólya
ensembles [34, 38] are polynomial ensembles with a specific derivative
structure, corresponding to $f_{m}(x)=x^{m-1}$ and
$g_{m}(x)=\displaystyle\frac{{\rm d}^{m-1}}{{\rm d}x^{m-1}}g(x)$ for some
function $g$.

In this paper, we consider a class of complex-valued measures on
$\mathbb{R}^{N}$, $N\in\mathbb{N}$, of the following form

${\rm d}\mu_{N}(\vec{x})\equiv{\rm
d}\mu_{N}(x_{1},\ldots,x_{N}):=\frac{1}{N!}\det\left(L_{N}(x_{m},x_{k})\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm
d}x_{k},$ (1.7)

where the kernel $L_{N}$ can be represented as a double contour integral of
the form

$L_{N}(x,x^{\prime})=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\Sigma_{N}}{\rm
d}u\int_{\ell_{N}}{\rm d}v\frac{W_{N}(v)}{W_{N}(u)}\frac{{\rm
e}^{-vx+ux^{\prime}}}{v-u}.$ (1.8)

Here, $W_{N}$, $\ell_{N}$, and ${\Sigma_{N}}$ need to satisfy the following
assumptions.

######  Assumption 1.1.

There exists a vertical strip
$\mathcal{S}_{N}:=\\{z\in\mathbb{C}:\alpha_{N}<\mathrm{Re}\,z<\beta_{N}\\}$ in
the complex plane such that the following holds:

  1. 1.

$W_{N}:\mathcal{S}_{N}\to\mathbb{C}$ is analytic, not identically zero and
such that $W_{N}(z)=O(|z|^{-\epsilon})$ as $z\to\infty$ in $\mathcal{S}_{N}$,
for some $\epsilon>0$;

  2. 2.

$\ell_{N}$ is a vertical line in $\mathcal{S}_{N}$, oriented upwards, i.e.
$\ell_{N}=c_{N}+\mathrm{i}\mathbb{R}$ with $\alpha_{N}<c_{N}<\beta_{N}$;

  3. 3.

${\Sigma_{N}}$ is a closed positively oriented curve in $\mathcal{S}_{N}$
without self-intersections, lying at the left of $\ell_{N}$, which has $N$
(not necessarily distinct) zeros of $W_{N}$ in its interior, and none on its
image; we denote these zeros as $a_{1},\ldots,a_{N}$.

Note that we allow $W_{N}$ to have other zeros in $\mathcal{S}_{N}$ than
$a_{1},\ldots,a_{N}$, and that the zeros $a_{1},\ldots,a_{N}$ do not need to
be distinct. The expression (1.8) is independent of the choice of
${\Sigma_{N}}$ and $\ell_{N}$, as long as the above assumptions are satisfied.
Under these assumptions, we will show that the complex-valued measures (1.7)
are biorthogonal: they can be written in the form (1.1), but they are not
necessarily positive. We will use the term biorthogonal measure instead of
biorthogonal ensemble to indicate that the measures are in general not
probability measures. We also define, in analogy with (1.2), marginals for
such measures,

$\rho_{n,N}(x_{1},\ldots,x_{n}):=\frac{1}{(N-n)!}\int_{\mathbb{R}^{N-n}}\det\Big{(}L_{N}(x_{m},x_{k})\Big{)}_{m,k=1}^{N}\prod_{k=n+1}^{N}\mathrm{d}x_{k}.$
(1.9)

Note that we use the same normalization as for the correlation functions given
in (1.2), however we avoid the term correlation functions because the measure
$\mu_{N}$ is not necessarily positive.

Our first result contains the essence of the biorthogonal structure of the
measures (1.7).

######  Theorem 1.2.

Let $W_{N},{\Sigma_{N}},\ell_{N}$ satisfy Assumption 1.1, and let $\mu_{N}$ be
defined by (1.7).

  1. 1.

The kernel $L_{N}$ given by (1.8) satisfies the properties

$\int_{\mathbb{R}}L_{N}(x,t)L_{N}(t,x^{\prime}){\rm
d}t=L_{N}(x,x^{\prime}),\qquad\int_{\mathbb{R}}L_{N}(x,x){\rm d}x=N,$ (1.10)

and, with $\rho_{n,N}$ as in (1.9), we have

$\rho_{n,N}(x_{1},\ldots,x_{n})=\det\Big{(}L(x_{m},x_{k})\Big{)}_{m,k=1}^{n},\qquad
n=1,\ldots,N.$ (1.11)

  2. 2.

If $a_{1},\ldots,a_{N}$ are distinct, we have

$L_{N}(x,x^{\prime})=\sum_{m=1}^{N}{\rm e}^{a_{m}x^{\prime}}\psi_{m}({\rm
e}^{x}),\quad\text{where}\quad\psi_{m}(y):=\frac{1}{2\pi\mathrm{i}W_{N}^{\prime}(a_{m})}\int_{\ell_{N}}\frac{W_{N}(v)y^{-v}}{v-a_{m}}{\rm
d}v,$ (1.12)

and we have the biorthogonality relations

$\int_{\mathbb{R}}{\rm e}^{a_{m}x}\psi_{k}({\rm e}^{x}){\rm
d}x=\delta_{mk},\qquad k,m=1,\ldots,N.$ (1.13)

The measure (1.7) is then a biorthogonal measure, given explicitly by

${\rm d}\mu_{N}(\vec{x})=\frac{1}{N!}\det\left({\rm
e}^{a_{m}x_{k}}\right)_{k,m=1}^{N}\det\left(\psi_{m}({\rm
e}^{x_{k}})\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm d}x_{k}.$ (1.14)

  3. 3.

In the confluent case $a_{1}=\cdots=a_{N}=a$, the measure (1.7) is also
biorthogonal and given explicitly by

${\rm d}\mu_{N}(\vec{x})=\frac{1}{Z_{N}}\Delta(\vec{x})\det\left(\phi_{m}({\rm
e}^{x_{k}})\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm
d}x_{k},\quad\Delta(\vec{x})=\prod_{1\leq j,k\leq N}(x_{k}-x_{j}),$ (1.15)

where

$Z_{N}:=\int_{\mathbb{R}^{N}}\Delta(\vec{x})\det\left(\phi_{m}({\rm
e}^{x_{k}})\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm d}x_{k},$ (1.16)

and

$\phi_{m}(y):=\frac{y^{a}}{2\pi\mathrm{i}}\int_{\ell_{N}}\frac{W_{N}(v)y^{-v}}{(v-a)^{N-m+1}}{\rm
d}v.$ (1.17)

######  Remark 1.3.

As we will explain in detail in Sections 5, 6 and 7, positive biorthogonal
measures of the form (1.7)–(1.8) occur in a variety of random matrix models.
Three choices of $W$ are particularly relevant in this paper: they are given
by

$\begin{array}[]{rcl}W_{N}^{{\rm
LUE+}}(z;\vec{a},\nu)&=&\displaystyle\frac{\prod_{j=1}^{N}(z-a_{j})}{(z-1)^{N+\nu}},\\\
\\\ W_{N}^{{\rm GUE+}}(z;\vec{a},\tau)&=&\displaystyle{\rm e}^{\tau
z^{2}/2}\prod_{m=1}^{N}(z-a_{m})\\\ \\\ W_{N}^{{\rm
GLUE+}}(z;\vec{a},\tau)&=&\displaystyle{\rm e}^{\tau
z^{2}/2}\frac{\prod_{k=1}^{N}(z-a_{k})}{(1-z)^{N}}.\end{array}$ (1.18)

They are associated, respectively, to the biorthogonal ensembles given by the
eigenvalues of an LUE (Laguerre Unitary Ensemble) matrix with external source,
a GUE (Gaussian Unitary Ensemble) with external source, and a sum of an LUE
matrix with external source and a GUE matrix. We anticipate that these three
models describe the small temperature limits of three different polymer
models: Log Gamma, O’Connell-Yor and O’Connell-Yor with boundary sources. More
details about these small temperature limits are given at the end of the
introduction.

######  Remark 1.4.

Observe that the integral expressions for $\psi_{m}$ and $\phi_{m}$ are
independent of the choice of the vertical line $\ell_{N}$ in the strip
$\mathcal{S}_{N}$. Recalling the definition of the direct and inverse Mellin
transforms,

$\mathcal{M}[g](v):=\int_{0}^{\infty}y^{v-1}g(y){\rm
d}y,\qquad\mathcal{M}^{-1}[G](y):=\frac{1}{2\pi\mathrm{i}}\int_{\ell_{N}}{G(v)y^{-v}}{\rm
d}v,$ (1.19)

we recognize $\psi_{m}$ as the inverse Mellin transforms of the function
$\frac{W_{N}(v)}{W_{N}^{\prime}(a_{m})(v-a_{m})}$ and $\phi_{m}$ as the
inverse Mellin transform of $W_{N}(v)/(v-a)^{N-m+1}$ multiplied by $y^{a}$. In
exponential variables, $\psi_{1},\ldots,\psi_{N}$ and
$\phi_{1},\ldots,\phi_{m}$ are Fourier integrals:

$\psi_{m}({\rm
e}^{x})=\frac{1}{2\pi\mathrm{i}W_{N}^{\prime}(a_{m})}\int_{\ell_{N}}\frac{W_{N}(v)}{v-a_{m}}{\rm
e}^{-vx}{\rm d}v=\frac{{\rm e}^{-cx}}{2\pi
W_{N}^{\prime}(a_{m})}\int_{\mathbb{R}}\frac{W_{N}(c+\mathrm{i}t)}{c+\mathrm{i}t-a_{m}}{\rm
e}^{-\mathrm{i}tx}{\rm d}t$ (1.20)

for $\alpha_{N}<c<\beta_{N}$ and $x\in\mathbb{R}$, and similarly for
$\phi_{1},\ldots,\phi_{m}$.

######  Remark 1.5.

The confluent form of the biorthogonal measure has the structure of a complex-
valued (not necessarily positive) polynomial ensemble, and more specifically
of a Pólya ensemble, since $-\frac{\partial}{\partial x}\phi_{m}({\rm
e}^{x})=\phi_{m+1}({\rm e}^{x})$. It will become clear from the proof that
explicit biorthogonal expressions for the measure can be given not only in the
completely confluent case where all of the values $a_{1},\ldots,a_{N}$
coincide, but also in cases where some of them occur with multiplicity bigger
than one.

######  Remark 1.6.

In exponential variables $s_{j}={\rm e}^{x_{j}}$, $j=1,\ldots,N$, the
biorthogonal measure (1.14) becomes

${\rm
d}\widehat{\mu}_{N}(\vec{s}):=\frac{1}{N!}\det\left(s_{k}^{a_{m}-1}\right)_{k,m=1}^{N}\det\left(\psi_{m}(s_{k})\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm
d}s_{k},\qquad\vec{s}\in(0,\infty)^{N},$ (1.21)

with associated kernel

$\widehat{L}_{N}(s,s^{\prime})=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\Sigma_{N}}{\rm
d}u\int_{\ell_{N}}{\rm d}v\frac{W_{N}(v)}{W_{N}(u)}\frac{{s}^{-v-1}s^{\prime
u}}{v-u}.$ (1.22)

In the special case where $a_{m}=m$, $m=1,\ldots,N$, the first determinant
becomes a Vandermonde determinant, and then we have the polynomial ensemble

${\rm d}\widehat{\mu}_{N}(\vec{s}):=\frac{1}{N!}\Delta(\vec{s})\
\det\left(\psi_{m}(s_{k})\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm
d}s_{k},\qquad\vec{s}\in(0,\infty)^{N}.$ (1.23)

Note that the mechanism to obtain a polynomial ensemble is different here than
in the confluent case, as it takes place for equi-spaced $a_{1},\ldots,a_{N}$,
and in exponential variables. For specific choices of $W_{N}$ which are ratios
of products of Gamma functions, this measure is positive and describes the
distribution of singular values for products of random matrices and Muttalib-
Borodin ensembles, as we will explain at the end of Section 7.

The above biorthogonal structure of the measures (1.7) gives us access to
determinantal representations for averages of multiplicative statistics of the
form $\prod_{k=1}^{N}(1-\sigma(x_{k}))$. The simplest is an expression in
terms of an $N\times N$ determinant. In addition, we have several Fredholm
determinant expressions. Specific Fredholm determinants of these types appear
in random matrix theory and in the literature of polymer models and Whittaker
processes, see for instance [12, 13, 16]. In our next result, we will gather
all these different determinantal representations for average multiplicative
statistics. This will allow us in particular to identify models of a different
nature, random matrix ensembles and polymer models, as instances of our class
of biorthogonal measures.

We will denote henceforth $\mu_{N}[\sigma]$ for the average multiplicative
statistic associated to a function $\sigma$:

$\mu_{N}[\sigma]:=\int_{\mathbb{R}^{N}}\prod_{k=1}^{N}(1-\sigma(x_{k})){\rm
d}\mu_{N}(x_{1},\ldots,x_{N}).$ (1.24)

Given a curve $\gamma$ in the complex plane, we recall the definition of the
Fredholm determinant of an operator via the Fredholm series associated to its
kernel:

$\det(1-K)_{L^{2}(\gamma)}=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\int_{\gamma^{k}}\det\left(K(z_{j},z_{m})\right)_{j,m=1}^{k}\prod_{j=1}^{k}{\rm
d}z_{k}.$ (1.25)

The relevant curves for us will be $\gamma=\mathbb{R}$, $\gamma=(0,+\infty)$,
and $\gamma={\Sigma_{N}}$. As already mentioned, for now, we write
$\det(1-K)_{L^{2}(\gamma)}$ whenever (1.25) is convergent, and we do not
require the associated operator to be trace-class. Later on, we will see that
suitable conjugations $h(x)K(x,x^{\prime})h(x^{\prime})^{-1}$ of the relevant
kernels $K$ give rise to rank $N$ and thus trace-class operators on
$L^{2}(\gamma)$ (in that case the series truncates after the term $k=N$),
which will allow us to use classical properties for Fredholm determinants.

A first identity expresses $\mu_{N}[\sigma]$ in terms of the Fredholm
determinant with kernel $L_{N}$, and holds in general under Assumptions 1.1,
for bounded and measurable $\sigma$. A second identity requires stronger decay
of $W_{N}$ and sufficiently fast decay of $\sigma$ at $-\infty$, and involves
the Fredholm determinant with kernel

$H_{N}^{\sigma}(y,y^{\prime}):=\int_{\mathbb{R}}\sigma(x)\Psi_{1}({\rm
e}^{y+x})\Psi_{2}({\rm e}^{y^{\prime}+x}){\rm d}x,\qquad y,y^{\prime}>0,$
(1.26)

where

$\Psi_{1}(s):=\frac{1}{2\pi\mathrm{i}}\int_{{\Sigma_{N}}}\frac{s^{u}}{W_{N}(u)}{\rm
d}u,\qquad\Psi_{2}(s):=\frac{1}{2\pi\mathrm{i}}\int_{\ell_{N}}s^{-v}W_{N}(v){\rm
d}v=\mathcal{M}^{-1}[W_{N}](s).$ (1.27)

A third identity holds only for the specific choice
$\sigma_{t}(x)=\frac{1}{1+{\rm e}^{-x-t}}$ for $t\in\mathbb{R}$, requires that
all zeros $a_{1},\ldots,a_{N}$ are contained in a vertical strip of width
$<1$, i.e. $a_{\max}-a_{\min}<1$, with

$a_{\min}:=\min\\{\mathrm{Re}\,a_{1},\ldots,\mathrm{Re}\,a_{N}\\},\qquad
a_{\max}:=\max\\{\mathrm{Re}\,a_{1},\ldots,\mathrm{Re}\,a_{N}\\},$ (1.28)

and involves the Fredholm determinant with kernel

$K_{N,t}(u,u^{\prime}):=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\ell_{N}}{\rm
d}v\frac{\pi{\rm
e}^{t(v-u)}}{\sin\pi(u-v)}\frac{W_{N}(v)}{W_{N}(u)}\frac{1}{v-u^{\prime}},\qquad
u,u^{\prime}\in{\Sigma_{N}}.$ (1.29)

######  Theorem 1.7.

Let $W_{N},{\Sigma_{N}},\ell_{N}$ satisfy Assumption 1.1, let
$\sigma:\mathbb{R}\to\mathbb{C}$ be bounded and measurable, and let $\mu_{N}$
be defined by (1.7)-(1.8).

  1. 1.

In the case where $a_{1},\ldots,a_{N}$ are distinct, we have, with $\psi_{k}$
as in (1.12),

$\mu_{N}[\sigma]=\det\left(\int_{\mathbb{R}}\left(1-\sigma(x)\right){\rm
e}^{a_{m}x}\psi_{k}({\rm e}^{x}){\rm d}x\right)_{m,k=1}^{N}.$

If $a_{1}=\cdots=a_{N}=a$, then with $\phi_{k}$ as in (1.17),

$\mu_{N}[\sigma]=\frac{\det\left(\displaystyle\int_{\mathbb{R}}\left(1-\sigma(x)\right)x^{m-1}\phi_{k}({\rm
e}^{x}){\rm
d}x\right)_{m,k=1}^{N}}{\det\left(\displaystyle\int_{\mathbb{R}}x^{m-1}\phi_{k}({\rm
e}^{x}){\rm d}x\right)_{m,k=1}^{N}}.$

  2. 2.

We have the Fredholm series representation

$\mu_{N}[\sigma]=\det(1-\sigma L_{N})_{L^{2}(\mathbb{R})},$ (1.30)

where $L_{N}$ is given by (1.8).

  3. 3.

If there exists $\epsilon>0$ such that

$W_{N}(v)=O(|v|^{-1-\epsilon}),\quad
v\to\infty,\,v\in\mathcal{S}_{N},\qquad\sigma(x)=O\left({\rm
e}^{(a_{\max}-a_{\min}+\epsilon)x}\right),\quad x\to-\infty,$

we have the alternative Fredholm series representation

$\mu_{N}[\sigma]=\det(1-H_{N}^{\sigma})_{L^{2}(0,\infty)},$ (1.31)

with $H_{N}^{\sigma}$ as in (1.26).

  4. 4.

If there exists $\epsilon>0$ such that

$W_{N}(v)=O(|v|^{-1-\epsilon}),\quad v\to\infty,\,v\in\mathcal{S}_{N},$

$a_{\max}-a_{\min}<1$ and $\sigma_{t}(x)=\frac{1}{1+{\rm e}^{-x-t}}$ for
$t\in\mathbb{R}$, we have yet another Fredholm determinant representation:

$\mu_{N}[\sigma_{t}]=\det(1+K_{N,t})_{L^{2}({\Sigma_{N}})},$ (1.32)

with $K_{N,t}$ given by (1.29).

######  Remark 1.8.

The kernel $H_{N}^{\sigma}$ has the structure of a finite-temperature type
deformation of the kernel $L_{N}$. Indeed, $L_{N}$ is a special case of
$H_{N}^{\sigma}$, corresponding to the case when $\sigma$ is the indicator
function of the interval $[0,+\infty)$. Such deformations appear frequently,
for example, in the study of free fermions at finite temperature [47], the KPZ
equation [3], and in the study of Schur processes with periodic boundary
conditions [9, 10].

