# Random Sorting Networks: edge limit

Vadim Gorin and Jiaming Xu

###### Abstract.

A sorting network is a shortest path from $12\dots n$ to $n\dots 21$ in the
Cayley graph of the symmetric group $\mathfrak{S}_{n}$ spanned by adjacent
transpositions. The paper computes the edge local limit of the uniformly
random sorting networks as $n\to\infty$. First, we find the asymptotic
distribution of the first occurrence of a given swap $(k,k+1)$ and identify it
with the law of the smallest positive eigenvalue of a $2k\times 2k$ aGUE (an
aGUE matrix has purely imaginary Gaussian entries that are independently
distributed subject to skew-symmetry). Second, we give two different formal
definitions of a spacing — the time distance between the occurrence of a given
swap $(k,k+1)$ in a uniformly random sorting network. Two definitions lead to
two different expressions for the asymptotic laws expressed in terms of
derivatives of Fredholm determinants.

##  1\. Introduction

###  1.1. Motivation

The main object in this article is the uniformly random sorting network which
we abbreviate as RSN. Let us start by giving basic definitions. Consider the
permutation group of $n$ elements $\mathfrak{S}_{n}$. We often use the one-row
notation for the elements of $\mathfrak{S}_{n}$ representing them as sequences
$(a_{1}\ a_{2}\ ...\ a_{n})$. We let $\tau_{k}$ be the adjacent swap
$(k,k+1)$, $k=1,2,\dots,n-1$. We have

$(a_{1}\ a_{2}\ \dots\ a_{n})\cdot\tau_{k}=(a_{1}\ a_{2}\ a_{k-1}\ a_{k+1}\
a_{k}\ a_{k+2}\ \dots\ a_{n}).$

A sorting network of size $n$ is a shortest sequence of swaps, whose product
equals the reverse permutation $(n\ n-1\ ...\ 2\ 1)$. By counting inversions
in permutations, one shows that the length $N$ of such shortest sequence is
$N=\binom{n}{2}$. Let $\Omega_{n}$ denote the set of all sorting networks of
size $n$. Thus, elements of $\Omega_{n}$ are sequences $(s_{1},\dots,s_{N})$
such that

$\tau_{s_{1}}\tau_{s_{2}}\cdots\tau_{s_{N}}=(n\ n-1\ ...\ 2\ 1).$

Sorting networks can be drawn as wiring diagrams, see Figure 1. Whenever
$s_{t}=k$, we say that swap $(k,k+1)$ occurs at time $t$; in the wiring
diagram this corresponds to two wires intersecting at the point
$(t,k+\tfrac{1}{2})$.

Figure 1. A sorting network $(s_{1}\dots s_{N})$, $N={n\choose 2}$ can be
represented as a diagram of $n$ wires, with wires at heights $k$ and $k+1$
being swapped at time $i$ whenever $s_{i}=k$. The figure shows the wiring
diagram of the sorting network $(2,1,3,2,4,3,4,1,2,1)$ with $n=5$. The blue
double arrow shows a spacing in row 3, which is a time interval between two
adjacent swaps $\tau_{3}$ and it has length 3 in our example.

The study of sorting networks is a fruitful topic, with the first mathematical
results going back to [S] where Stanley computed the number of elements in
$\Omega_{n}$:

$|\Omega_{n}|=\frac{{n\choose 2}!}{\prod_{j=1}^{n-1}(2n-1-2j)^{j}}.$

A recent point of interest is the study of uniformly random sorting networks
initiated by Angel, Holroyd, Romik, and Virag in [AHRV]. The probabilistic
results can be split into two groups: global and local. On the global side,
[AHRV] proved that the density of swaps approximates the semi-circle
law111There is no known direct connection to the Wigner semi-circle law of the
random matrix theory and the match seems coincidental. as $n\to\infty$; [AHRV]
predicted and [DVi, D] proved (among other results) that individual
trajectories in the wiring diagram become (random) sine curves; see also [K,
RVV] for related results.

Our paper belongs to the second group of results, [AGH, R, ADHV, GR], which
studies local limits. The most basic local characteristic of the sorting
network is _spacing_ in row $k$, which is a distance between two occurrences
of the same swap $s_{i}=k$ in the sorting network, cf. Figure 1. [R] and [GR]
discovered a link between the asymptotic laws of spacings as $n\to\infty$ and
random matrix theory. Namely, [R] matched the asymptotic law of the spacing in
row $1$ with the distance between two eigenvalues of $2\times 2$ matrix of
Gaussian Orthogonal Ensemble (GOE) of real symmetric random matrices. [GR]
dealt with spacings in row $\alpha n$ with $0<\alpha<1$ and showed that after
proper rescaling they converge to the _Gaudin–Mehta law_ , which is the
universal asymptotic law for spacings between eigenvalues of real symmetric
matrices of very large sizes.

Comparing the results of [R] and [GR], we see that the former links the
spacings in the extreme row (i.e. at the end-point of the edge) to $2\times 2$
real symmetric matrices, while the latter links the spacings in the bulk rows
to the real symmetric matrices of infinite sizes. The observed gap in the
asymptotics between $2\times 2$ and $\infty\times\infty$ matrices motivated
the central question of our paper: we would like to understand, whether
distributions related to $k\times k$ random matrices can be also found in the
asymptotic laws for random sorting networks.

The answer presented in this text is both Yes and No. On one side, we are so
far unable to find any connections of sorting networks to eigenvalues of real
symmetric $k\times k$ matrices (which would be the most direct interpolation
between [R] and [GR]). On the other hand, by slightly adjusting our point
view, we find that the law of the asymptotic spacing in row $k$ can be
expressed in terms of the eigenvalues of $2k\times 2k$ random _anti-symmetric_
Gaussian matrices — they are known in the literature as aGUE, since their
analysis reveals connections to the tools used for the Gaussian Unitary
Ensemble222However, up to multiplication by $\mathbf{i}$, matrix elements of
aGUE are real, rather than complex., see [M, Chapter 13] and [FN].

###  1.2. Main Results

Here is the precise definition of the random matrix object appearing in our
asymptotic results.

######  Definition 1.1.

Let $Y$ be an $\ell\times\ell$ matrix whose entries are i.i.d. standard
Gaussian random variables $N(0,1)$. Then

$M_{\ell}=\frac{\mathbf{i}}{2}\,(Y-Y^{T})$

is called $\ell\times\ell$ anti-symmetric GUE or aGUE for being short.

Note that the eigenvalues of $M_{\ell}$ are real and come in pairs: $\lambda$
is an eigenvalue if and only if so is $-\lambda$. When $\ell$ is odd,
$M_{\ell}$ is necessary degenerate, i.e. it has a zero eigenvalue; for even
$\ell$, zero is not an eigenvalue almost surely.

We also would like to deal with eigenvalues of all $M_{\ell}$ together.

######  Definition 1.2.

Let $M$ be an infinite aGUE, i.e, an $\infty\times\infty$ matrix whose
$l^{\text{th}}$ corner is a $l\times l$ aGUE. The aGUE–corners process is a
point process $\pazocal{X}$ on $\mathbb{Z}_{\geq 0}\times\mathbb{R}_{\geq 0}$:
we put a particle at $(l,u)$ whenever $u$ is one of the
$\lfloor\frac{l}{2}\rfloor$ positive eigenvalues of the top-left $l\times l$
corner of $M$.

######  Remark 1.3.

aGUE is studied in [FN], and it turns out that its corners process
$\pazocal{X}$ has a determinantal structure. We give the correlation kernel of
$\pazocal{X}$ in Theorem 2.10.

######  Definition 1.4.

$\mathbf{T}_{FS}(k)$ is the smallest positive eigenvalue of a $2k\times 2k$
aGUE.