######  Remark 1.9.

The result of part 4 may at first sight seem too specific to be interesting,
but as we will point out later, it is precisely this identity that will allow
us to make the connection with the Log Gamma polymer, since Fredholm
determinants with kernels of the form (1.29) are intimately connected with
($\alpha$-)Whittaker processes and Sklyanin measures, see e.g. [14, 15]. In
particular, a Fredholm determinant of this type is known to be equal to the
Laplace transform of the Log Gamma polymer partition function [15]. It should
be noted that Fredholm determinant expressions like (1.32), but with unbounded
contours ${\Sigma_{N}}$ containing an infinite number of zeros of $W_{N}$, are
also frequently encountered in the literature of polymers and Whittaker
processes, see e.g. [12, 13] and the discussion in [6, Section 2]. However,
such expressions with kernels defined on large contours do not fit directly in
our framework.

Next, we introduce a deformation of the biorthogonal measure (1.7). Given a
bounded measurable function $\sigma:\mathbb{R}\to\mathbb{C}$, we define the
measures

${\rm
d}\nu_{N,t}(\vec{x}):=\frac{1}{Z_{N,t}}\det\Big{(}L_{N}(x_{m},x_{k})\Big{)}_{m,k=1}^{N}\prod_{k=1}^{N}\big{(}1-\sigma_{t}(x_{k})\big{)}{\rm
d}x_{k},\quad t\in\mathbb{R},\ \sigma_{t}(x):=\sigma(x+t),$ (1.33)

where

$Z_{N,t}:=\int_{\mathbb{R}^{N}}\det\Big{(}L_{N}(x_{m},x_{k})\big{)}_{m,k=1}^{N}\prod_{k=1}^{N}\big{(}1-\sigma_{t}(x_{k})\big{)}{\rm
d}x_{k},$ (1.34)

so that $\displaystyle\int_{\mathbb{R}^{N}}{\rm
d}\nu_{N,t}(x_{1},\ldots,x_{N})=1$, and where we assume that $\sigma,N,t$ are
such that $Z_{N,t}\neq 0$.

For $\sigma(x)=1_{(0,\infty)}(x)$, the deformed measure is simply the original
biorthogonal measure, restricted to $(-\infty,-t)^{N}$ and renormalized or, in
other words, the original biorthogonal ensemble conditioned on
$\max\\{x_{1},\ldots,x_{N}\\}\leq-t$, in case the measure is positive. For
general $\sigma$, if the biorthogonal measure is positive, it can be seen as a
pushed particle system [43] in which the particles $x_{1},\ldots,x_{N}$ are
pushed to regions where $\sigma_{t}(x)$ is small; it can be constructed via a
procedure of marking and conditioning of the original biorthogonal ensemble
[23].

By definition, the deformed measure (1.33) is also a biorthogonal measure.
However, the associated kernel in general does not admit an explicit double
integral form like in (1.8), which makes the deformed measure harder to
analyze. Nevertheless, the deformed measure encodes $\mu_{N}[\sigma]$ in a
simple way: as we will see, logarithmic derivatives of average multiplicative
statistics $\mu_{N}[\sigma]$ of the non-deformed measure can be expressed in
terms of average linear statistics of the deformed measure. Average linear
statistics of determinantal measures are much easier to analyze than average
multiplicative statistics, since they can be expressed in terms of the one-
point functions. Similarly to the one-point function $\rho_{1,N}(x)$ defined
by (1.2) and given by (1.3), we define the one-point function of the deformed
measure as

$\kappa_{N,t}(x_{1})=\frac{N}{Z_{N,t}}(1-\sigma_{t}(x_{1}))\int_{\mathbb{R}^{N-1}}\det\left(L_{N}(x_{m},x_{k})\right)_{m,k=1}^{N}\prod_{k=2}^{N}(1-\sigma_{t}(x_{k})){\rm
d}x_{k}.$ (1.35)

It is then straightforward to verify that

$\bigintsss_{\mathbb{R}^{N}}\left(\sum_{k=1}^{N}f(x_{k})\right){\rm
d}\nu_{N,t}(x_{1},\ldots,x_{N})=\int_{\mathbb{R}}f(x)\kappa_{N,t}(x){\rm d}x,$
(1.36)

for any bounded and measurable $f:\mathbb{R}\to\mathbb{C}$. Later on, it will
be convenient to interpret (1.33) as an absolutely continuous measure on
$\mathbb{R}^{N}$ with respect to the deformed measure $(1-\sigma_{t}(x)){\rm
d}x$ instead of the Lebesgue measure. With respect to this deformed measure,
the one-point function becomes

$\widetilde{\kappa}_{N,t}(x_{1})=\frac{N}{Z_{N,t}}\int_{\mathbb{R}^{N-1}}\det\big{(}L_{N}(x_{m},x_{k})\big{)}_{m,k=1}^{N}\prod_{k=2}^{N}(1-\sigma_{t}(x_{k})){\rm
d}x_{k}.$ (1.37)

We have $\widetilde{\kappa}_{N,t}(x)=\kappa_{N,t}(x)/(1-\sigma_{t}(x))$ if
$\sigma_{t}(x)\neq 0$, but $\widetilde{\kappa}_{t}(x)$ has the advantage that
it is also defined when $\sigma_{t}(x)=1$.

######  Theorem 1.10.

Let $W_{N},{\Sigma_{N}},\ell_{N}$ satisfy Assumption 1.1, let
$\sigma:\mathbb{R}\to\mathbb{C}$ be bounded and $C^{1}$ with bounded
derivative, and define $\sigma_{t}(x):=\sigma(x+t)$. Let ${\rm d}\mu_{N}$ and
${\rm d}\nu_{N,t}$ be defined by (1.7) and (1.33). If $\mu_{N}[\sigma_{t}]\neq
0$, we have the identity

$\frac{{\rm d}}{{\rm
d}t}\log\mu_{N}[\sigma_{t}]=\int_{\mathbb{R}}\sigma^{\prime}(x+t)\widetilde{\kappa}_{N,t}(x){\rm
d}x.$ (1.38)

If $\sigma(x)=\displaystyle\frac{1}{1+{\rm e}^{-x}}$, the above identity
simplifies to

$\frac{{\rm d}}{{\rm d}t}\log\mu_{N}[\sigma]=\int_{\mathbb{R}}\frac{1}{1+{\rm
e}^{-x-t}}\kappa_{N,t}(x){\rm d}x.$ (1.39)

######  Remark 1.11.

We believe that the last identity (1.39) will be useful to study the
asymptotic behavior of average multiplicative statistics $\mu_{N}[\sigma]$ as
$N\to\infty$. Via (1.39), these asymptotics are encoded simply in the one-
point function $\kappa_{N,t}(x)$ of the deformed biorthogonal measure. Even if
the deformed biorthogonal measure is not particularly simple, there is a
variety of techniques available to study one-point functions, like Coulomb gas
techniques and characterizations via equilibrium problems. For specific
choices of $W_{N}$, see Sections 5, 6 and 7, the above result implies that the
one-point function $\kappa_{N,t}$ encodes tail asymptotics and large
deviations of polymer partition functions. The condition that $\sigma$ is
$C^{1}$ is convenient for the proof of (1.38), but it is not a necessary
condition, and one can generalize the identity to piecewise $C^{1}$-functions.
For instance, for $\sigma_{t}=1_{(-t,+\infty)}$, the identity (1.38) continues
to hold if we replace $\sigma^{\prime}$ by a Dirac $\delta$-function, such
that

$\frac{{\rm d}}{{\rm
d}t}\log\mu_{N}[1_{(-t,\infty)}]=\widetilde{\kappa}_{N,t}(-t).$ (1.40)

#### Outline.

The rest of this paper is organized as follows. In Section 2, we will study
the biorthogonal structure of the measures $\mu_{N}$ and prove Theorem 1.2. In
Section 3, we will establish the determinantal expressions for the
multiplicative statistics $\mu_{N}[\sigma]$, and prove Theorem 1.7. In Section
4, we will study the deformed biorthogonal measures $\nu_{N,t}$ and prove
Theorem 1.10. Finally, Sections 5, 6, and 7 are devoted to applications in
polymer and random matrix models. There, we will study the biorthogonal
measures (1.7)–(1.8) associated to functions of the form

$\begin{array}[]{rcl}\displaystyle W_{n,N}^{{\rm
Log}\,\Gamma}(z;\vec{\alpha},\vec{a})&=&\displaystyle\frac{\prod_{j=1}^{n}\Gamma(\alpha_{j}-z)}{\prod_{k=1}^{N}\Gamma(z-a_{k})},\\\
\\\ \displaystyle W_{N}^{{\rm OY}}(z;\vec{a},\tau)&=&\displaystyle\frac{{\rm
e}^{\frac{\tau z^{2}}{2}}}{\prod_{k=1}^{N}\Gamma(z-a_{k})},\\\ \\\
\displaystyle W_{N}^{{\rm
Mixed}}(z;\vec{\alpha},\vec{a},\tau)&=&\displaystyle{\rm e}^{\tau
z^{2}/2}\frac{\prod_{j=1}^{N}\Gamma(\alpha_{j}-z)}{\prod_{k=1}^{N}\Gamma(z-a_{k})}\end{array}$
(1.41)

and prove that these measures characterize, respectively, the partition
functions of the Log Gamma, the O’Connell-Yor, and the mixed polymer, see
Corollaries 5.2, 6.1, and 7.3. Moreover, we also prove that in a certain
“small temperature” regime, these biorthogonal measures degenerate to the ones
described in Remark 1.3 and associated to matrix models, see Propositions 5.4,
6.3 and 7.6. For the Log Gamma and O’Connell-Yor polymers, this is not
surprising, since the small temperature degeneration of the Log Gamma polymer
is a last passage percolation model well-known to be characterized in terms of
the LUE with external source [39, 17, 27], while the small temperature
degeneration of the O’Connell-Yor polymer is a model of Brownian queues well-
known to be characterized by the GUE with external source [7, 35].
Nevertheless, it reveals how the zero temperature limit takes place on the
level of the biorthogonal measures. For the mixed polymer, to the best of our
knowledge, the degeneration to the matrix model given by the sum of LUE with
external source and GUE is completely new.

##  2 Biorthogonal structure of the measures

We will suppose throughout this section that $W_{N},{\Sigma_{N}},\ell_{N}$
satisfy Assumption 1.1, and that $L_{N}$ is defined by (1.8). The goal of this
section is to prove Theorem 1.2.

To start, we focus on the situation where the zeros $a_{1},\ldots,a_{N}$ of
$W_{N}$ inside ${\Sigma_{N}}$ are all distinct. Then, we prove the following.

######  Proposition 2.1.

If $a_{1},\ldots,a_{N}$ are all distinct, we have

$L_{N}(x,x^{\prime})=\sum_{m=1}^{N}{\rm e}^{a_{m}x^{\prime}}\psi_{m}({\rm
e}^{x}),$ (2.1)

with $\psi_{m}$ as in (1.20), and we have the orthogonality relation (1.13).
Consequently, $L_{N}$ has the reproducing property

$\int_{\mathbb{R}}L_{N}(x,t)L_{N}(t,x^{\prime}){\rm d}t=L_{N}(x,x^{\prime}).$
(2.2)

###### Proof.

We can evaluate the $u$-integral in (1.8) using the residue theorem, and in
this way, we immediately obtain (2.1).

Next, since

$\psi_{m}({\rm
e}^{x})=\frac{1}{2\pi\mathrm{i}W_{N}^{\prime}(a_{m})}\int_{\ell_{N}}\frac{W_{N}(v){\rm
e}^{-xv}}{v-a_{m}}{\rm d}v,$

for any vertical line $\ell_{N}$ (avoiding $a_{m}$) in the strip
$\mathcal{S}_{N}=\\{z\in\mathbb{C}:\alpha_{N}<\mathrm{Re}\,z<\beta_{N}\\}$,
and since $W_{N}(z)=O(|z|^{-\epsilon})$ as $z\to\infty$, it is easy to verify
that, for any $\delta>0$,

$\psi_{m}({\rm e}^{x})=O\left({\rm
e}^{-(\alpha_{N}+\delta)x}\right)\quad\mbox{as
$x\to-\infty$},\qquad\psi_{m}({\rm e}^{x})=O\left({\rm
e}^{-(\beta_{N}-\delta)x}\right)\quad\mbox{as $x\to+\infty$}.$ (2.3)

The integrals $\displaystyle\int_{-\infty}^{+\infty}{\rm
e}^{a_{k}x}\psi_{m}({\rm e}^{x}){\rm d}x$, $k=1,\ldots,N$ are thus well-
defined, since $a_{1},\ldots,a_{N}\in\mathcal{S}_{N}$.

To obtain the biorthogonality relations, recalling the direct and inverse
Mellin transform from (1.19), we observe that

$\int_{-\infty}^{+\infty}{\rm e}^{a_{k}x}\psi_{m}({\rm e}^{x}){\rm
d}x=\int_{0}^{\infty}s^{a_{k}-1}\psi_{m}(s){\rm
d}s=\frac{1}{W_{N}^{\prime}(a_{k})}\left(\mathcal{M}\circ\mathcal{M}^{-1}\right)\left[\frac{W_{N}(\cdot)}{\cdot-
a_{m}}\right](a_{k}).$

Since $W_{N}$ has simple zeros at the points $a_{1},\ldots,a_{N}$, the above
expression is $0$ for $k\neq m$, and to $1$ for $k=m$.

Finally, it is straightforward to deduce the reproducing property (2.2) from
(2.1) and (1.13). ∎

As a consequence of (2.1), using the multiplicativity of the determinant, we
can rewrite (1.7) as

$\displaystyle{\rm d}\mu_{N}(\vec{x})$
$\displaystyle=\frac{1}{N!}\det\left(\sum_{j=1}^{N}{\rm
e}^{a_{j}x_{k}}\psi_{j}({\rm e}^{x_{m}})\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm
d}x_{k}$ $\displaystyle=\frac{1}{N!}\det\left({\rm
e}^{a_{j}x_{k}}\right)_{k,j=1}^{N}\det\left(\psi_{j}({\rm
e}^{x_{m}})\right)_{j,m=1}^{N}\prod_{k=1}^{N}{\rm d}x_{k},$ (2.4)

which is (1.14), such that part 2 of Theorem 1.2 is proved.

######  Proposition 2.2.

For arbitrary $a_{1},\ldots,a_{N}$, $L_{N}$ satisfies the reproducing property
(2.2) and we have $\displaystyle\int_{\mathbb{R}}L_{N}(x,x){\rm d}x=N$.

###### Proof.

The reproducing property is already proved in the case where all
$a_{1},\ldots,a_{N}$ are distinct. If not all $a_{1},\ldots,a_{N}$ are
distinct, write

$W_{N}(z)=W_{0}(z)\prod_{m=1}^{N}(z-a_{m}).$

For the sake of clarity, let us temporarily make explicit the dependence of
$L_{N}$ on the parameters $a_{1},\ldots,a_{N}$ in our notations, and write

$L_{N}(x,x^{\prime};\vec{a})\equiv
L_{N}(x,x^{\prime})=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\Sigma_{N}}{\rm
d}u\int_{\ell_{N}}{\rm
d}v\frac{W_{0}(v)\prod_{m=1}^{N}(v-a_{m})}{W_{0}(u)\prod_{m=1}^{N}(u-a_{m})}\frac{{\rm
e}^{-vx+ux^{\prime}}}{v-u}.$

Given $W_{0}$ and ${\Sigma_{N}}$, the kernel $L_{N}$ is continuous in
$\vec{a}$, as long as $a_{1},\ldots,a_{N}$ remain inside ${\Sigma_{N}}$. We
already know that $L_{N}$ has the reproducing property if $a_{1},\ldots,a_{N}$
are distinct:

$\int_{\mathbb{R}}L_{N}(x,t;\vec{a})L_{N}(t,x^{\prime};\vec{a}){\rm
d}t=L_{N}(x,x^{\prime};\vec{a}).$ (2.5)

Both sides of this identity are continuous in $\vec{a}$, thus the identity
continues to hold when $a_{1},\ldots,a_{N}$ are not distinct. Similarly, by
the biorthogonality relation (1.13) and by (2.1), we have that

$\int_{\mathbb{R}}L_{N}(x,x;\vec{a}){\rm d}x=N$

for $a_{1},\ldots,a_{N}$ distinct, and by continuity this continues to hold in
general. ∎

Our next result is a determinant expression for the marginals $\rho_{n,N}$
defined in (1.9). It follows from a standard integrate-out trick, we record
the proof here for the convenience of the reader, and because it is found in
many classical references for Hermitian kernels only.

######  Proposition 2.3.

The marginals $\rho_{n,N}$ of the measure $\mu_{N}$ defined by (1.7), satisfy
the determinantal identity

$\rho_{n,N}(x_{1},\ldots,x_{n})=\det\Big{(}L_{N}(x_{j},x_{k})\Big{)}_{j,k=1}^{n},\quad
n=1,\ldots,N.$ (2.6)

###### Proof.

We need to prove that

$\int_{\mathbb{R}^{N-n}}\det\Big{(}L_{N}(x_{j},x_{m})\Big{)}_{j,m=1}^{N}\mathrm{{\rm
d}}x_{n+1}\cdots{\rm
d}x_{N}=(N-n)!\det\Big{(}L_{N}(x_{j},x_{k})\Big{)}_{j,k=1}^{n},$ (2.7)

for $n=1,\ldots,N-1$. Let $k\in\\{2,\ldots,N\\}$, and write
$M:=\Big{(}L_{N}(x_{i},x_{j})\Big{)}_{i,j=1}^{k}$. In the rest of this proof,
we will write $M^{a,b}$ for the $(k-1)\times(k-1)$ matrix obtained from $M$ by
removing the $a$-th line and the $b$-th column, and similarly
$M^{a_{1}a_{2},b_{1}b_{2}}$ for the $(k-2)\times(k-2)$ matrix obtained from
$M$ by removing the two corresponding rows and columns. We now expand $\det M$
with respect to the last row, to obtain

$\displaystyle\int_{\mathbb{R}}\det M{\rm d}x_{k}$
$\displaystyle=\int_{\mathbb{R}}\sum_{j=1}^{k}(-1)^{j+k}L_{N}(x_{k},x_{j})\det
M^{k,j}{\rm d}x_{k}$
$\displaystyle=\left(\int_{\mathbb{R}}L_{N}(x_{k},x_{k}){\rm
d}x_{k}\right)\det
M^{k,k}+\int_{\mathbb{R}}\sum_{j=1}^{k-1}(-1)^{j+k}L_{N}(x_{k},x_{j})\det
M^{k,j}{\rm d}x_{k}.$

Observing that the integral in the first term is equal to $N$ (recall (1.10))
and expanding $\det M^{k,j}$ once more, now with respect to the last column,
we find

$\displaystyle\int_{\mathbb{R}}\det M{\rm d}x_{k}$ $\displaystyle=N\det
M^{k,k}+\int_{\mathbb{R}}\sum_{j=1}^{k-1}(-1)^{j+k}L_{N}(x_{k},x_{j})\sum_{m=1}^{k-1}(-1)^{m+k-1}L_{N}(x_{m},x_{k})\det
M^{km,jk}{\rm d}x_{k}$ $\displaystyle=N\det
M^{k,k}-\sum_{j,m=1}^{k-1}(-1)^{j+m}L_{N}(x_{m},x_{j})\det M^{km,jk}$ (2.8)
$\displaystyle=(N-k+1)\det M^{k,k},$ (2.9)

where we used the reproducing property (1.10) in the second line, and where
the last line is verified again by row or column expansion. This implies that

$\int_{\mathbb{R}}\det\left(L_{N}(x_{i},x_{j})\right)_{i,j=1}^{k}{\rm
d}x_{k}=(N-k+1)\det\Big{(}L_{N}(x_{i},x_{j})\Big{)}_{i,j=1}^{k-1},$

for every $k\in\\{2,\ldots,N\\}$. Repeatedly applying this identity for
$k=N,N-1,\ldots,n+1$, we obtain (2.7). ∎

Propositions 2.2 and 2.3 together imply part 1 of Theorem 1.2.