The determinantal structure for the distribution of eigenvalues of $M_{2k}$
(see [M, Chapter 13] and [FN]) leads to an expression for the distribution of
$\mathbf{T}_{FS}(k)$ as a series, which can be identified with a Fredholm
determinant:

$\mathbb{P}\left[\mathbf{T}_{\rm{FS}}(k)>t\right]=1+\sum_{l=1}^{\infty}\frac{(-1)^{l}}{l!}\int_{0}^{t}\dots\int_{0}^{t}\det[K^{(k)}(u_{i},u_{j})]\,\mathrm{d}u_{1}\cdots\mathrm{d}u_{l},$

where $K^{(k)}$ is a correlation kernel, expressed through the Hermite333We
use the “physicist’s” normalization, so that $H_{i}(x)$, $i=0,1,\dots$, is a
monic polynomial of degree $i$ such that
$\int_{\mathbb{R}}H_{i}(x)H_{j}(x)e^{-x^{2}}dx=\delta_{ij}j!2^{j}\sqrt{\pi}$.
polynomials $H_{i}(x)$ as

(1.1)
$K^{(k)}(u_{1},u_{2})=e^{-\tfrac{1}{2}u_{1}^{2}-\tfrac{1}{2}u_{2}^{2}}\sum_{\ell=0}^{k-1}\frac{H_{2\ell}(u_{1})H_{2\ell}(u_{2})}{\sqrt{\pi}(2\ell)!2^{2\ell-1}}.$

Vadim: I do not want to replace
$e^{-\frac{1}{2}u_{1}^{2}-\frac{1}{2}u_{2}^{2}}$ in the kernel by
$e^{-u_{1}^{2}}$. The reason is that I would like to keep the kernel
symmetric. We can just add a remark somewhere latter that we will be dealing
with an alternative equivalent form of th kernel.

We can now state our first theorem.

######  Theorem 1.5 (First swapping law).

Fix $k\in\mathbb{Z}_{\geq 1}$, and let $\mathbf{T}_{FS,n}(k)$ denote the first
occurrence time of the swap $(k,k+1)$ in a uniformly random sorting network
$(s_{1},\dots,s_{N})\in\Omega_{n}$, $N={n\choose 2}$:

$\mathbf{T}_{FS,n}(k)=\min\\{t\geq 1:\,s_{t}=k\\}.$

Then we have the following convergence in law:

$\lim_{n\rightarrow\infty}\frac{\mathbf{T}_{FS,n}(k)}{n^{\frac{3}{2}}}\,=\frac{1}{2}\mathbf{T}_{FS}(k).$

In order to connect the first swaps to the spacings, we are going to use
translation invariance of the random sorting networks (cf. [AHRV, Theorem 1,
(i)]), which is based on the following combinatorial fact:

######  Proposition 1.6.

Let $N={n\choose 2}$. Then $(s_{1},...,s_{N})\in\Omega_{n}$ if and only if
$(s_{2},...,s_{N},n-s_{1})\in\Omega_{n}$.

Applying Proposition 1.6 one readily proves that for all $1\leq r\leq r+t\leq
N$,

$\mathbb{P}\left[\mathbf{T}_{\rm{FS}}(k)>t\right]=\mathbb{P}\left[\text{there
are no swaps }(k,k+1)\text{ at times }r+1,r+2,\dots,r+t\right],$

which can be used to identify the distribution of a spacing in row $k$. Before
doing so, we need to specify the definition of a spacing. In fact, there are
two natural definitions, which lead to distinct random variables. For the
definition it is convenient to extend a sorting network to a $2N$–periodic
$\mathbb{Z}$–indexed sequence:

######  Definition 1.7.

Given $(s_{1}\dots s_{N})\in\Omega_{n}$, we extend it to a sequence
$(s_{t})_{t\in\mathbb{Z}}$ by requiring $s_{t+N}=n-s_{t}$ for all
$t\in\mathbb{Z}$. We call $(s_{t})_{t\in\mathbb{Z}}$ the _periodic extension_
of $(s_{1}\dots s_{N})\in\Omega_{n}$.