If $a_{1}=\cdots=a_{N}=a$, we can still evaluate the $u$-integral in the
double contour integral expression for $L_{N}$ using the residue theorem:

$L_{N}(x,x^{\prime})=\frac{1}{2\pi\mathrm{i}}\int_{\ell_{N}}{\rm
Res}\left(\frac{{\rm
e}^{ux^{\prime}}}{W_{0}(u)(u-a)^{N}(v-u)};u=a\right)W_{N}(v){\rm e}^{-vx}{\rm
d}v.$

The residue takes the form ${\rm
e}^{ax^{\prime}}\sum_{j=1}^{N}c_{j}(v)x^{\prime j-1}$, with
$c_{j}(v)=\frac{1}{v-a}P_{N-j}\left(\frac{1}{v-a}\right)$, for some polynomial
$P_{N-j}$ of degree $N-j$. It follows that

$L_{N}(x,x^{\prime})={\rm e}^{ax^{\prime}}\sum_{j=1}^{N}x^{\prime
j-1}\frac{1}{2\pi\mathrm{i}}\int_{\ell_{N}}W_{N}(v)c_{j}(v){\rm e}^{-vx}{\rm
d}v.$

Hence, (1.7) is equal to

$\frac{1}{N!}\det\left({\rm
e}^{ax_{k}}\sum_{j=1}^{N}x_{k}^{j-1}\int_{\ell_{N}}W_{N}(v)c_{j}(v){\rm
e}^{-vx_{m}}{\rm d}v\right)_{k,m=1}^{N}\prod_{k=1}^{N}{\rm d}x_{k}\\\
=\frac{1}{N!}\det\left(x_{k}^{j-1}\right)_{k,j=1}^{N}\det\left(\frac{1}{2\pi\mathrm{i}}\int_{\ell_{N}}c_{j}(v)W_{N}(v){\rm
e}^{-vx_{m}}{\rm d}v\right)_{j,m=1}^{N}\prod_{k=1}^{N}{\rm e}^{ax_{k}}{\rm
d}x_{k}.$ (2.10)

In the latter expression, we can replace
$\displaystyle\int_{\ell_{N}}c_{j}(v)W_{N}(v){\rm e}^{-vx_{m}}{\rm d}v$,
$j=1,\ldots,N$, by any set of functions with the same span, like for instance
$\displaystyle\int_{\ell_{N}}\frac{(v-a)^{j-1}W_{N}(v)}{(v-a)^{N}}{\rm
e}^{-vx_{m}}{\rm d}v$, $j=1,\ldots,N$, at the price of replacing the
normalization constant $N!$ by a different constant $Z_{N}$. We then obtain
that

$\displaystyle{\rm d}\mu_{N}(\vec{x})$
$\displaystyle=\frac{1}{N!}\det\left(x_{k}^{j-1}\right)_{k,j=1}^{N}\det\left(\frac{1}{2\pi\mathrm{i}}\int_{\ell_{N}}c_{j}(v)W_{N}(v){\rm
e}^{-vx_{k}}{\rm d}v\right)_{j,k=1}^{N}\prod_{k=1}^{N}{\rm e}^{ax_{k}}{\rm
d}x_{k}$
$\displaystyle=\frac{1}{Z_{N}}\det\left(x_{k}^{j-1}\right)_{k,j=1}^{N}\det\left(\frac{1}{2\pi\mathrm{i}}\int_{\ell_{N}}\frac{(v-a)^{j-1}W_{N}(v)}{(v-a)^{N}}{\rm
e}^{-vx_{k}}{\rm d}v\right)_{j,k=1}^{N}\prod_{k=1}^{N}{\rm e}^{ax_{k}}{\rm
d}x_{k}$
$\displaystyle=\frac{1}{Z_{N}}\det\left(x_{k}^{j-1}\right)_{k,j=1}^{N}\det\left(\phi_{j}(x_{k})\right)_{j,k=1}^{N}\prod_{k=1}^{N}{\rm
d}x_{k}$

This proves part 3 of Theorem 1.2.

In a similar way, one can obtain explicit expressions for the biorthogonal
measures in cases where some, but not all, of the zeros $a_{1},\ldots,a_{N}$
are confluent.

##  3 Determinant identities

In this section, we will prove the determinant identities from Theorem 1.7. We
will start by expressing $\mu_{N}[\sigma]$ as an $N\times N$ determinant, and
afterwards we proceed towards the different Fredholm determinant identities.
Again, throughout the section, we assume that ${\Sigma_{N}},\ell_{N},W_{N}$
satisfy Assumption 1.1.

###  3.1 Matrix determinants

The finite size determinants of part 1 of Theorem 1.7 are obtained after a
straightforward application of Andréief’s identity.

######  Proposition 3.1.

Let $\sigma:\mathbb{R}\to\mathbb{C}$ be bounded and measurable.

  1. 1.

If $a_{1},\ldots,a_{N}$ are distinct, we have

$\mu_{N}[\sigma]=\det\left(\int_{\mathbb{R}}(1-\sigma(x)){\rm
e}^{a_{m}x}\psi_{k}({\rm e}^{x}){\rm d}x\right)_{m,k=1}^{N}.$

  2. 2.

If $a_{1},\ldots,a_{N}=a$, we have

$\mu_{N}[\sigma]=\frac{\displaystyle\det\left(\int_{\mathbb{R}}(1-\sigma(x))x^{m-1}\phi_{k}({\rm
e}^{x}){\rm
d}x\right)_{m,k=1}^{N}}{\displaystyle\det\left(\int_{\mathbb{R}}x^{m-1}\phi_{k}({\rm
e}^{x}){\rm d}x\right)_{m,k=1}^{N}}.$

###### Proof.

Let $a_{1},\ldots,a_{N}$ be distinct. From the expression (1.14) for
$\mu_{N}$, we have

$\mu_{N}[\sigma]=\frac{1}{N!}\int_{\mathbb{R}^{N}}\det\left({\rm
e}^{a_{j}x_{k}}\right)_{k,j=1}^{N}\det\left(\psi_{j}({\rm
e}^{x_{m}})\right)_{j,m=1}^{N}\prod_{j=1}^{N}\left(1-\sigma(x_{j})\right){\rm
d}x_{j}.$

We now apply Andréief’s identity on the right hand side and obtain the result.

If $a_{1}=\cdots=a_{N}=a$, we have

$\mu_{N}[\sigma]=\frac{1}{Z_{N}}\int_{\mathbb{R}^{N}}\det\left(x_{j}^{k-1}\right)_{k,j=1}^{N}\det\left(\phi_{j}({\rm
e}^{x_{m}})\right)_{j,m=1}^{N}\prod_{j=1}^{N}\left(1-\sigma(x_{j})\right){\rm
d}x_{j},$

with

$Z_{N}=\int_{\mathbb{R}^{N}}\det\left(x_{j}^{k-1}\right)_{k,j=1}^{N}\det\left(\phi_{j}({\rm
e}^{x_{m}})\right)_{j,m=1}^{N}\prod_{j=1}^{N}{\rm d}x_{j}.$

Applying Andréief’s identity to the above two expressions, we obtain the
result. ∎

###  3.2 Fredholm determinants

The following lemma is another classical result concerning biorthogonal
ensembles. Once again, we recall the proof for convenience, and to convince
the reader that it does not require positivity of the biorthogonal measure.

######  Proposition 3.2.

The average multiplicative statistics defined in (1.24) satisfy the identity

$\mu_{N}[\sigma]=1+\sum_{n=1}^{N}\frac{(-1)^{n}}{n!}\int_{\mathbb{R}^{n}}\det\Big{(}\sigma(x_{j})L_{N}(x_{j},x_{m})\Big{)}_{j,m=1}^{n}{\rm
d}x_{1}\cdots{\rm d}x_{n}.$ (3.1)

###### Proof.

We expand the left hand side of the identity as

$\displaystyle\mu_{N}[\sigma]$
$\displaystyle=\int\prod_{j=1}^{N}\left(1-\sigma(x_{j})\right){\rm
d}\mu_{N}(x_{1},\ldots,x_{N})$
$\displaystyle=1+\sum_{n=1}^{N}(-1)^{n}\binom{N}{n}\int_{\mathbb{R}^{N}}\prod_{j=1}^{n}\sigma(x_{j}){\rm
d}\mu_{N}(x_{1},\ldots,x_{N})$
$\displaystyle=1+\sum_{n=1}^{N}(-1)^{n}\frac{1}{n!(N-n)!}\int_{\mathbb{R}^{N}}\prod_{j=1}^{n}\sigma(x_{j})\det\Big{(}L_{N}(x_{k},x_{m})\Big{)}_{k,m=1}^{N}{\rm
d}x_{1}\cdots{\rm d}x_{N}.$

Integrating out the variables $x_{n+1},\ldots,x_{N}$, using Proposition 2.3
(or the equivalent identity (2.7)), we obtain that the latter is equal to

$\displaystyle
1+\sum_{n=1}^{N}\frac{(-1)^{n}}{n!}\int_{\mathbb{R}^{n}}\det\left(\sigma(x_{j})L_{N}(x_{j},x_{m})\right)_{j,m=1}^{n}{\rm
d}x_{1}\cdots{\rm d}x_{n},$ (3.2)

∎

We recognize the right hand side of the identity (3.1) as the truncated
Fredholm series $\det(1-\sigma L_{N})_{L^{2}(\mathbb{R})}$, and the identity
(1.30) is proven.

To prove the other identities in Theorem 1.7 (points 3 and 4), we need to rely
on properties of Fredholm determinants. For that reason, we will now construct
trace-class operators associated to suitable conjugations of the kernels we
considered before, such that we can interpret the truncated Fredholm series as
proper Fredholm determinants associated to (finite rank) trace-class
operators.

The kernel

$L_{N}(x,x^{\prime})=\sum_{m=1}^{N}{\rm e}^{a_{m}x^{\prime}}\psi_{m}({\rm
e}^{x})$

does not directly define an integral operator on $L^{2}(\mathbb{R})$, because
neither ${\rm e}^{a_{m}x^{\prime}}$ nor $\psi_{m}({\rm e}^{x})$ are (in
general) in $L^{2}(\mathbb{R})$. We can however define a new kernel of the
form

$\widetilde{L}_{N}(x,x^{\prime}):=\frac{h(x)}{h(x^{\prime})}L_{N}(x,x^{\prime}),$
(3.3)

such that the Fredhom series $\det(1-\sigma L_{N})_{L^{2}(\mathbb{R})}$ and
$\det(1-\sigma\widetilde{L}_{N})_{L^{2}(\mathbb{R})}$ are equal by definition,
but the integral operator $\mathcal{L}_{N}$ defined by

$\mathcal{L}_{N}[f](x):=\int_{\mathbb{R}}\widetilde{L}_{N}(x,x^{\prime})f(x^{\prime}){\rm
d}x^{\prime},$ (3.4)

is a genuine integral operator acting on $L^{2}(\mathbb{R})$, of rank $N$ and
in particular trace-class. A suitable choice for $h$ turns out to be

$h(x)=\begin{cases}{\rm e}^{(a_{\max}+\epsilon)x},&x\geq 0,\\\ {\rm
e}^{(a_{\min}-\epsilon)x},&x<0,\end{cases}$ (3.5)

for sufficiently small $\epsilon>0$, with $a_{\min}$ and $a_{\max}$ as in
(1.28). Then, the asymptotic behavior (2.3) for $\psi({\rm e}^{x})$ as
$x\to\pm\infty$ implies indeed that $\mathcal{L}_{N}$ is a bounded linear
operator on $L^{2}(\mathbb{R})$. Similarly, we define for $H_{N}^{\sigma}$
from (1.26) the conjugated kernel

$\widetilde{H}_{N}^{\sigma}(y,y^{\prime}):={\rm
e}^{-(a_{\max}+\epsilon)(y-y^{\prime})}H_{N}^{\sigma}(y,y^{\prime}),$ (3.6)

and the associated integral operator

$\mathcal{H}_{N}^{\sigma}[f](y)=\int_{0}^{\infty}\widetilde{H}_{N}^{\sigma}(y,y^{\prime})f(y^{\prime}){\rm
d}y^{\prime},$

acting on $L^{2}(0,\infty)$.

######  Proposition 3.3.

Let $W_{N}$ satisfy Assumption 1.1 and be such that
$W_{N}(v)=O(|v|^{-1-\epsilon})$ as $v\to\infty$ in $\mathcal{S}_{N}$. Let
$\sigma:\mathbb{R}\to\mathbb{C}$ be bounded and measurable, and such that
there exists $\epsilon>0$ such that

$\sigma(x)=O\left({\rm e}^{(a_{\max}-a_{\min}+\epsilon)x}\right),\qquad
x\to-\infty.$

The operators $\mathcal{H}_{N}^{\sigma}$ and $\sigma\mathcal{L}_{N}$ are
trace-class, and we have the identity

$\det(1-\mathcal{H}_{N}^{\sigma})_{L^{2}(0,\infty)}=\det(1-\sigma\mathcal{L}_{N})_{L^{2}(\mathbb{R})}.$
(3.7)

###### Proof.

Given $\Psi_{1}$ and $\Psi_{2}$ as in (1.27), define two operators

$\mathcal{A}:L^{2}(\mathbb{R})\to
L^{2}(0,\infty)\quad\text{and}\quad\mathcal{B}:L^{2}(0,\infty)\to
L^{2}(\mathbb{R})$

with kernels

$A(y,x):=\frac{{\rm e}^{-(a_{\max}+\epsilon)y}}{h(x)}\Psi_{1}\left({\rm
e}^{y+x}\right),\quad B(x,y^{\prime})={\rm
e}^{(a_{\max}+\epsilon)y^{\prime}}h(x)\sigma(x)\Psi_{2}\left({\rm
e}^{y^{\prime}+x}\right).$

Observe also that $\Psi_{2}$ is well-defined since
$W_{N}(v)=O(|v|^{-1-\epsilon})$ as $v\to\infty$ in $\mathcal{S}_{N}$. Using
Assumption 1.1 and the definition (1.27) of $\Psi_{1}$, it is straightforward
to check by deforming $\Sigma_{N}$ to a sufficiently narrow curve encircling
$a_{1},\ldots,a_{N}$ that, for any $\epsilon>0$,

$\Psi_{1}({\rm e}^{y+x})=O\left({\rm e}^{(a_{\min}-\epsilon/2)(y+x)}\right)\
\mbox{as $y+x\to-\infty$,}\quad\Psi_{1}({\rm e}^{y+x})=O\left({\rm
e}^{(a_{\max}+\epsilon/2)(y+x)}\right)\ \mbox{as $y+x\to+\infty$},$

such that

$A(y,x)=O\left({\rm e}^{-\epsilon|x+y|/2}\right),\qquad x+y\to\pm\infty.$

It is now straightforward to check that $\mathcal{A}$ is a Hilbert-Schmidt
operator.

Similarly, for sufficiently small $\epsilon$, we can deform $\ell_{N}$ to the
vertical line with real part $a_{\max}+\frac{3}{2}\epsilon$, and hence

$\Psi_{2}({\rm e}^{y+x})=O\left({\rm
e}^{-(a_{\max}+\frac{3}{2}\epsilon)(y+x)}\right)\quad\mbox{as
$y+x\to\pm\infty$},$

such that

$B(x,y)=O\left({\rm e}^{-\frac{\epsilon}{2}y}{\rm
e}^{-(a_{\max}+\frac{3}{2}\epsilon)x}\sigma(x)h(x)\right),\qquad
x+y\to\pm\infty,$

which implies that $\mathcal{B}$ is also a Hilbert-Schmidt operator, under the
condition that

$\sigma(x)={\rm e}^{(a_{\max}-a_{\min}+\epsilon)x},\qquad x\to-\infty.$

As compositions of Hilbert-Schmidt operators, $\mathcal{A}\mathcal{B}$ and
$\mathcal{B}\mathcal{A}$ are trace-class operators, and we have

$\det(1-\mathcal{A}\mathcal{B})_{L^{2}(0,\infty)}=\det(1-\mathcal{B}\mathcal{A})_{L^{2}(\mathbb{R})}.$
(3.8)

It is easy to see that $\mathcal{H}_{N}^{\sigma}=\mathcal{A}\mathcal{B}$ by
comparing the kernels of both operators, and we will now prove that
$\sigma\mathcal{L}_{N}=\mathcal{B}\mathcal{A}$, which implies the result.

Using the integral representation of $\Psi_{1}$ and $\Psi_{2}$ from (1.27), we
compute the kernel of the operator $\mathcal{B}\mathcal{A}$ acting on
$L^{2}(\mathbb{R})$ as follows:

$\displaystyle(BA)(x,x^{\prime})$
$\displaystyle=\frac{1}{(2\pi\mathrm{i})^{2}}\sigma(x)\frac{h(x)}{h(x^{\prime})}\int_{0}^{\infty}{\rm
d}y\int_{{\Sigma_{N}}}{\rm d}u\frac{{\rm
e}^{(y+x^{\prime})u}}{W_{N}(u)}\int_{\ell_{N}}{\rm d}vW_{N}(v){\rm
e}^{-(y+x)v}$
$\displaystyle=\frac{1}{(2\pi\mathrm{i})^{2}}\sigma(x)\frac{h(x)}{h(x^{\prime})}\int_{{\Sigma_{N}}}{\rm
d}u\frac{{\rm e}^{x^{\prime}u}}{W_{N}(u)}\int_{\ell_{N}}{\rm d}vW_{N}(v){\rm
e}^{-xv}\int_{0}^{\infty}{\rm d}y{\rm e}^{y(u-v)}$
$\displaystyle=\sigma(x)\frac{h(x)}{h(x^{\prime})}L_{N}(x,x^{\prime})=\sigma(x)\widetilde{L}_{N}(x,x^{\prime}),$

such that indeed $\mathcal{B}\mathcal{A}=\sigma\mathcal{L}_{N}$. ∎

######  Remark 3.4.

If $\sigma$ is differentiable, we can use integration by parts to rewrite
$H_{N}^{\sigma}(y,y^{\prime})$ as

$\displaystyle H_{N}^{\sigma}(y,y^{\prime})$
$\displaystyle=\int_{\mathbb{R}}{\rm d}r\sigma(r)\int_{{\Sigma_{N}}}\frac{{\rm
d}u}{2\pi\mathrm{i}}\frac{1}{W_{N}(u)}{\rm
e}^{(y+r)u}\int_{\ell_{N}}\frac{{\rm d}v}{2\pi\mathrm{i}}W_{N}(v){\rm
e}^{-(y^{\prime}+r)v}$
$\displaystyle=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{{\mathbb{R}}}{\rm
d}r\sigma^{\prime}(r)\int_{{\Sigma_{N}}}{\rm d}u\int_{\ell_{N}}{\rm
d}v\frac{W_{N}(v)}{W_{N}(u)}{\rm e}^{(y+r)u}{\rm
e}^{-(y^{\prime}+r)v}\frac{1}{v-u}$ $\displaystyle=\int_{{\mathbb{R}}}{\rm
d}r\sigma^{\prime}(r)L_{N}\left(y^{\prime}+r,y+r\right).$

Fredholm determinants with kernels of this type can be useful to obtain
underlying differential equations and integro-differential equations, like for
instance in [18].

We constructed $\mathcal{L}_{N}$ and $\mathcal{H}_{N}^{\sigma}$ in such a way
that

$\det(1-\sigma\mathcal{L}_{N})_{L^{2}(0,\infty)}=\det(1-\sigma
L_{N})_{L^{2}(\mathbb{R})},\qquad\det(1-\mathcal{H}_{N}^{\sigma})_{L^{2}(0,\infty)}=\det(1-H_{N}^{\sigma})_{L^{2}(\mathbb{R})},$

where with the left hand sides, we mean proper Fredholm determinants, and with
the right hand sides, we mean Fredholm series. As a consequence, combining
Proposition 3.3 with Proposition 3.2, we obtain (1.31).

The result below generalizes Lemma 8.8 in [12], and will imply (1.32).

######  Proposition 3.5.

Let $W_{N}$ satisfy Assumption 1.1 and be such that
$W_{N}(v)=O(|v|^{-1-\epsilon})$ as $v\to\infty$ in $\mathcal{S}_{N}$. Let
$t\in\mathbb{R}$, and let $\sigma_{t}(x)=\frac{1}{1+{\rm e}^{-x-t}}$. If
$a_{\max}-a_{\min}<1$, we have the identity

$\det\left(1-\mathcal{H}_{N}^{\sigma_{t}}\right)_{L^{2}(\mathbb{R})}=\det\left(1+\mathcal{K}_{N,t}\right)_{L^{2}({\Sigma_{N}})},$
(3.9)

with $\mathcal{K}_{N,t}$ the integral operator acting on $L^{2}(\Sigma_{N})$
with kernel $K_{N,t}$ defined in (1.29).

###### Proof.

Using the fact that $\frac{1}{v-u^{\prime}}=\displaystyle\int_{0}^{\infty}{\rm
d}y\,{\rm e}^{-y(v-u^{\prime})}$ when $\mathrm{Re}(v-u^{\prime})>0$, we can
write $K_{N,t}$ from (1.29) as the composition of two Hilbert-Schmidt
operators, $\mathcal{K}_{N,t}=\mathcal{C}\mathcal{D}$, with

$\mathcal{C}:L^{2}(0,\infty)\to
L^{2}({\Sigma_{N}})\;\text{and}\;\mathcal{D}:L^{2}({\Sigma_{N}})\to
L^{2}(0,\infty)$

with kernels

$C(u,y)=\frac{{\rm e}^{yc}}{2\pi\mathrm{i}}\int_{\ell_{N}}\frac{\pi{\rm
e}^{t(v-u)}}{\sin\pi(u-v)}\frac{W_{N}(v)}{W_{N}(u)}{\rm e}^{-yv}{\rm
d}v,\quad\quad D(y,u^{\prime})=\frac{1}{2\pi\mathrm{i}}{\rm
e}^{y(u^{\prime}-c)},$

where $\ell_{N}$ is the vertical line with real part $c$, with
$a_{\max}<c<a_{\min}+1$, and where $\Sigma_{N}$ lies at the left of
$\ell_{N}$. It is then straightforward to verify that $\mathcal{D}$ is a
Hilbert-Schmidt operator. For $\mathcal{C}$, the same is true, but note that
we need here that $0<\mathrm{Re}\,(v-u)<1$ for every $v\in\ell_{N}$,
$u\in{\Sigma_{N}}$ in order to avoid zeros of $\sin\pi(u-v)$.

Now we use the fact that
$\det(1+\mathcal{C}\mathcal{D})_{L^{2}({\Sigma_{N}})}=\det(1+\mathcal{D}\mathcal{C})_{L^{2}(0,\infty)}$,
and compute the kernel of $\mathcal{D}\mathcal{C}$ as (note that the minus
sign in front compensates the modified sign of the argument of $\sin$)

$(DC)(y,y^{\prime})=-\frac{1}{(2\pi\mathrm{i})^{2}}\int_{{\Sigma_{N}}}{\rm
d}u\ {\rm e}^{y(u-c)}\int_{\ell_{N}}{\rm d}v\frac{\pi{\rm
e}^{t(v-u)}}{\sin\pi(v-u)}\frac{W_{N}(v)}{W_{N}(u)}{\rm
e}^{-y^{\prime}(v-c)}.$

We then substitute the identity

$\frac{\pi}{\sin\pi s}=\int_{\mathbb{R}}\frac{{\rm e}^{-sx}}{1+{\rm
e}^{-x}}{\rm d}x,\qquad 0<\mathrm{Re}\,s<1,$

where we use once more that $0<\mathrm{Re}\,(v-u)<1$ for every $v\in\ell_{N}$,
$u\in{\Sigma_{N}}$, and change the order of integration, to obtain

$\displaystyle(DC)(y,y^{\prime})$ $\displaystyle=-{\rm
e}^{c(y^{\prime}-y)}\int_{{\mathbb{R}}}\frac{{\rm d}x}{1+{\rm
e}^{-x}}\int_{{\Sigma_{N}}}\frac{{\rm
d}u}{2\pi\mathrm{i}}\frac{1}{W_{N}(u)}{\rm
e}^{(y+x-t)u}\int_{\ell_{N}}\frac{{\rm d}v}{2\pi\mathrm{i}}W_{N}(v){\rm
e}^{-(y^{\prime}+x-t)v}$ $\displaystyle=-{\rm
e}^{c(y^{\prime}-y)}\int_{{\mathbb{R}}}\frac{{\rm d}x}{1+{\rm
e}^{-x}}\int_{{\Sigma_{N}}}\frac{{\rm
d}u}{2\pi\mathrm{i}}\frac{1}{W_{N}(u)}{\rm
e}^{(y+x-t)u}\int_{\ell_{N}}\frac{{\rm d}v}{2\pi\mathrm{i}}W_{N}(v){\rm
e}^{-(y^{\prime}+x-t)v}$ $\displaystyle=-{\rm
e}^{c(y^{\prime}-y)}\int_{{\mathbb{R}}}\frac{{\rm d}r}{1+{\rm
e}^{-r-t}}\int_{{\Sigma_{N}}}\frac{{\rm
d}u}{2\pi\mathrm{i}}\frac{1}{W_{N}(u)}{\rm
e}^{(y+r)u}\int_{\ell_{N}}\frac{{\rm d}v}{2\pi\mathrm{i}}W_{N}(v){\rm
e}^{-(y^{\prime}+r)v},$

which is exactly $-\widetilde{H}_{N}^{\sigma_{t}}(y,y^{\prime})$ by (1.26) and
(1.27), if $c=a_{\max}+\epsilon$. ∎

##  4 Logarithmic derivatives and deformed biorthogonal measures

In this section, we will prove Theorem 1.10. The proof of this result is
essentially a special case of a more general result in [23]. Besides
Assumption 1.1, we assume here that $\sigma:\mathbb{R}\to\mathbb{C}$ is
$C^{1}$, bounded, and with bounded derivative, and that $t$ is such that
$\mu_{N}[\sigma_{t}]=\det(1-\sigma_{t}\mathcal{L}_{N})_{L^{2}(\mathbb{R})}\neq
0$, with $\sigma_{t}(x)=\sigma(x+t)$.

For simplicity, we will first assume that $a_{1},\ldots,a_{N}$ are distinct.
Later, we will use a continuity argument to prove our identities in general.
By Proposition 2.1, we know that $\widetilde{L}_{N}$ can be written in the
form

$\widetilde{L}_{N}(x;x^{\prime})=\sum_{k=1}^{N}F_{k}(x)G_{k}(x^{\prime}),\qquad\int_{\mathbb{R}}F_{k}(x)G_{j}(x){\rm
d}x=\delta_{jk},\quad\forall\,j,k=1,\ldots,N,$ (4.1)

with $F_{k}(x)=h(x)\psi_{k}({\rm e}^{x})$ and $G_{k}(x)={\rm
e}^{a_{k}x}/h(x)$. The asymptotic behavior (2.3) of $\psi_{k}$ and the
definition (3.5) imply that $F_{k}(x)$ and $G_{k}(x)$ decay rapidly as
$x\to\pm\infty$. Consequently, the operator $\mathcal{L}_{N}$ is a bounded
linear operator, which is moreover finite rank and thus trace-class. By the
Jacobi identity, we have

$\frac{{\rm d}}{{\rm d}t}\log\mu_{N}[\sigma_{t}]=\frac{{\rm d}}{{\rm
d}t}\log\det(1-\sigma_{t}\mathcal{L}_{N})_{L^{2}(\mathbb{R})}=-\mathrm{Tr}\,\left(\left(\frac{{\rm
d}}{{\rm
d}t}\sigma_{t}\right)\mathcal{L}_{N}(1-\sigma_{t}\mathcal{L}_{N})^{-1}\right)_{L^{2}(\mathbb{R})},$
(4.2)

where we observe that $(\frac{{\rm d}}{{\rm
d}t}\sigma_{t})\mathcal{L}_{N}(1-\sigma_{t}\mathcal{L}_{N})^{-1}=\sigma_{t}^{\prime}\mathcal{L}_{N}(1-\sigma_{t}\mathcal{L}_{N})^{-1}$
is a trace-class operator if $\sigma_{t}^{\prime}$ is bounded. We thus have

$\frac{{\rm d}}{{\rm
d}t}\log\mu_{N}[\sigma_{t}]=-\mathrm{Tr}\,\left(\sigma_{t}^{\prime}\mathcal{L}_{N}(1-\sigma_{t}\mathcal{L}_{N})^{-1}\right)_{L^{2}(\mathbb{R})}.$
(4.3)

By definition (3.4) and by Proposition 2.2, we directly verify that
$\mathcal{L}_{N}^{2}=\mathcal{L}_{N}$, such that $\mathcal{L}_{N}$ is a rank
$N$ (not necessarily Hermitian) projection operator. We denote
$\mathcal{F}_{N}$ for the linear span of $F_{1},\ldots,F_{N}$ and
$\mathcal{G}_{N}$ for the linear span of $G_{1},\ldots,G_{N}$.

######  Proposition 4.1.

$\mathcal{L}_{N}$ is the unique projection operator on $L^{2}(\mathbb{R})$
with range $\mathcal{F}_{N}$ and kernel $\mathcal{G}_{N}^{\perp}$.

###### Proof.

It is straightforward to verify from (3.4)–(3.3) and (4.1) that
$\mathcal{L}_{N}[f]=f$ for $f\in\mathcal{F}_{N}$ and that
$\mathcal{L}_{N}[f]=0$ for $f\in\mathcal{G}_{N}^{\perp}$. Moreover,
$\mathcal{F}_{N}\oplus\mathcal{G}_{N}^{\perp}=L^{2}(\mathbb{R})$. Indeed,
suppose that this is not the case; then there exists a non-zero $f\in
L^{2}(\mathbb{R})$ such that $f\perp h$ for every
$h\in\mathcal{F}_{N}\oplus\mathcal{G}_{N}^{\perp}$. Consequently,
$f\in\mathcal{F}_{N}^{\perp}\cap\mathcal{G}_{N}$. but
$\mathcal{F}_{N}^{\perp}\cap\mathcal{G}_{N}=\\{0\\}$, by the orthogonality
relation $\int_{\mathbb{R}}F_{j}(x)G_{k}(x){\rm d}x=\delta_{jk}$, which gives
a contradiction. Using the above properties, we obtain that $\mathcal{L}_{N}$
is the unique projection with range $\mathcal{F}_{N}$ and kernel
$\mathcal{G}_{N}^{\perp}$. ∎

Let us now consider a kernel $M_{t}(x,x^{\prime})$ of the operator
$\mathcal{L}_{N}(1-\sigma_{t}\mathcal{L}_{N})^{-1}$ acting on
$L^{2}(\mathbb{R})$, such that

$\mathcal{L}_{N}(1-\sigma_{t}\mathcal{L}_{N})^{-1}[f](x)=\int_{\mathbb{R}}M_{t}(x,x^{\prime})f(x^{\prime}){\rm
d}x^{\prime}.$

Then we define an operator $\mathcal{P}_{t}$ acting on
$L^{2}\left(\mathbb{R},(1-\sigma_{t}(x)){\rm d}x\right)$, with respect to the
deformed Lebesgue measure $(1-\sigma_{t}(x)){\rm d}x$, as follows:

$\mathcal{P}_{t}[g](x)=\mathcal{L}_{N}(1-\sigma_{t}\mathcal{L}_{N})^{-1}[(1-\sigma_{t})g]=\int_{\mathbb{R}}M_{t}(x,x^{\prime})g(x^{\prime})(1-\sigma_{t}(x^{\prime})){\rm
d}x^{\prime}.$ (4.4)

######  Proposition 4.2.

The operator $\mathcal{P}_{t}$ is the unique projection operator on
$L^{2}\left((1-\sigma_{t}(x)){\rm d}x\right)$ with range $\mathcal{F}_{N}$ and
kernel $\mathcal{G}_{N}^{\perp}$.

###### Proof.

Note first that the orthogonal complement in $\mathcal{G}_{N}^{\perp}$ here
stands for orthogonality with respect to the deformed Lebesgue measure
$(1-\sigma_{t}(x)){\rm d}x$.

Let us first prove that
$\mathcal{F}_{N}\oplus\mathcal{G}_{N}^{\perp}=L^{2}\left((1-\sigma_{t}(x)){\rm
d}x\right)$. Similarly as before, suppose that this is not true, then there is
a non-zero $g\in L^{2}\left(\mathbb{R},(1-\sigma_{t}(x)){\rm d}x\right)$ in
the orthogonal complement of $\mathcal{F}_{N}\oplus\mathcal{G}_{N}^{\perp}$.
This means that $g\in\mathcal{F}_{N}^{\perp}\cap\mathcal{G}_{N}$. In other
words, $f=g(1-\sigma_{t})\in L^{2}(\mathbb{R},{\rm d}x)$ belongs to the
$L^{2}(\mathbb{R},{\rm d}x)$-orthogonal complement of $\mathcal{F}_{N}$, which
has trivial intersection with $\mathcal{G}_{N}$. This gives a contradiction,
and we conclude that
$\mathcal{F}_{N}\oplus\mathcal{G}_{N}^{\perp}=L^{2}\left((1-\sigma_{t}(x)){\rm
d}x\right)$.

Let $g\in\mathcal{F}_{N}$, such that
$(1-\sigma_{t})g=(1-\sigma_{t}\mathcal{L}_{N})g$. Then,

$\mathcal{P}_{t}[g]=\mathcal{L}_{N}(1-\sigma_{t}\mathcal{L}_{N})^{-1}[(1-\sigma_{t})g]=\mathcal{L}_{N}[g]=g.$

Finally, let $g\in\mathcal{G}_{N}^{\perp}$, such that
$\mathcal{L}_{N}[(1-\sigma_{t})g]=0$ and
$(1-\sigma_{t})g=\left(1-\sigma_{t}\mathcal{L}_{N}\right)[(1-\sigma_{t})g]$.
Thus,

$\mathcal{P}_{t}[g]=\mathcal{L}_{N}[(1-\sigma_{t})g]=0.$

Combining the above properties, we obtain the result. ∎

Recall from (4.4) that the $L^{2}\left(\mathbb{R},(1-\sigma_{t}(x)){\rm
d}x\right)$-operator $\mathcal{P}_{t}$ has integral kernel
$M_{t}(x,x^{\prime})$, i.e.

$\mathcal{P}_{t}[g]=\mathcal{L}_{N}(1-\sigma_{t}\mathcal{L}_{N})^{-1}[(1-\sigma_{t})g]=\int_{\mathbb{R}}M_{t}(x,x^{\prime})g(x^{\prime})(1-\sigma_{t}(x^{\prime})){\rm
d}x,$

and we can take $M_{t}$ of the form

$M_{t}(x,x^{\prime})=\sum_{m=1}^{N}\widetilde{F}_{m}(x)\widetilde{G}_{m}(x^{\prime}),\quad\text{with}\quad\int_{\mathbb{R}}\widetilde{F}_{j}(x)\widetilde{G}_{k}(x)(1-\sigma_{t}(x)){\rm
d}x=\delta_{jk},\quad j,k=1,\ldots,N,$

where $\widetilde{F}_{1},\ldots,\widetilde{F}_{N}$ span the range
$\mathcal{F}_{N}$ of $\mathcal{L}_{N}$, and
$\widetilde{G}_{1},\ldots,\widetilde{G}_{N}$ span $\mathcal{G}_{N}$, since
$\mathcal{G}_{N}^{\perp}$ (in $L^{2}(\mathbb{R},(1-\sigma_{t}(x)){\rm d}x)$)
is the kernel of $\mathcal{P}_{t}$. The biorthogonal measure

${\rm
d}\widetilde{\nu}_{N,t}(x_{1},\ldots,x_{N}):=\frac{1}{N!}\det\left(M_{t}(x_{j},x_{k})\right)_{j,k=1}^{N}\prod_{k=1}^{N}(1-\sigma_{t}(x_{k})){\rm
d}x_{k}$

can then be rewritten as

$\displaystyle{\rm d}\widetilde{\nu}_{N,t}(\vec{x})$
$\displaystyle=\frac{1}{N!}\det\left(\sum_{m=1}^{N}\widetilde{F}_{m}(x_{j})\widetilde{G}_{m}(x_{k})\right)_{j,k=1}^{N}\prod_{k=1}^{N}(1-\sigma_{t}(x_{k}){\rm
d}x_{k}$
$\displaystyle=\frac{1}{N!}\det\left(\widetilde{F}_{m}(x_{j})\right)_{j,m=1}^{N}\det\left(\widetilde{G}_{m}(x_{k})\right)_{m,k=1}^{N}\prod_{k=1}^{N}(1-\sigma_{t}(x_{k})){\rm
d}x_{k}$
$\displaystyle=\frac{1}{Z_{N,t}}\det\left(F_{m}(x_{j})\right)_{j,m=1}^{N}\det\left(G_{m}(x_{k})\right)_{m,k=1}^{N}\prod_{k=1}^{N}(1-\sigma_{t}(x_{k})){\rm
d}x_{k}$
$\displaystyle=\frac{1}{N!}\det\left(\sum_{m=1}^{N}F_{m}(x_{j})G_{m}(x_{k})\right)_{j,k=1}^{N}\prod_{k=1}^{N}(1-\sigma_{t}(x_{k})){\rm
d}x_{k}$
$\displaystyle=\frac{1}{Z_{N,t}}\det\left(L_{N}(x_{j},x_{k})\right)_{j,k=1}^{N}\prod_{k=1}^{N}(1-\sigma_{t}(x_{k})){\rm
d}x_{k}={\rm d}\nu_{N,t}(\vec{x}).$

We can now conclude from (1.38) that

$\frac{{\rm d}}{{\rm
d}t}\log\mu_{N}[\sigma_{t}]=-\mathrm{Tr}\,\left(\sigma_{t}^{\prime}\mathcal{L}_{N}(1-\sigma_{t}\mathcal{L}_{N})^{-1}\right)_{L^{2}(\mathbb{R})}=\int_{\mathbb{R}}\sigma_{t}^{\prime}(x)M_{t}(x,x){\rm
d}x,$

and recalling (1.35) and (1.37), we moreover have
$M_{t}(x,x)=\widetilde{\kappa}_{N,t}(x)$, and
$M_{t}(x,x)=\frac{\kappa_{N,t}(x)}{1-\sigma_{t}(x)}$ if $\sigma_{t}(x)\neq 1$.
The above identities imply Theorem 1.10 in the case of distinct
$a_{1},\ldots,a_{N}$, and the general identities follow as before by
continuity.

##  5 Log Gamma polymer partition function and LUE with external source

In the last three sections of this paper, we give an overview of concrete
polymer and random matrix models which fit in our general framework, and to
which the results presented in Section 1 can be applied. A concise table with
a (probably non-exhaustive) list of models which can be described in terms of
our general class of biorthogonal measures is given at the end, see Table 1.

#### The model.

The Log Gamma polymer was introduced by Seppäläinen in [50]. Let $n,N>1$ be
two positive integers. We consider the lattice

$\Big{\\{}(j,k):j\in\\{1,2,\ldots,n\\},k\in\\{1,2,\ldots,N\\}\Big{\\}},$

and assign to each point $(j,k)$ in the lattice independently a random
variable $d_{j,k}\sim\Gamma^{-1}(\gamma_{j,k})$ with inverse Gamma
distribution,

$\mathbb{P}(d_{j,k}\leq
y)=\frac{1}{\Gamma(\gamma_{j,k})}\int_{0}^{y}x^{-\gamma_{j,k}-1}{\rm
e}^{-1/x}{\rm d}x.$

The mean of $d_{j,k}$ thus exists only for $\gamma_{j,k}>1$ and is then equal
to $\frac{1}{\gamma_{j,k}-1}$, while the variance is finite only if
$\gamma_{j,k}>2$, and then equal to
$\frac{1}{(\gamma_{j,k}-1)^{2}(\gamma_{j,k}-2)}$. The parameters
$\gamma_{j,k}$ take the form $\gamma_{j,k}=\alpha_{j}-a_{k}$, where
$\vec{\alpha}=(\alpha_{j})_{j=1}^{n}$ and $\vec{a}=(a_{k})_{k=1}^{N}$ are two
vectors of parameters such that $\alpha_{j}-a_{k}>0$ for all $j=1,\ldots,n$
and $k=1,\ldots,N$.

$(1,1)$$(n,N)$$\alpha_{1}$$\alpha_{2}$$\alpha_{3}$$\alpha_{4}$$\alpha_{5}$$\alpha_{6}$$\alpha_{7}$$\alpha_{8}$$\alpha_{9}$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$
Figure 1: The Log Gamma polymer lattice for $n=9$, $N=6$, with a possible up-
right path. The values $\alpha_{j},a_{k}$ determine the weight of the point
$(j,k)$ in an up-right path.

The Log Gamma polymer partition function $Z_{n,N}^{{\rm
Log}\Gamma}({\vec{\alpha},\vec{a}})$ is the random variable

$Z_{n,N}^{{\rm
Log}\Gamma}(\vec{\alpha},\vec{a}):=\sum_{\pi:(1,1)\nearrow(n,N)}\prod_{(j,k)\in\pi}d_{j,k},$
(5.1)

where the sum is taken over all up-right directed lattice paths $\pi$ from
$(1,1)$ to $(n,N)$, and the product over all lattice points in an up-right
directed lattice path. See Figure 1.

Heuristically, given the fact that the mean and variance of the random
variables $d_{j,k}$ are small for $\gamma_{j,k}$ large, we can expect that for
large values of $\gamma_{j,k}$ , the weights $\prod_{(j,k)\in\pi}d_{j,k}$ of
different paths $\pi$ will then typically be small and not show large
deviations, such that many different paths will contribute to the leading
order of the partition function (large entropy). For $\gamma_{j,k}$ small, we
expect on the other hand large fluctuations, and then we expect that the
partition function will be dominated by few paths with large weights (small
entropy), such that the model degenerates to a last passage percolation or
corner growth model. See [50, 15] for more details about this model.

#### Fredholm determinant identity.

Suppose that $n\geq N$ and that there exist constants
$\delta_{1},\delta_{2}\in(0,1)$ such that
$\delta_{1}<\min\\{\delta_{2},1-\delta_{2}\\}<1/2,$ and such that

$0\leq a_{j}<\delta_{1},\quad\alpha_{k}>\delta_{2},\quad\forall
j=1,\ldots,N,\;\forall k=1,\ldots,n.$ (5.2)

Under these assumptions, Borodin, Corwin, and Remenik in [15, Corollary 1.8]
proved the identity

$\mathbb{E}\left[{\rm e}^{-{\rm e}^{t}Z_{n,N}^{{\rm
Log}\Gamma}({\vec{\alpha}},{\vec{a}})}\right]=\det(1+K_{N,t})_{L^{2}({\Sigma_{N}})},$
(5.3)

where $K_{N,t}$ (now also depending on $n,\vec{\alpha},{\vec{a}}$) is as in
(1.29), with ${\Sigma_{N}}$ the circle of radius $\delta_{1}$ around the
origin, $\ell_{N}$ the vertical line $\mathrm{Re}\,z=\delta_{2}$, and $W_{N}$
given by

$W_{N}(z)=W_{n,N}^{{\rm
Log}\,\Gamma}(z;\vec{\alpha},\vec{a}):=\frac{\prod_{j=1}^{n}\Gamma(\alpha_{j}-z)}{\prod_{k=1}^{N}\Gamma(z-a_{k})}.$
(5.4)

Assumption 1.1 is valid for this choice of $W_{N}$ if $n>N$, and also for
$n=N$ if $\sum_{k=1}^{N}(\alpha_{j}+a_{j}-2\delta_{2})<0$. The stronger decay
assumption for $W_{N}$ needed in parts 3 and 4 of Theorem 1.7 is also valid if
$n>N$, and if $n=N$ provided that
$\sum_{k=1}^{N}(\alpha_{j}+a_{j}-2\delta_{2})<-1$. This is easily seen by
using the following variant of Stirling’s approximation (see, e.g., [28,
formula (5.11.9)]),

$\Gamma\left(x+\mathrm{i}y\right)\sim\sqrt{2\pi}|y|^{x-\frac{1}{2}}{\rm
e}^{-\frac{\pi}{2}|y|},\qquad x,y\in\mathbb{R},\ y\to\pm\infty.$ (5.5)

The homogenenous Log Gamma polymer occurs when $\alpha_{k}=\alpha>0$ for
$k=1,\ldots,n$ and $a_{m}=0$ for $m=1,\ldots,N$, and is of particular
interest. Observe that this case meets the assumptions also when $n=N$ and
$\alpha<2$, since we can then take $\delta_{2}>\alpha/2$. The stronger decay
needed in parts 3 and 4 of Theorem 1.7 is also valid for $\alpha<2$ if $n=N$
is sufficiently large.

######  Remark 5.1.

The analogue of the kernel $K_{N,t}$ in [15] was defined without squaring
$2\pi\mathrm{i}$ in the denominator, but this is due to an extra factor
$\frac{1}{2\pi\mathrm{i}}$ in the definition of the scalar product on
$L^{2}({\Sigma_{N}})$ there.

#### Associated biorthogonal measures.

As anticipated in the introduction, Remark 1.9, combining the Fredholm
determinant identity (5.3) with Theorem 1.7, part 4, we can relate the
partition function of the Log Gamma polymer to an explicit biorthogonal
measure. In the following corollary and below, given $b\in\mathbb{C}$,
$[b]_{k}$ will denote a $k$-dimensional vector with $b$ in each entry and
$\vec{e}_{m}$ will denote the $N$-dimensional vector containing $1$ in
position $m$ and $0$ elsewhere.

######  Corollary 5.2.

Consider two sets of parameters
$\vec{\alpha}=\left(\alpha_{j}\right)_{j=1}^{n}$ and
$\vec{a}=\left(a_{k}\right)_{k=1}^{N}$ with $n\geq N$ and such that
$\alpha_{j}-a_{k}>0$ for all $j=1,\ldots,n$ and $k=1,\ldots,N$. Suppose
moreover that there exist two positive constants $\delta_{1},\delta_{2}$ such
that $\delta_{1}<\min\\{\delta_{2},1-\delta_{2}\\}<1/2$ and such that
equations (5.2) are satisfied. In the case where $n=N$ we assume in addition
that $\sum_{k=1}^{N}(\alpha_{j}+a_{j}-2\delta_{2})<-1$. Let $W_{N}$ be as in
(5.4), $\Sigma_{N}$ the circle of radius $\delta_{1}$ around the origin, and
$\ell_{N}$ the vertical line $\mathrm{Re}z=\delta_{2}$. Then:

  1. 1.

$\mathbb{E}\left[{\rm e}^{-{\rm e}^{t}Z_{n,N}^{{\rm
Log}\Gamma}({\vec{\alpha}},{\vec{a}})}\right]=\mu_{N}[\sigma_{t}],\qquad\sigma_{t}(x)=\frac{1}{1+{\rm
e}^{-x-t}},$ (5.6)

where the measure $\mu_{N}$ is given by equations (1.7)–(1.8).

  2. 2.

$\frac{{\rm d}}{{\rm d}t}\log\mathbb{E}\left[{\rm e}^{-{\rm
e}^{t}Z_{n,N}^{{\rm
Log}\Gamma}({\vec{\alpha}},{\vec{a}})}\right]=\int_{\mathbb{R}}\sigma_{t}(x)\kappa_{N,t}(x){\rm
d}x,$ (5.7)

where $\kappa_{N,t}$ is the one-point function (1.35) of the deformed
biorthogonal measure (1.33).

  3. 3.

In the case where the parameters $\vec{a}$ are all distinct, the biorthogonal
measure is given explicitly, in terms of Meijer $G$-function [28], by

${\rm d}\mu_{n,N}^{{\rm
Log}\Gamma}(\vec{x};\vec{\alpha},\vec{a})=\frac{1}{Z_{N}}\det\left({\rm
e}^{a_{m}x_{k}}\right)_{k,m=1}^{N}\det\left(G^{\,*,n}_{0,0}\lparen\begin{smallmatrix}n+N\\\
-\end{smallmatrix}|\,{\vec{\alpha}};[1]_{m}+{\vec{a}}-\vec{e}_{m}\rparen{{\rm
e}^{-x_{k}}}\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm d}x_{k}.$ (5.8)

In the confluent case $a_{1}=\cdots=a_{N}=0$, the biorthogonal measure is
given by

${\rm d}\mu_{n,N}^{{\rm
Log}\Gamma}(\vec{x};\vec{\alpha},\vec{0})=\frac{1}{Z_{N}}\Delta(\vec{x})\det\left(G^{\,*,n}_{0,0}\lparen\begin{smallmatrix}n+N\\\
-\end{smallmatrix}|\,{\vec{\alpha}};[0]_{N-m+1};[1]_{m-1}\rparen{{\rm
e}^{-x_{k}}}\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm d}x_{k}.$ (5.9)

###### Proof.

The first part is a direct consequence of the Fredholm determinant identity
(5.3) and Theorem 1.7, while the second part is a corollary of Theorem 1.10.
As for the third part, one can compute explicitly the functions $\psi_{m}$
from (1.12) as

$\displaystyle\psi_{m}(y)$
$\displaystyle=\frac{1}{2\pi\mathrm{i}\left(W_{n,N}^{{\rm
Log}\Gamma}\right)^{\prime}(a_{m};\vec{\alpha},\vec{a})}\int_{\ell_{N}}\frac{\prod_{j=1}^{n}\Gamma(\alpha_{j}-v)\
y^{-v}}{(v-a_{m})\prod_{j=1}^{N}\Gamma(v-a_{j})}{\rm d}v$
$\displaystyle=\frac{1}{2\pi\mathrm{i}\left(W_{n,N}^{{\rm
Log}\Gamma}\right)^{\prime}(a_{m};\vec{\alpha},\vec{a})}\int_{\ell_{N}}\frac{\prod_{j=1}^{n}\Gamma(\alpha_{j}-v)\
y^{-v}}{\Gamma(v-a_{m}+1)\prod_{j\neq m}\Gamma(v-a_{j})}{\rm d}v,$

and we recognize the right hand side as a Meijer $G$-function [28]:

$\psi_{m}(y)=\frac{1}{\left(W_{n,N}^{{\rm
Log}\Gamma}\right)^{\prime}(a_{m};\vec{\alpha},\vec{a})}G^{\,*,n}_{0,0}\lparen\begin{smallmatrix}n+N\\\
-\end{smallmatrix}|\,{\vec{\alpha}};[1]_{N}+{\vec{a}}-\vec{e}_{m}\rparen{1/y}.$
(5.10)

In the confluent case where $a_{1}=\cdots=a_{N}=0$, we have similarly that the
functions $\phi_{m}$ in (1.17) are given by

$\displaystyle\phi_{m}(y)$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\int_{\ell_{N}}\frac{\prod_{j=1}^{n}\Gamma(\alpha_{j}-v)}{\Gamma(v)^{N}}\frac{y^{-v}}{v^{N-m+1}}{\rm
d}v$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\int_{\ell_{N}}\frac{\prod_{j=1}^{n}\Gamma(\alpha_{j}-v)}{\Gamma(v)^{m-1}\Gamma(v+1)^{N-m+1}}{y^{-v}}{\rm
d}v$ $\displaystyle=G^{\,*,n}_{0,0}\lparen\begin{smallmatrix}n+N\\\
-\end{smallmatrix}|\,{\vec{\alpha}};[0]_{N-m+1};[1]_{m-1}\rparen{1/y}.$

∎

#### LUE with external source.

Let $A$ be a deterministic positive-definite Hermitian $N\times N$ matrix with
eigenvalues $a_{1},\ldots,a_{N}\in[0,1)$, and consider the probability
distribution

$\frac{1}{Z_{N}^{\prime}}(\det M)^{\nu}{\rm e}^{-{\rm Tr}((I-A)M)}{\rm
d}M,\qquad\nu>0,\quad{\rm d}M=\prod_{j=1}^{N}{\rm d}M_{jj}\ \prod_{1\leq
j<k\leq N}{\rm d}\mathrm{Re}\,M_{jk}\,{\rm d}\mathrm{Im}\,M_{jk},$

on the space of $N\times N$ positive-definite Hermitian matrices. This model
is known as the chiral Unitary Ensemble or LUE with external source and is
connected to sample covariance matrices, see e.g. [4, 26, 29, 30]. For $A=0$,
this is the classical LUE or Wishart-Laguerre ensemble.

The eigenvalues of $M$ have the joint probability distribution

$\frac{1}{Z_{N}}\Delta(\vec{x})\ \det\left(x_{j}^{\nu}{\rm
e}^{-(1-a_{k})x_{j}}\right)_{j,k=1}^{N}\prod_{j=1}^{N}{\rm d}x_{j},\qquad
x_{1},\ldots,x_{N}>0.$ (5.11)

This is a determinantal point process with kernel (see [26, Formula (4.2)] or
[30, Formula (5.158)])

$L_{N}^{\rm LUE+}(x,x^{\prime};\vec{a},\nu):=\frac{{\rm
e}^{x^{\prime}-x}}{(2\pi\mathrm{i})^{2}}\int_{\gamma_{1}}{\rm
d}u\int_{\gamma_{2}}{\rm
d}v\left(\frac{u}{v}\right)^{N+\nu}\prod_{j=1}^{N}\frac{v-1+a_{j}}{u-1+a_{j}}\frac{{\rm
e}^{-ux^{\prime}+vx}}{v-u},$ (5.12)

where $\gamma_{1}$ encircles $1-a_{1},\ldots,1-a_{N}$ anti-clockwise,
$\gamma_{2}$ encircles $0$ anti-clockwise, and they don’t intersect. Changing
$u\mapsto 1-u$, $v\mapsto 1-v$, this is equal to

$L_{N}^{\rm
LUE+}(x,x^{\prime};\vec{a},\nu):=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\Sigma_{N}}{\rm
d}u\int_{\ell_{N}}{\rm
d}v\left(\frac{u-1}{v-1}\right)^{N+\nu}\prod_{j=1}^{N}\frac{v-a_{j}}{u-a_{j}}\frac{{\rm
e}^{-vx+ux^{\prime}}}{v-u},$ (5.13)

where $\Sigma_{N}$ is a positively oriented circle around $a_{1},\ldots,a_{N}$
with radius $<1$, and $\ell_{N}$ a negatively oriented circle around $1$,
which can moreover be deformed (after changing the orientation) to a vertical
line lying between $\Sigma_{N}$ and $1$ if $\nu>0$. This kernel is of the form
(1.8) with

$W_{N}(z)\equiv W_{N}^{\rm
LUE+}(z;\vec{a},\nu):=\frac{\prod_{j=1}^{N}(z-a_{j})}{(z-1)^{N+\nu}}.$ (5.14)

Biorthogonal systems associated to this kernel can be constructed in terms of
multiple Laguerre polynomials, see [8]. We will now explain that (5.13), with
$\nu=n-N$ a non-negative integer, is the _zero-temperature limit_ of the Log
Gamma polymer biorthogonal measure.

#### Zero-temperature limit.

Let us consider the Log Gamma polymer with parameters

$\alpha_{1}=\cdots=\alpha_{n}=T\to 0$

and re-scaled $a$-parameters $Ta_{1},\ldots,Ta_{N}$ with $0\leq
a_{1},\ldots,a_{N}<1$. We then have

$\displaystyle W^{{\rm Log}\Gamma}_{n,N}(T\zeta;[T]_{n},T\vec{a})$
$\displaystyle=\frac{\Gamma(T(1-\zeta))^{n}}{\prod_{m=1}^{N}\Gamma(T(\zeta-
a_{m}))}$ $\displaystyle=T^{N-n}\frac{\prod_{m=1}^{N}(\zeta-
a_{m})}{(1-\zeta)^{n}}\frac{\Gamma(1+T(1-\zeta))^{n}}{\prod_{m=1}^{N}\Gamma(1+T(\zeta-
a_{m}))}.$

and hence

$\lim_{T\to 0}\frac{W^{{\rm Log}\Gamma}_{n,N}(Tv;[T]_{n},T\vec{a})}{W^{{\rm
Log}\Gamma}_{n,N}(Tu;[T]_{n},T{\vec{a}})}=\frac{W_{N}^{\rm
LUE+}(v;{\vec{a}},n-N)}{W_{N}^{\rm LUE+}(u;{\vec{a}},n-N)},\qquad\mbox{for
$v\in\ell_{N}$, $u\in\Sigma_{N}$}.$ (5.15)

A more careful analysis leads us to the following.

######  Proposition 5.3.

We have the point-wise limit

$\lim_{T\to 0}\frac{1}{T}L_{n,N}^{{\rm
Log}\Gamma}\left(\frac{x}{T},\frac{x^{\prime}}{T};[T]_{n},T{\vec{a}}\right)=L_{N}^{\rm
LUE+}\left(x,x^{\prime};{\vec{a}},\nu=n-N\right),\qquad
x,x^{\prime}\in\mathbb{R},$ (5.16)

and there exists $C_{n,N}(\vec{a})$ such that for $\epsilon,T>0$ sufficiently
small,

$\left|\frac{1}{T}L_{n,N}^{{\rm
Log}\Gamma}\left(\frac{x}{T},\frac{x^{\prime}}{T};[T]_{n},T{\vec{a}}\right)\right|\leq
C_{n,N}(\vec{a})h(x)\tilde{h}(x^{\prime}),$ (5.17)

with

$h(x)={\rm
e}^{-(a_{\max}+2\epsilon)x},\qquad\tilde{h}(x^{\prime})=\begin{cases}{\rm
e}^{(a_{\min}-\epsilon)x^{\prime}},&x^{\prime}<0\\\ {\rm
e}^{(a_{\max}+\epsilon)x^{\prime}},&x^{\prime}\geq 0\end{cases}.$ (5.18)

###### Proof.

It is straightforward to verify that

$\displaystyle\frac{1}{T}L_{n,N}^{{\rm
Log}\Gamma}\left(\frac{x}{T},\frac{x^{\prime}}{T};[T]_{n},T{\vec{a}}\right)=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\Sigma_{N}}{\rm
d}s\int_{\ell_{N}}{\rm d}w\frac{W_{n,N}^{{\rm
Log}\Gamma}(Tw;[T]_{n},T\vec{a})}{W_{n,N}^{{\rm
Log}\Gamma}(Ts;[T]_{n},T\vec{a})}\frac{{\rm e}^{-wx+sx^{\prime}}}{w-s}$
$\displaystyle\quad=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\Sigma_{N}}{\rm
d}s\int_{\ell_{N}}{\rm
d}w\left(\frac{s-1}{w-1}\right)^{n}\frac{\Gamma(1+T(1-w))^{n}}{\Gamma(1+T(1-s))^{n}}\prod_{j=1}^{N}\frac{w-a_{j}}{s-a_{j}}\frac{\Gamma(1+T(s-a_{j}))}{\Gamma(1+T(w-a_{j}))}\frac{{\rm
e}^{-wx+sx^{\prime}}}{w-s},$

where $\Sigma_{N}$ is a contour surrounding $a_{1},\ldots,a_{N}$ and no other
zeros of $W_{n,N}^{{\rm Log}\Gamma}(Tu)$, and $\ell_{N}$ lies at the right of
$\Sigma_{N}$ and at the left of $1$, and where we used
$\Gamma(z+1)=z\Gamma(z)$ to pass to the second line. Hence,

$\displaystyle\frac{1}{T}L_{n,N}^{{\rm
Log}\Gamma}\left(\frac{x}{T},\frac{x^{\prime}}{T};[T]_{n},T{\vec{a}}\right)$
$\displaystyle\quad=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\Sigma_{N}}{\rm
d}s\int_{\ell_{N}}{\rm d}w\frac{W_{N}^{\rm LUE+}(w;\vec{a},n-N)}{W_{N}^{\rm
LUE+}(s;\vec{a},n-N)}\left(\frac{\Gamma(1+T(1-w))^{n}}{\Gamma(1+T(1-s))^{n}}\prod_{j=1}^{N}\frac{\Gamma(1+T(s-a_{j}))}{\Gamma(1+T(w-a_{j}))}\right)\frac{{\rm
e}^{-wx+sx^{\prime}}}{w-s}.$

The integrand converges point-wise to $0$ as $T\to 0$. Moreover, the part
between brackets is uniformly bounded for $T$ sufficiently small,
$s\in\Sigma_{N}$, and $w\in\ell_{N}$, provided that we take $\ell_{N}$ close
to $1$. To see the uniformity in $w$, one has to use (5.5). We can then use
Lebesgue’s dominated convergence theorem to conclude that we have the point-
wise convergence (5.16).

To obtain the bound (5.17), we take
$\ell_{N}=a_{\max}+2\epsilon+\mathrm{i}\mathbb{R}$, and $\Sigma_{N}$ within
the strip $a_{\min}-\epsilon\leq\mathrm{Re}\,s\leq a_{\max}+\epsilon$, with
$\epsilon>0$ sufficiently small so that $a_{\max}+2\epsilon<1$. For $T>0$
sufficiently small, it then follows that

$\displaystyle\left|\frac{1}{T}L_{n,N}^{{\rm
Log}\Gamma}\left(\frac{x}{T},\frac{x^{\prime}}{T};[T]_{n},T{\vec{a}}\right)\right|$
$\displaystyle\leq
2\frac{h(x)\tilde{h}(x^{\prime})}{(2\pi)^{2}}\int_{\Sigma_{N}}|{\rm
d}s|\int_{\ell_{N}}|{\rm d}w|\left|\frac{W_{N}^{\rm
LUE+}(w;\vec{a},n-N)}{W_{N}^{\rm LUE+}(s;\vec{a},n-N)}\right|\frac{1}{|w-s|}$
$\displaystyle\leq C_{n,N}({\vec{a}})h(x)\widetilde{h}(x^{\prime}).$

The latter inequality holds only if $n>N$, since $\left|\frac{W_{N}^{\rm
LUE+}(w;\vec{a},n-N)}{W_{N}^{\rm LUE+}(s;\vec{a},n-N)}\right|\frac{1}{|w-s|}$
is not integrable for $n=N$. One can however refine the above estimates by
taking into account the decay of
$\frac{\Gamma(1+T(1-w))^{n}}{\Gamma(1+T(1-s))^{n}}\prod_{j=1}^{N}\frac{\Gamma(1+T(s-a_{j}))}{\Gamma(1+T(w-a_{j}))}$
as $w\to\infty$ on $\ell_{N}$, such that the same inequality remains to hold.
∎

######  Corollary 5.4.

As $T\to 0$, we have the convergence

$\mathbb{E}\left[{\rm e}^{-{\rm e}^{t/T}Z_{n,N}^{{\rm
Log}\Gamma}([T]_{n},T{\vec{a}})}\right]\longrightarrow\mathbb{P}_{\mathrm{LUE+}}\left(\max\\{x_{1},\ldots,x_{N}\\}\leq-t;\vec{a},\nu=N-n\right),$
(5.19)

where the expectation on the left is with respect to the distribution of the
Log Gamma polymer partition function, and the probability on the right is with
respect to the LUE with external source (5.11).

###### Proof.

As a consequence of the previous result, we have point-wise convergence of the
determinants

$\det\left(\frac{\sigma_{t/T}(x_{j}/T)}{T}L_{n,N}^{{\rm
Log}\Gamma}\left(\frac{x_{j}}{T},\frac{x_{k}}{T};[T]_{n},T{\vec{a}}\right)\right)_{j,k=1}^{N}\to\det\left(1_{(-t,+\infty)}(x)L_{N}^{\rm
LUE+}\left(x_{j},x_{k};{\vec{a}},n-N\right)\right)_{j,k=1}^{N}$

as $T\to 0$, where $\sigma_{t/T}(x/T)=\frac{1}{1+{\rm e}^{-\frac{x+t}{T}}}$.
Moreover, we have the domination

$\left|\det\left(\frac{\sigma_{t/T}(x_{j}/T)}{T}L_{n,N}^{{\rm
Log}\Gamma}\left(\frac{x_{j}}{T},\frac{x_{k}}{T};[T]_{n},T{\vec{a}}\right)\right)_{j,k=1}^{N}\right|\leq
C_{n,N}^{N}\prod_{j=1}^{N}h(x_{j})\tilde{h}(x_{j})\sigma_{t/T}(x_{j}/T)\ |\det
O(1)|,$

where $O(1)$ here denotes an $N\times N$ matrix with uniformly bounded
entries. Integrating over $\mathbb{R}^{N}$ yields $\mu_{n,N}^{{\rm
Log}\Gamma}[\sigma_{t/T}(./T)]\to\mu_{N}^{\rm LUE+}[1_{(t,+\infty)}]$, which
implies the result by (5.6). ∎

######  Remark 5.5.

The right hand side of (5.19) is the largest eigenvalue distribution in the
LUE with external source, and it is well-known [39, 17, 27] that this
distribution characterizes last passage percolation with exponential weights,
which is the zero temperature limit of the Log Gamma polymer. The above
results are thus not surprising, but reveal how the zero-temperature limit
takes place on the level of the associated biorthogonal measures. Moreover, it
now becomes clear that (5.6) is the natural finite temperature generalization
of the results from [39, 17, 27] which express the Laplace transform of last
passage percolation in terms of the largest eigenvalue of the LUE with
external source: the biorthogonal measures (5.8) and (5.9) play a similar role
in the Log Gamma polymer as the LUE with external source in last passage
percolation.

######  Remark 5.6.

Corollary 5.4 is consistent with known large $N$ asymptotic results from [15,
6], stating that suitably rescaled Log Gamma polymer partition function has
Tracy-Widom and Baik-Ben Arous-Péché fluctuations. These types of behavior are
well-known to occur also in the large $N$ limits of the LUE and the LUE with
external source, respectively.

##  6 O’Connell-Yor polymer partition function and GUE with external source

#### The model.

The O’Connell-Yor polymer [49], or semi-discrete directed polymer, is a semi-
discrete version of the Log Gamma polymer. In order to define it, we fix a
real positive number $\tau>0$, a positive integer $N>0$ and real numbers
$a_{1},\ldots,a_{N}$. For any $i=1,\ldots,N$, $B_{i}$ will denote a one-
dimensional Brownian motion with drift $a_{i}$. In other words, for any $s\geq
0$, $B_{i}(s)\sim a_{i}s+B(s)$, where $B(s)$ is a standard one-dimensional
Brownian motion. Moreover, we assume that $B_{1},\ldots,B_{N}$ are
independent. A semi-discrete up-right path between $(0,1)$ and $(\tau,N)$
consists of horizontal segments that can jump vertically between certain
points $(s,k)$ and $(s,k+1)$, see Figure 2. The energy of such a path $\phi$
is given by the sum of the Brownian increments between the jump points,

$E(\phi)=B_{1}(s_{1})+(B_{2}(s_{2})-B_{2}(s_{1}))+\ldots+(B_{N}(\tau)-B_{N}(s_{N-1})).$

The O’Connell-Yor polymer partition function is then defined as

$Z_{N}^{\rm OY}(\vec{a},\tau):=\int_{0<s_{1}<\ldots<s_{N-1}<\tau}{\rm
e}^{E(\phi)}{\rm d}s_{1}\cdots{\rm d}s_{N-1}.$ (6.1)

$(0,1)$$(\tau,N)$$(s_{1},1)$$(s_{1},2)$$(s_{2},2)$$(s_{2},3)$$(s_{3},3)$$(s_{3},4)$$(s_{4},4)$$(s_{4},5)$$(s_{5},5)$$(s_{5},6)$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$
Figure 2: A possible semi-discrete up-right path, with $N=6$, and jump points
$s_{1},\ldots,s_{5}$. The weight of a path with jump at $(s_{k},k)$ in the
O’Connell-Yor polymer depends on the value of $a_{k}$.

#### Fredholm determinant identity.

Suppose that $|a_{i}|<\delta_{1}<1/2$ for all $i=1,\ldots,N$, and take
$\delta_{2}\in(0,1)$ such that $\delta_{2}>2\delta_{1}$. Then, Borodin and
Corwin [14] proved the identity

$\mathbb{E}\left[{\rm e}^{-{\rm e}^{t}Z_{N}^{\rm
OY}({\vec{a}},\tau)}\right]=\det(1+K_{N,t})_{L^{2}({\Sigma_{N}})},$ (6.2)

with $K_{N,t}$ as in (1.29), with ${\Sigma_{N}}$ the circle of radius
$\delta_{1}$ around the origin, $\ell_{N}$ the vertical line
$\mathrm{Re}\,z=\delta_{2}$, and $W_{N}$ given by

$W_{N}(z)=W_{N}^{\rm OY}(z;\vec{a},\tau):=\frac{{\rm e}^{\frac{\tau
z^{2}}{2}}}{\prod_{k=1}^{N}\Gamma(z-a_{k})}.$ (6.3)

It is easy to check that $W_{N}$ satisfies Assumption 1.1, because of the
decay of ${\rm e}^{\frac{\tau z^{2}}{2}}$ as $\mathrm{Im}\,z\to\pm\infty$.

#### Associated biorthogonal measures.

In the same fashion as for the Log Gamma polymer, we can combine the identity
(6.2) with part 4 of Theorem 1.7 to relate the partition function of the
O’Connell-Yor polymer to an explicit biorthogonal measure. In the homogeneous
case, the following corollary was already established by Imamura and Sasamoto
in [36, Theorem 2].

######  Corollary 6.1.

Consider a set of real parameters $\vec{a}=(a_{i})_{i=1}^{N}$ with
$|a_{i}|<\delta_{1}<1/2$ for all $i=1,\ldots,N$, and take $\delta_{2}\in(0,1)$
such that $\delta_{2}>2\delta_{1}$. Let $W_{N}$ be as in (6.3), ${\Sigma_{N}}$
the positively oriented circle of radius $\delta_{1}$ around $0$, and
$\ell_{N}$ the vertical line with real part $\delta_{2}$. Then:

  1. 1.

$\mathbb{E}\left[{\rm e}^{-{\rm e}^{t}Z_{N}^{{\rm
OY}}({\vec{a}},\tau)}\right]=\mu_{N}[\sigma_{t}],\qquad\sigma_{t}(x)=\frac{1}{1+{\rm
e}^{-x-t}},$ (6.4)

where the measure $\mu_{N}$ is given by equations (1.7)–(1.8).

  2. 2.

$\frac{{\rm d}}{{\rm d}t}\log\mathbb{E}\left[{\rm e}^{-{\rm e}^{t}Z_{N}^{{\rm
OY}}({\vec{a}},\tau)}\right]=\int_{\mathbb{R}}\sigma_{t}(x)\kappa_{N,t}(x){\rm
d}x,$ (6.5)

where $\kappa_{N,t}$ is the one-point function (1.35) of the deformed
biorthogonal measure (1.33).

  3. 3.

In the case where the parameters $\vec{a}$ are all distinct, the biorthogonal
measure $\mu_{N}$ is given explicitly as

${\rm d}\mu_{N}^{{\rm OY}}(\vec{x};\vec{a},\tau)=\frac{1}{Z_{N}}\det\left({\rm
e}^{a_{m}x_{k}}\right)_{k,m=1}^{N}\det\left(\frac{1}{2\pi\mathrm{i}}\int_{\frac{1}{2}+\mathrm{i}\mathbb{R}}\frac{{\rm
e}^{\tau\frac{t^{2}}{2}}{\rm
e}^{-tx_{k}}}{(t-a_{m})\prod_{k=1}^{N}\Gamma(t-a_{k})}{\rm
d}t\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm d}x_{k}.$

In the confluent case $a_{1}=\cdots=a_{N}=a$, it is given by

${\rm d}\mu_{N}^{{\rm
OY}}(\vec{x};\vec{a},\tau)=\frac{1}{Z_{N}}\det\left(x_{k}^{m-1}\right)_{k,m=1}^{N}\det\left(\frac{1}{2\pi
i}\int_{\frac{1}{2}+\mathrm{i}\mathbb{R}}\frac{{\rm e}^{-x_{k}(t-a)}{\rm
e}^{\tau\frac{t^{2}}{2}}}{\Gamma(t-a)^{N}(t-a)^{N-m+1}}{\rm
d}t\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm d}x_{k}.$

#### GUE with external source.

The $N\times N$ GUE with external source can be defined as

$\frac{1}{Z_{N}}{\rm e}^{-{\rm Tr}\,\left(\tau M^{2}/2-AM\right)}{\rm
d}M,\quad{\rm d}M=\prod_{j=1}^{N}{\rm d}M_{jj}\ \prod_{1\leq j<k\leq N}{\rm
d}\mathrm{Re}\,M_{jk}\,{\rm d}\mathrm{Im}\,M_{jk},$

where $A$ is a deterministic Hermitian $N\times N$ matrix, and $\tau>0$. For
$A=0$ and $\tau=1$, this is the classical GUE. In general, the random matrix
$M$ is the sum of a GUE matrix with the deterministic matrix $A$, see [19, 20,
40, 8]. The eigenvalue distribution is given by

$\frac{1}{Z_{N}}\Delta(\vec{x})\ \det\left({\rm e}^{-\frac{\tau
x_{j}^{2}}{2}+a_{k}x_{j}}\right)_{j,k=1}^{N}\prod_{j=1}^{N}{\rm d}x_{j},\qquad
x_{1},\ldots,x_{N}\in\mathbb{R},$ (6.6)

if $a_{1},\ldots,a_{N}$ are distinct, and is explicit in confluent cases as
well. For general $a_{1},\ldots,a_{N}$, the eigenvalue distribution is a
biorthogonal ensemble with kernel $L_{N}$ of the form (1.8), with $W_{N}$
given by

$W_{N}^{\rm GUE+}(z;{\vec{a}},{\tau})=\prod_{m=1}^{N}(z-a_{m})\ {\rm
e}^{{\tau}z^{2}/2}.$ (6.7)

see e.g. [30]. We will denote the associated kernel, defined by (1.8), as
$L_{N}^{\rm GUE+}$. Biorthogonal systems associated to this measure can be
constructed in terms of multiple Hermite polynomials, see [8].

####  $T\to 0$ limit.

Consider the O’Connell-Yor polymer partition function with rescaled parameters
tau parameter $\tau/T^{2}$ and with re-scaled $a_{m}$-parameters
$Ta_{1},\ldots,Ta_{N}$. Then, the kernel of the associated biorthogonal
measure $L_{N}$ is given by

$L_{N}^{\rm
OY}\left(x,x^{\prime};T\vec{a},\tau/T^{2}\right)=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\Sigma_{N}}{\rm
d}u\int_{\ell_{N}}{\rm d}v\frac{W_{N}^{\rm
OY}(v;T\vec{a},\tau/T^{2})}{W_{N}^{\rm OY}(u;T{\vec{a}},\tau/T^{2})}\frac{{\rm
e}^{-vx+ux^{\prime}}}{v-u},$

or equivalently by

$\frac{1}{T}L_{N}^{\rm
OY}(x/T,x^{\prime}/T;T\vec{a},\tau/T^{2})=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\widetilde{\Sigma}_{N}}{\rm
d}s\int_{\widetilde{\ell}_{N}}{\rm d}t\frac{\widetilde{W}_{N}^{\rm
OY}(t;{\vec{a}},\tau)}{\widetilde{W}_{N}^{\rm OY}(s;{\vec{a}},\tau)}\frac{{\rm
e}^{-tx+sx^{\prime}}}{t-s},$

with

$\widetilde{W}_{N}^{\rm OY}(z;\vec{a},\tau)=\frac{{\rm e}^{\tau
z^{2}/2}}{\prod_{k=1}^{N}\Gamma(T(z-a_{k}))}=\frac{T^{N}\prod_{k=1}^{N}(z-a_{k})}{\prod_{k=1}^{N}\Gamma((1+T(z-a_{k})))}{\rm
e}^{\tau z^{2}/2},$

and where

$\tilde{\ell}_{N}:=T^{-1}\ell_{N},\quad\widetilde{\Sigma}_{N}:=T^{-1}\Sigma_{N}.$

It is easy to see that

$\lim_{\tau\to\infty}\frac{\widetilde{W}_{N}^{\rm
OY}(v;{\vec{a}},\tau)}{\widetilde{W}_{N}^{\rm
OY}(u;{\vec{a}},\tau)}=\frac{W_{N}^{\rm GUE+}(v;{\vec{a}},\tau)}{W_{N}^{\rm
GUE+}(u;{\vec{a}},\tau)},$

where $W_{N}^{\rm GUE+}$ is defined in equation (6.7) above. We can now prove
the following results similarly as for the Log Gamma polymer, with the
simplification that the integrands can be dominated more easily thanks to the
Gaussian factors.

######  Proposition 6.2.

We have the point-wise limit

$\lim_{T\to 0}T^{-1}L_{N}^{{\rm
OY}}\left(x/T,x^{\prime}/T;T{\vec{a}},\tau/T^{2}\right)=L_{N}^{\rm
GUE+}\left(x,x^{\prime};\vec{a},\tau\right),\qquad x,x^{\prime}\in\mathbb{R},$
(6.8)

and there exists $C_{N}(\vec{a})$ such that for $\epsilon>0$ sufficiently
small and $T>0$ sufficiently small,

$\left|T^{-1}L_{N}^{{\rm
OY}}\left(x/T,x^{\prime}/T;T{\vec{a}},\tau/T^{2}\right)\right|\leq
C_{N}(\vec{a})h(x)\tilde{h}(x^{\prime}),$ (6.9)

with $h,\tilde{h}$ as in (5.18).

######  Corollary 6.3.

As $T\to 0$, we have the convergence

$\mathbb{E}\left[{\rm e}^{-{\rm e}^{t/T}Z_{N}^{{\rm
OY}}(T{\vec{a}},\tau/T^{2})}\right]\longrightarrow\mathbb{P}_{\mathrm{GUE+}}\left(\max\\{x_{1},\ldots,x_{N}\\}\leq-t;\vec{a},\tau\right),$

where the expectation on the left is with respect to the distribution of the
O’Connell-Yor polymer partition function, and the probability on the right is
with respect to the GUE with external source, equation (6.6).

######  Remark 6.4.

Like for the Log Gamma polymer, this result should not come as a surprise. The
zero temperature version of the O’Connell-Yor polymer is a model of Brownian
queues which can be characterized by the GUE [7, 35]. The first identity in
(6.4) is the finite temperature analogue of these characterizations, in which
the role of the GUE (with external source) is played by the biorthogonal
measure $\mu_{N}$.

##  7 Mixed polymer partition function

#### The model.

The mixed polymer, or O’Connell-Yor polymer with boundary sources, is a
mixture of the Log Gamma and the O’Connell-Yor polymer. It was introduced by
Borodin, Corwin, Ferrari, and Vető in [13]. Given parameters
$n,N\in\mathbb{N}$, $\tau>0$, $\alpha_{1},\ldots,\alpha_{n}$, and
$a_{1},\ldots,a_{N}$ with $\alpha_{j}-a_{k}>0$ for every $j=1,\ldots,n$ and
$k=1,\ldots,N$, the mixed polymer partition function is defined as

$Z_{n,N}^{\rm Mixed}(\vec{\alpha},\vec{a},\tau)=\sum_{k=1}^{N}Z_{k,N}^{\rm
Log\Gamma}(\vec{\alpha},a_{1},\ldots,a_{k})Z_{N-k}^{\rm
OY}(a_{k+1},\ldots,a_{N},\tau).$ (7.1)

It can be interpreted as the partition function for up-right paths consisting
of a discrete (Log Gamma polymer type) path from $(1,1)$ to a point $(N,k)$,
concatenated with a semi-discrete (O’Connell-Yor polymer type) path from
$(N,k)$ to $(N+\tau,n)$, where $N,n,\tau$ are given, and $k$ depends on the
chosen path, see Figure 3.

#### Fredholm determinant identity.

As far as we know, a Laplace transform formula of the form (5.3) or (6.2) has
not appeared in the literature for the general mixed polymer partition
function, although there are Fredholm determinant identities for the Laplace
transform that look alike [13], but with the kernel defined through
integration over an unbounded contour instead of ${\Sigma_{N}}$ (recall Remark
1.9). It is not clear whether such unbounded contour integral Fredholm
determinant expressions can be related directly to our class of biorthogonal
measures. There is however another general Fredholm determinant identity
obtained by Imamura and Sasamoto in [37, Proposition 4.5]:

$\mathbb{E}\left[{\rm e}^{-{\rm e}^{t}Z_{N,N}^{\rm
Mixed}({\vec{\alpha}},{\vec{a}},\tau)}\right]=\det(1-\sigma_{t}M)_{L^{2}(\mathbb{R})},$
(7.2)

where $\sigma_{t}(x)=\frac{1}{1+{\rm e}^{-x-t}}$, $M$ is given by

$M(x,x^{\prime})=\sum_{k=1}^{N}\Phi_{k}(x)\Psi_{k}(x^{\prime}),$ (7.3)

and

$\displaystyle\Phi_{k}(x)=\frac{1}{2\pi\mathrm{i}}\int_{{\Sigma_{N}}}{\rm
d}u\frac{{\rm e}^{ux-\tau
u^{2}/2}}{u-a_{k}}\prod_{j=1}^{k-1}\frac{u-\alpha_{j}}{u-a_{j}}\prod_{j=1}^{N}\frac{\Gamma(1+u-a_{j})}{\Gamma(1+\alpha_{j}-u)},$
(7.4)
$\displaystyle\Psi_{k}(x)=\frac{\alpha_{k}-a_{k}}{2\pi}\int_{\mathbb{R}}{\rm
d}w\frac{{\rm e}^{-\mathrm{i}wx-\tau
w^{2}/2}}{\alpha_{k}-\mathrm{i}w}\prod_{j=1}^{k-1}\frac{\mathrm{i}w-a_{j}}{\mathrm{i}w-\alpha_{j}}\prod_{j=1}^{N}\frac{\Gamma(1+\alpha_{j}-\mathrm{i}w)}{\Gamma(1+\mathrm{i}w-a_{j})}.$
(7.5)

For the above identity to hold, ${\Sigma_{N}}$ has to enclose all the points
$a_{1},\ldots,a_{N}$ but none of the poles of
$\prod_{m=1}^{N}\Gamma(1+u-a_{m})$. Note that this is only possible when there
are no $a_{k}$-values which differ by an integer. Below, we assume that this
is the case. We mention however that our final results Corollary 7.3,
Proposition 7.5, and Corollary 7.6 hold without this assumption. This can be
seen by a continuity argument.

$(1,1)$$(\tau+N,N)$$(s_{3},3)$$(s_{3},4)$$(s_{4},4)$$(s_{4},5)$$(s_{5},5)$$(s_{5},6)$$\alpha_{1}$$\alpha_{2}$$\alpha_{3}$$\alpha_{4}$$\alpha_{5}$$\alpha_{6}$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$
Figure 3: An up-right path in the mixed polymer model, corresponding to
$n=N=6$, $k=3$. The values $\alpha_{j},a_{k}$ determine the weight of the
point $(j,k)$ in the discrete part of the lattice, while the values $a_{k}$
determine the weight of jump points $(s_{k},k)$ in the lattice.

#### Associated biorthogonal measures.

We will show, starting from equation (7.2), that the mixed polymer partition
function is also related to an explicit biorthogonal measure of the form
(1.14). We start with the following

######  Lemma 7.1.

Let $N\in\mathbb{N},\\{a_{1},\ldots,a_{N}\\}$ and
$\\{\alpha_{1},\ldots,\alpha_{N}\\}$ two sets of real parameters such that
$\alpha_{j}-a_{k}>0$ for all $j,k=1,\cdots,N$. Then the two families of
functions $\\{\Phi_{k},\;k=1,\ldots,N\\}$ and $\\{\Psi_{k},\;k=1,\ldots N\\}$,
defined in (7.4)–(7.5), are biorthogonal:

$\int_{\mathbb{R}}\Phi_{k}(x)\Psi_{m}(x){\rm d}x=\delta_{km},\qquad
k,m=1,\ldots,N.$

###### Proof.

We first observe that $\Psi_{m}(x)$ is a Fourier transform:

$\Psi_{m}(x)=\frac{\alpha_{m}-a_{m}}{2\pi}\mathcal{F}[g_{m}]\left(\frac{x}{2\pi}\right),\quad
g_{m}(w)=\frac{{\rm e}^{-\tau
w^{2}/2}}{\alpha_{m}-\mathrm{i}w}\prod_{j=1}^{m-1}\frac{\mathrm{i}w-a_{j}}{\mathrm{i}w-\alpha_{j}}\prod_{j=1}^{N}\frac{\Gamma(1+\alpha_{j}-\mathrm{i}w)}{\Gamma(1+\mathrm{i}w-a_{j})},$

where

$\mathcal{F}[g](t):=\int_{\mathbb{R}}g(w){\rm e}^{-2\pi\mathrm{i}wt}{\rm
d}t,\quad\mathcal{F}^{-1}[G](w)=\int_{\mathbb{R}}G(t){\rm
e}^{2\pi\mathrm{i}wt}{\rm d}t.$

It follows that

$\displaystyle\int_{\mathbb{R}}\Phi_{k}(x)\Psi_{m}(x){\rm d}x$
$\displaystyle=-\frac{\alpha_{m}-a_{m}}{4\pi^{2}\mathrm{i}}\int_{{\Sigma_{N}}}{\rm
d}u\frac{1}{(u-a_{k})(u-\alpha_{k})g_{k}(-\mathrm{i}u)}\int_{\mathbb{R}}{\rm
d}x{\rm e}^{ux}\mathcal{F}[g_{m}]\left(\frac{x}{2\pi}\right){\rm d}x$
$\displaystyle=-\frac{\alpha_{m}-a_{m}}{2\pi\mathrm{i}}\int_{{\Sigma_{N}}}{\rm
d}u\frac{1}{(u-a_{k})(u-\alpha_{k})g_{k}(-\mathrm{i}u)}\int_{\mathbb{R}}{\rm
d}s{\rm e}^{2\pi su}\mathcal{F}[g_{m}]\left(s\right){\rm d}s$
$\displaystyle=-\frac{\alpha_{m}-a_{m}}{2\pi\mathrm{i}}\int_{{\Sigma_{N}}}{\rm
d}u\frac{1}{(u-a_{k})(u-\alpha_{k})g_{k}(-\mathrm{i}u)}\left(\mathcal{F}^{-1}\circ\mathcal{F}\right){[g_{m}]}\left(-\mathrm{i}u\right)$
$\displaystyle=-\frac{\alpha_{m}-a_{m}}{2\pi\mathrm{i}}\int_{{\Sigma_{N}}}{\rm
d}u\frac{1}{(u-a_{k})(u-\alpha_{k})}\frac{g_{m}\left(-\mathrm{i}u\right)}{g_{k}(-\mathrm{i}u)}.$

If $m=k$, this is equal to
$-\frac{\alpha_{m}-a_{m}}{2\pi\mathrm{i}}\int_{{\Sigma_{N}}}{\rm
d}u\frac{1}{(u-a_{k})(u-\alpha_{k})}=1$. If $m>k$, the integrand is analytic
such that the integrand is $0$. If $m<k$, the integrand is a rational function
without zeros in the exterior of ${\Sigma_{N}}$ and decaying like $O(1/u^{2})$
as $u\to\infty$, such that we can deform ${\Sigma_{N}}$ to a large circle to
see that the integral is $0$. ∎

Using the techniques from Section 3.1, we can then conclude that the kernel
$M$ defined in (7.3) is a kernel satisfying the reproducing property, which
defines the $N$-point biorthogonal measure

${\rm
d}\widetilde{\mu}_{N}(x_{1},\ldots,x_{N}):=\frac{1}{N!}\det\left(M(x_{m},x_{k})\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm
d}x_{k},$ (7.6)

which can be equivalently rewritten as

${\rm
d}\widetilde{\mu}_{N}(\vec{x})=\frac{1}{N!}\det\left(\Phi_{j}(x_{m})\right)_{m,j=1}^{N}\det\left(\Psi_{j}(x_{k})\right)_{j,k=1}^{N}\prod_{k=1}^{N}{\rm
d}x_{k}.$ (7.7)

We will now show that the above biorthogonal measure is of the form (1.7).

######  Proposition 7.2.

Let $N\in\mathbb{N},\\{a_{1},\ldots,a_{N}\\}$ and
$\\{\alpha_{1},\ldots,\alpha_{N}\\}$ two sets of real parameters such that
$\alpha_{j}-a_{k}>0$ for all $j,k=1,\cdots,N$. The measure (7.7) is equal to
(1.7) and (2.1), with ${\Sigma_{N}}$ in the left half plane and enclosing
$a_{1},\ldots,a_{N}$ but none of the poles of
$\prod_{j=1}^{N}\Gamma(1+v-a_{j})$, $\ell_{N}$ a vertical line at the right of
$\Sigma_{N}$, and $W_{N}$ given by

$W_{N}(z)\equiv W_{N}^{\rm Mixed}(z;\vec{\alpha},\vec{a},\tau):={\rm e}^{\tau
z^{2}/2}\frac{\prod_{j=1}^{N}\Gamma(\alpha_{j}-z)}{\prod_{k=1}^{N}\Gamma(z-a_{k})}.$
(7.8)

###### Proof.

To show that $\widetilde{\mu}_{N}$ is equal to $\mu_{N}$ with $W_{N}$ given by
(7.8), it is sufficient to prove that these measures are equal in case
$a_{1},\ldots,a_{N}$ are distinct, by continuity. In that case, we can easily
evaluate the functions $\Phi_{1},\ldots,\Phi_{N}$ using the residue theorem,
which implies that they have the same linear span as ${\rm
e}^{a_{1}x},\ldots,{\rm e}^{a_{N}x}$. Similarly, we show that
$\Psi_{1},\ldots,\Psi_{N}$ have the same linear span as $\psi_{1}({\rm
e}^{x}),\ldots,\psi_{N}({\rm e}^{x})$ defined in (1.12): we have

$\displaystyle\Psi_{k}(x)$
$\displaystyle=\frac{\alpha_{k}-a_{k}}{2\pi\mathrm{i}}\int_{\mathbb{R}}{\rm
d}w{\rm e}^{-\mathrm{i}wx-\tau
w^{2}/2}\frac{\prod_{j=1}^{k-1}(\mathrm{i}w-a_{j})}{\prod_{j=1}^{k}{(\mathrm{i}w-\alpha_{j})}}\prod_{j=1}^{N}\frac{\Gamma(1+\alpha_{j}-\mathrm{i}w)}{\Gamma(1+\mathrm{i}w-a_{j})}$
$\displaystyle=\pm\frac{\alpha_{k}-a_{k}}{2\pi\mathrm{i}}\int_{\mathbb{R}}{\rm
d}w{\rm e}^{-\mathrm{i}wx-\tau
w^{2}/2}\frac{\prod_{j=1}^{k-1}(\mathrm{i}w-a_{j})\
\prod_{j=k+1}^{N}(\mathrm{i}w-\alpha_{j})}{\prod_{j=1}^{N}{(\mathrm{i}w-\alpha_{j})}\
}\prod_{j=1}^{N}\frac{\Gamma(\alpha_{j}-\mathrm{i}w)}{\Gamma(\mathrm{i}w-a_{j})}$
$\displaystyle=\pm\frac{\alpha_{k}-a_{k}}{2\pi}\int_{\ell_{N}}{\rm d}u{\rm
e}^{-ux+\tau
u^{2}/2}\frac{\prod_{j=1}^{k-1}(u-a_{j})\prod_{j=k+1}^{N}{(u-\alpha_{j})}}{\prod_{j=1}^{N}{(u-a_{j})}}\prod_{j=1}^{N}\frac{\Gamma(\alpha_{j}-u)}{\Gamma(u-a_{j})}$
$\displaystyle=\pm\frac{\alpha_{k}-a_{k}}{2\pi}\int_{\ell_{N}}{\rm d}u{\rm
e}^{-ux}\frac{\prod_{j=1}^{k-1}(u-a_{j})\prod_{j=k+1}^{N}{(u-\alpha_{j})}}{\prod_{j=1}^{N}{(u-a_{j})}}W_{N}(u).$

On the other hand, we already know that $\psi_{1}({\rm
e}^{x}),\ldots,\psi_{N}({\rm e}^{x})$ span every function of the form

$\int_{\ell_{N}}{\rm d}w{\rm
e}^{-ux}\frac{P_{N-1}(u)}{\prod_{j=1}^{N}{(u-a_{j})}}W_{N}(u),$

with $P_{N-1}$ a polynomial of degree $\leq N-1$. We conclude that
$\Psi_{1}(x),\ldots,\Psi_{N}(x)$ and $\psi_{1}({\rm
e}^{x}),\ldots,\psi_{N}({\rm e}^{x})$ have the same linear span, such that
(7.7) can be rewritten as

${\rm d}\widetilde{\mu}_{N}(\vec{x}):=\frac{1}{N!}\det\left({\rm
e}^{a_{j}x_{m}}\right)_{m,j=1}^{N}\det\left(\psi_{j}({\rm
e}^{x_{k}})\right)_{j,k=1}^{N}\prod_{k=1}^{N}{\rm d}x_{k},$

which is equal to ${\rm d}\mu_{N}(\vec{x})$ by (1.14). The result if proved. ∎

Combining the above results with Theorem 1.10, we obtain the following
characterization of the mixed polymer partition function in terms of the
biorthogonal measures $\mu_{N}$ and $\nu_{N,t}$.

######  Corollary 7.3.

Let $N\in\mathbb{N}$, and let $\\{a_{1},\ldots,a_{N}\\}$,
$\\{\alpha_{1},\ldots,\alpha_{N}\\}$ two sets of real parameters such that
$\alpha_{j}-a_{k}>0$ for all $j,k=1,\cdots,N$. Then:

  1. 1.

$\mathbb{E}\left[{\rm e}^{-{\rm e}^{t}Z_{N,N}^{{\rm
Mixed}}({\vec{\alpha}},{\vec{a}},\tau)}\right]=\mu_{N}[\sigma_{t}],\qquad\sigma_{t}(x)=\frac{1}{1+{\rm
e}^{-x-t}},$ (7.9)

where the measure $\mu_{N}$ is given by equations (1.7), (1.8) with $W_{N}$
defined by (7.8), with $\Sigma_{N}$ a loop in the left half plane enclosing
$a_{1},\ldots,a_{N}$ but none of the poles of
$\prod_{j=1}^{N}\Gamma(1+v-a_{j})$, and $\ell_{N}$ a vertical line at the
right of $\Sigma_{N}$.

  2. 2.

$\frac{{\rm d}}{{\rm d}t}\log\mathbb{E}\left[{\rm e}^{-{\rm
e}^{t}Z_{N,N}^{{\rm
Mixed}}({\vec{\alpha}},{\vec{a}},\tau)}\right]=\int_{\mathbb{R}}\sigma_{t}(x)\kappa_{N,t}(x){\rm
d}x,$ (7.10)

where $\kappa_{N,t}$ is the one-point function (1.35) of the deformed
biorthogonal measure (1.33).

  3. 3.

In the case where the parameters $\vec{a}$ are all distinct, the biorthogonal
measure $\mu_{N}$ is given explicitly as

${\rm d}\mu_{N}^{{\rm
Mixed}}(\vec{x};\vec{\alpha},\vec{a},\tau)=\frac{1}{Z_{N}}\det\left({\rm
e}^{a_{m}x_{k}}\right)_{k,m=1}^{N}\det\left(\frac{1}{2\pi\mathrm{i}}\int_{\frac{1}{2}+\mathrm{i}\mathbb{R}}\frac{{\rm
e}^{\tau\frac{t^{2}}{2}}\prod_{j=1}^{n}\Gamma(\alpha_{j}-t)}{(t-a_{m})\prod_{k=1}^{N}\Gamma(t-a_{k})}{{\rm
e}}^{-tx_{k}}{\rm d}t\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm d}x_{k}.$

In the confluent case $a_{1}=\cdots=a_{N}=a$, it is given by

${\rm d}\mu_{N}^{{\rm
Mixed}}(\vec{x};\vec{\alpha},\vec{a},\tau)=\frac{1}{Z_{N}}\det\left(x_{k}^{m-1}\right)_{k,m=1}^{N}\det\left(\frac{1}{2\pi
i}\int_{\frac{1}{2}+\mathrm{i}\mathbb{R}}\frac{{\rm
e}^{\tau\frac{t^{2}}{2}}\Gamma(\alpha-t)^{n}{{\rm
e}}^{-(t-a)x_{k}}}{(t-a)^{N-m+1}\Gamma(t-a)^{N}}\mathrm{d}t\right)_{m,k=1}^{N}\prod_{k=1}^{N}{\rm
d}x_{k}.$

######  Remark 7.4.

Corollary 7.3 contains both Corollary 5.2 with $n=N$ and Corollary 6.1 as
degenerate cases: the Log Gamma polymer corresponds to $\tau=0$, while the
O’Connell-Yor polymer corresponds to $\alpha_{1}=\ldots=\alpha_{N}\to\infty$.
As already mentioned, the above identities are finite temperature
generalizations of classical characterizations of last passage percolation and
Brownian queues models in terms of the GUE and LUE with external sources. The
biorthogonal measures $\mu_{N}$ replace the GUE and LUE with external source
in the corresponding finite temperature models. The identity (7.9) can also be
seen as a finite $N$ version of the characterization from [3] of the narrow
wedge solution of the KPZ equation (or continuum directed polymer) in terms of
the Airy point process. In this sense, the biorthogonal measures $\mu_{N}$
play the same role in (semi-)discrete directed polymer models as the Airy
point process in the continuum directed polymer model.

#### Sum of LUE with external source with GUE.

Consider an $N\times N$ matrix $M$ from the LUE with external source, with
distribution (5.11) and $\nu=0$, and consider an $N\times N$ GUE matrix $H$,
with distribution (6.6) with $A=0$. Set $Q=M+\sqrt{\tau N}H$. The eigenvalue
distribution of $Q$ can be computed using [24, Theorem 2.3 (b)], or more
directly using the re-scaled variant of this identity given in [22, Equation
(1.6)]. This result gives an integral expression for the correlation kernel of
the eigenvalues of $Q$ in terms of the eigenvalue correlation of $M$, valid
for any random matrix $M$ whose eigenvalue distribution is a polynomial
ensemble, and thus in particular for our LUE with external source random
matrix $M$. It states that the eigenvalue correlation kernel
$K_{N}(x,x^{\prime})$ of $Q$ is given by

$K_{N}(x,x^{\prime})=\frac{1}{2\pi\mathrm{i}\tau}\int_{\mathrm{i}\mathbb{R}}{\rm
d}s\int_{\mathbb{R}}{\rm d}t\;k_{N}(s,t){\rm
e}^{\frac{1}{2\tau}\left((x-s)^{2}-(x^{\prime}-t)^{2}\right)},$ (7.11)

where

$k_{N}(s,t)=\sum_{k=1}^{N}p_{k}(s)q_{k}(t),$

with $p_{k}$ a polynomial of degree $k-1$, $q_{1},\ldots,q_{N}$ in the linear
span of the functions $g_{1},\ldots,g_{N}$ from (1.1), and they are
biorthogonal:

$\int_{\mathbb{R}}p_{j}(s)q_{k}(s){\rm d}s=\delta_{jk},\qquad j,k=1,\ldots,N.$

In our situation, we can take $k_{N}(s,t)$ of the form (1.8) with
$W_{N}=W_{N}^{\rm LUE+}$ as in (5.14) with $\nu=0$. Then $K_{N}(x,x^{\prime})$
is given by

$\displaystyle K_{N}(x,x^{\prime})$
$\displaystyle=\frac{1}{2\pi\mathrm{i}\tau}\int_{\mathrm{i}\mathbb{R}}{\rm
d}s\int_{\mathbb{R}}{\rm d}t\;L_{N}^{{\rm LUE+}}(s,t{;\vec{a},0}){\rm
e}^{\frac{1}{2\tau}\left((x-s)^{2}-(x^{\prime}-t)^{2}\right)}$
$\displaystyle=\frac{1}{(2\pi\mathrm{i})^{3}\tau}\int_{\mathrm{i}\mathbb{R}}{\rm
d}s\int_{\mathbb{R}}{\rm d}t\int_{\Sigma_{N}}{\rm d}u\int_{\ell_{N}}{\rm
d}v\frac{W_{N}^{{\rm LUE+}}(v{;\vec{a},0})}{W_{N}^{{\rm
LUE+}}(u{;\vec{a},0})}\frac{{\rm e}^{-vs+ut}}{v-u}{\rm
e}^{\frac{1}{2\tau}\left((x-s)^{2}-(x^{\prime}-t)^{2}\right)}.$

Changing the order of integration, we obtain after a straightforward
computation that

$\displaystyle K_{N}(x,x^{\prime})$
$\displaystyle=\frac{1}{(2\pi\mathrm{i})^{3}\tau}\int_{\Sigma_{N}}{\rm
d}u\int_{\ell_{N}}{\rm d}v\frac{W_{N}^{{\rm LUE+}}(v{;\vec{a},0})}{W_{N}^{{\rm
LUE+}}(u{;\vec{a},0})}\frac{{\rm
e}^{\frac{\tau}{2}v^{2}-\frac{\tau}{2}u^{2}+ux^{\prime}-vx}}{v-u}$
$\displaystyle\qquad\qquad\qquad\times\ \left(\int_{\mathrm{i}\mathbb{R}}{\rm
d}s\;{\rm e}^{\frac{1}{2\tau}\left(s-(x+\tau
v)\right)^{2}}\right)\left(\int_{\mathbb{R}}{\rm d}t\;{\rm
e}^{-\frac{1}{2\tau}\left(t-(x^{\prime}+\tau v)\right)^{2}}\right)$
$\displaystyle=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\Sigma_{N}}{\rm
d}u\int_{\ell_{N}}{\rm d}v\frac{W_{N}^{{\rm LUE+}}(v{;\vec{a},0})}{W_{N}^{{\rm
LUE+}}(u{;\vec{a},0})}\frac{{\rm
e}^{\frac{\tau}{2}v^{2}-\frac{\tau}{2}u^{2}+ux^{\prime}-vx}}{v-u}.$

This is again of the form (1.8), but now with transformed function $W$ given
by where $W_{N}^{\rm GLUE+}$ is defined by

$W_{N}^{\rm GLUE+}(v;\vec{a},\tau):=\frac{{\rm e}^{\tau
z^{2}/2}\prod_{k=1}^{N}(z-a_{k})}{(1-z)^{N}}.$ (7.12)

We will now point out that this biorthogonal measure corresponds to the small
temperature limit of the biorthogonal measure associated to the mixed polymer.

#### Small temperature limit for the mixed polymer.

Consider the mixed polymer partition function with re-scaled
$\tau=\tau/T^{2}$, re-scaled $a_{m}$-parameters $Ta_{1},\ldots,Ta_{N}$, and
$\alpha_{1}=\cdots=\alpha_{N}={T}$, in the limit where $T\to 0$. Then, the
kernel of the associated biorthogonal measure $L_{N}$ is given by

$L_{N}^{\rm
Mixed}\left(x,x^{\prime};\left[T\right]_{N},T\vec{a},\tau/T^{2}\right)=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\Sigma_{N}}{\rm
d}u\int_{\ell_{N}}{\rm d}v\frac{W_{N}^{\rm
Mixed}(v;\left[T\right]_{N},T\vec{a},\tau/T^{2})}{W_{N}^{\rm
Mixed}(u;\left[T\right]_{N},T{\vec{a}},\tau/T^{2})}\frac{{\rm
e}^{-vx+ux^{\prime}}}{v-u},$

or equivalently by

$\frac{1}{T}L_{N}^{\rm
Mixed}(\frac{x}{T},\frac{x^{\prime}}{T};\left[T\right]_{N},T\vec{a},\tau/T^{2})=\frac{1}{(2\pi\mathrm{i})^{2}}\int_{\widetilde{\Sigma}_{N}}{\rm
d}s\int_{\widetilde{\ell}_{N}}{\rm d}t\frac{\widetilde{W}_{N}^{\rm
Mixed}(t;[1]_{N},{\vec{a}},\tau)}{\widetilde{W}_{N}^{\rm
Mixed}(s;[1]_{N},{\vec{a}},\tau,T)}\frac{{\rm e}^{-tx+sx^{\prime}}}{t-s},$

with

$\widetilde{W}_{N}^{\rm Mixed}(z;[1]_{N},\vec{a},\tau)=\frac{{\rm e}^{\tau
z^{2}/2}\Gamma((1-z)T)^{N}}{\prod_{k=1}^{N}\Gamma((z-a_{k})T)}=\frac{\Gamma(1+(1-z)T)^{N}\prod_{k=1}^{N}(z-a_{k})}{(1-z)^{N}\prod_{k=1}^{N}\Gamma(1+(z-a_{k})T)}{\rm
e}^{\tau z^{2}/2},$

and where

$\tilde{\ell}_{N}:=\frac{1}{T}\ell_{N},\quad\widetilde{\Sigma}_{N}:=\frac{1}{T}\Sigma_{N}.$

It is easy to see that

$\lim_{T\to 0}\frac{\widetilde{W}_{N}^{\rm
Mixed}(v;[1]_{N},{\vec{a}},\tau)}{\widetilde{W}_{N}^{\rm
Mixed}(u;[1]_{N},{\vec{a}},\tau)}=\frac{W_{N}^{\rm
GLUE+}(v;{\vec{a}},\tau)}{W_{N}^{\rm GLUE+}(u;{\vec{a}},\tau)}.$

In analogy to the zero temperature results for the Log Gamma polymer and the
O’Connell-Yor polymer, we obtain the following.

######  Proposition 7.5.

We have the point-wise limit

$\lim_{T\to 0}\frac{1}{T}L_{N}^{{\rm
Mixed}}\left(\frac{x}{T},\frac{x^{\prime}}{T};\left[T\right]_{N},T{\vec{a}},\tau/T^{2}\right)=L_{N}^{\rm
GLUE+}\left(x,x^{\prime};\vec{a},\tau\right),\qquad
x,x^{\prime}\in\mathbb{R},$ (7.13)

and there exists $C_{N}(\vec{a},\tau)$ such that for $\epsilon>0$ sufficiently
small and $T>0$ sufficiently small,

$\left|\frac{1}{T}L_{N}^{{\rm
Mixed}}\left(\frac{x}{T},\frac{x^{\prime}}{T};\left[T\right]_{N},T{\vec{a}},\tau/T^{2}\right)\right|\leq
C_{N}(\vec{a},\tau)h(x)\tilde{h}(x^{\prime}),$ (7.14)

with $h,\tilde{h}$ as in (5.18).

######  Corollary 7.6.

As $T\to\infty$, we have the convergence

$\mathbb{E}\left[{\rm e}^{-{\rm e}^{t/T}Z_{N,N}^{{\rm
Mixed}}\left(\left[T\right]_{N},T{\vec{a}},\tau/T^{2}\right)}\right]\longrightarrow\mathbb{P}_{\mathrm{GLUE+}}\left(\max\\{x_{1},\ldots,x_{N}\\}\leq-t;\vec{a},\tau\right),$

where the expectation on the left is with respect to the distribution of the
mixed polymer partition function, and the probability on the right is with
respect to the distribution of the eigenvalues $x_{1},\ldots,x_{N}$ of the
random matrix $Q$ constructed above as a sum of a GUE with an LUE with
external source matrix.

######  Remark 7.7.

To the best of our knowledge, the appearance of the sum of an LUE with
external source matrix with a GUE matrix is completely new. One can also think
of the sum $Q=M+\sqrt{\tau N}H$ as the matrix obtained by applying independent
Brownian motions to the independent entries of $M$. The resulting process for
the eigenvalues of this process is called Dyson’s Brownian motion, see e.g.
[40]. In the zero temperature limit, $\tau$ thus plays the role of time in
Dyson’s Brownian motion applied to the eigenvalue configuration of a random
matrix from the LUE with external source. If the measures ${\rm d}\mu_{N}^{\rm
Mixed}(\vec{x};\vec{\alpha},{\vec{a}},\tau)$ are positive polynomial
ensembles, such an interpretation in terms of Dyson’s Brownian motion remains
valid for finite temperature: let $\tau^{\prime}>\tau$, and suppose that
$\vec{x}$ is a random configuration of points at time $\tau$; we then obtain a
random configuration of points at time $\tau^{\prime}$ by applying Dyson’s
Brownian motion to $\vec{x}$ for a time period $\tau^{\prime}-\tau$.

######  Remark 7.8.

There are other random matrix models whose eigenvalue distributions admit
kernels of the form (1.22), but then in exponential variables, and with the
special choice $a_{m}=m-1$. In this situation, the biorthogonal ensembles
become polynomial ensembles, recall Remark 1.6. This is the case for Wishart-
type products of Ginibre matrices [1, 2, 31, 32, 45, 46, 51], for Wishart-type
products of truncated unitary matrices [41], and for Laguerre Muttalib-Borodin
ensembles [11, 33, 46]. The different double contour integral representations
for the kernels in these models can be found for instance in [21, Appendix A]
or in [24]. We summarize the relevant choices of $W_{N}$ in these models in
Table 1.

Model | ${\bf W_{N}(z)}$ | ${\bf f_{m}(x)}$ | ${\bf g_{m}(x)}$
---|---|---|---
Log $\Gamma$ polymer | $\frac{\prod_{j=1}^{n}\Gamma(\alpha_{j}-z)}{\prod_{k=1}^{N}\Gamma(z-a_{k})}$ | ${\rm e}^{a_{m}x}$ | $G^{\,*,n}_{0,0}\lparen\begin{smallmatrix}n+N\\\ -\end{smallmatrix}|\,{\vec{\alpha}};\vec{1}+{\vec{a}}-\vec{e}_{m}\rparen{{\rm e}^{-x}}$
Homogeneous Log $\Gamma$ polymer | $\frac{\Gamma(\alpha-z)^{n}}{\Gamma(z)^{N}}$ | $x^{m-1}$ | $G^{\,*,n}_{0,0}\lparen\begin{smallmatrix}n+N\\\ -\end{smallmatrix}|\,{\vec{\alpha}};(0)_{N-m+1};(1)_{m-1}\rparen{{\rm e}^{-x}}$
OY polymer | ${\rm e}^{\tau z^{2}/2}\frac{1}{\prod_{k=1}^{N}\Gamma(z-a_{k})}$ | ${\rm e}^{a_{m}x}$ | $\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{\frac{1}{2}+\mathrm{i}\mathbb{R}}\frac{{\rm e}^{\tau\frac{t^{2}}{2}}{{\rm e}}^{-tx}}{(t-a_{m})\prod_{k=1}^{N}\Gamma(t-a_{k})}\mathrm{d}t$
Homogeneous OY polymer | ${\rm e}^{\tau z^{2}/2}\frac{1}{\Gamma(z-a)^{N}}$ | $x^{m-1}$ | $\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{\frac{1}{2}+\mathrm{i}\mathbb{R}}\frac{{\rm e}^{\tau\frac{t^{2}}{2}}{{\rm e}}^{-(t-a)x}}{(t-a)^{N-m+1}\Gamma(t-a)^{N}}\mathrm{d}t$
Mixed polymer | ${\rm e}^{\tau z^{2}/2}\frac{\prod_{j=1}^{n}\Gamma(\alpha_{j}-z)}{\prod_{k=1}^{N}\Gamma(z-a_{k})}$ | ${\rm e}^{a_{m}x}$ | $\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{\mathrm{i}\mathbb{R}}\frac{{\rm e}^{\tau\frac{t^{2}}{2}}\prod_{j=1}^{n}\Gamma(\alpha_{j}-t)}{(t-a_{m})\prod_{k=1}^{N}\Gamma(t-a_{k})}{{\rm e}}^{-tx}\mathrm{d}t$
Homogeneous Mixed polymer | ${\rm e}^{\tau z^{2}/2}\frac{\Gamma(\alpha-z)^{n}}{\Gamma(z-a)^{N}}$ | $x^{m-1}$ | $\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{\mathrm{i}\mathbb{R}}\frac{{\rm e}^{\tau\frac{t^{2}}{2}}\Gamma(\alpha-t)^{n}{{\rm e}}^{-(t-a)x}}{(t-a)^{N-m+1}\Gamma(t-a)^{N}}\mathrm{d}t$
GUE | $z^{N}{\rm e}^{\tau z^{2}/2}$ | $x^{m-1}$ | $x^{m-1}{\rm e}^{-\tau x^{2}/2}$
GUE + external source | $\prod_{m=1}^{N}(z-a_{m})\ {\rm e}^{\tau z^{2}/2}$ | ${x^{m-1}}$ | ${{{\rm e}}^{-(\tau x^{2}/2-a_{m}x)}}$
LUE | $\frac{z^{N}}{(z-1)^{N+\nu}}$ | $x^{m-1}$ | $x^{m-1+\nu}{\rm e}^{-x}$
LUE + external source | $\frac{\prod_{m=1}^{N}(z-a_{m})}{(z-1)^{N+\nu}}$ | $x^{m-1}$ | $x^{\nu}{{\rm e}}^{-(1-a_{m})x}$
(LUE + external source) + GUE | $\frac{{\rm e}^{\tau z^{2}/2}\prod_{k=1}^{N}(z-a_{k})}{(1-z)^{N}}$ | $x^{m-1}$ | $\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{\mathrm{i}\mathbb{R}}\frac{{\rm e}^{\tau\frac{t^{2}}{2}}\prod_{j=1}^{N}(t-a_{j})}{(t-a_{m})(t-1)^{N}}{{\rm e}}^{-tx}\mathrm{d}t$
Ginibre products | $\frac{\prod_{k=0}^{n}\Gamma(1+\nu_{k}-z)}{\Gamma(1-N-z)}$ | ${\rm e}^{(m-1)x}$ | $G^{\,*,n}_{0,0}\lparen\begin{smallmatrix}n\\\ -\end{smallmatrix}|\,\vec{\nu}\rparen{{\rm e}^{-x}}$
Muttalib-Borodin LUE | $\frac{\Gamma(1-z)\Gamma(\nu+1-\theta z)}{\Gamma(1-N-z)}$ | ${\rm e}^{(m-1)x}$ | ${\rm e}^{\theta(m-1)x}{\rm e}^{-{\rm e}^{x}}$
Truncated unitary products | $\frac{\prod_{k=0}^{n}\Gamma(1+\nu_{k}-z)}{\prod_{k=0}^{n}\Gamma(1+\ell_{k}-N-z)}$ | ${\rm e}^{(m-1)x}$ | $G^{\,*,n}_{0,n}\lparen\begin{smallmatrix}n\\\ \vec{\mu}+\vec{\nu}\end{smallmatrix}|\,\vec{\nu}\rparen{{\rm e}^{-x}}$
Table 1: List of polymer and random matrix models associated to biorthogonal
measures with symbol $F(z)$, and a possible choice, not necessarily
biorthogonal, for the corresponding functions $f_{m}(x)$ and $g_{m}(x)$ in
(1.1), for $m=1,\ldots,N$. The choices for $g_{m}$ in the GUE with external
source and LUE with external source are only valid if $a_{1},\ldots,a_{N}$ are
distinct. The last three models correspond to the case $a_{m}=m-1$, and the
eigenvalues or squared singular values in these models correspond to the
exponential variable $y={\rm e}^{x}$.

### Acknowledgements

We thank Ivan Corwin and Milind Hegde for valuable conversations on Fredholm
determinant representations of polymer partition functions, which constituted
the starting point of this article. We are also grateful to Jiyuan Zhang for
useful discussions. M.C. acknowledge support by the European Union Horizon
2020 research and innovation program under the Marie Skłodowska-Curie RISE
2017 grant agreement no. 778010 IPaDEGAN and by the IRP “PIICQ”, funded by the
CNRS. T.C. acknowledges support by FNRS Research Project T.0028.23, and by
FNRS under EOS project O013018F.

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