For instance $(1\,2\,1)\in\Omega_{3}$ is extended to the infinite
$\mathbb{Z}$–indexed sequence $(\dots 2\,1\,2\,1\,2\,\dots)$ with $1$s at odd
positions and $2$s at even positions. The reason for this particular
definition is contained in the following straightforward corollary of
Proposition 1.6:

######  Corollary 1.8.

If $(s_{1},\dots,s_{N})$ is a uniformly random element of $\Omega_{n}$, then
its periodic extension $(s_{t})_{t\in\mathbb{Z}}$ is translation-invariant:
for each $a\in\mathbb{Z}$ we have a distributional identity

$(s_{t+a})_{t\in\mathbb{Z}}\stackrel{{\scriptstyle
d}}{{=}}(s_{t})_{t\in\mathbb{Z}}.$

######  Definition 1.9 (First definition of the spacing).

Fix $n=1,2,\dots$, $k\in\\{1,2,...,n-1\\}$ and $a\in\mathbb{Z}$. Let
$(s_{t})_{t\in\mathbb{Z}}$ be the periodic extension of a uniformly random
sorting network in $\Omega_{n}$. Set

$X=\max\\{t\leq a:s_{t}=k\\},\qquad Y=\min\\{t>a:s_{t}=k\\}.$

We define the spacing on the $k^{\text{th}}$ row of a sorting network of size
$n$ as $\mathrm{Sp}_{k,n}:=Y-X$.

######  Remark 1.10.

While the definition depends on the choice of $a$, Corollary 1.8 implies that
the distribution of $\mathrm{Sp}_{k,n}$ is the same for all $a\in\mathbb{Z}$.
Hence, we omit $a$ from the notation.

The $n\to\infty$ limit of $\mathrm{Sp}_{k,n}$ turns out to be connected to
$\mathbf{T}_{\rm{FS}}(k)$ of Definition 1.4.

######  Theorem 1.11.

Fix $k\in\mathbb{Z}_{\geq 1}$. We have the following convergence in
distribution:

$\lim_{n\to\infty}\frac{\mathrm{Sp}_{k,n}}{n^{\frac{3}{2}}}=\frac{1}{2}Z_{k},$

where $Z_{k}$ is a positive random variable of density $g_{k}(x)$, $x>0$,
given by

(1.2) $g_{k}(x)=x\,\frac{\partial^{2}}{\partial
x^{2}}\bigl{(}\mathbb{P}\left[\mathbf{T}_{\rm{FS}}(k)>x\right]\bigr{)}.$

Here is an alternative definition of the spacing.

######  Definition 1.12 (Second definition of the spacing).

Fix $n=1,2,\dots$, $k\in\\{1,2,...,n-1\\}$ and $a\in\mathbb{Z}$. Let
$(s_{t})_{t\in\mathbb{Z}}$ be the periodic extension of a uniformly random
sorting network in $\Omega_{n}$. Set

$Y=\min\\{t>a:s_{t}=k\\}.$

We define the spacing $\widehat{\mathrm{Sp}}_{k,n}$ on the $k^{\text{th}}$ row
of a sorting network of size $n$ as a random variable whose law is the
distribution of $Y$ conditional on the event $\\{s_{a}=k\\}$.

######  Remark 1.13.

As in Definition 1.9, the choice of $a$ does not affect the distribution of
$\widehat{\mathrm{Sp}}_{k,n}$.

Both definitions of the spacing have their own merits. Definition 1.12 is the
one preferred in theoretical physics and random matrix literature. However, a
disadvantage of Definition 1.12 is that it is hard to sample or observe
$\widehat{\mathrm{Sp}}_{k,n}$, as it fails to be a random variable on the
state space of all sorting networks; on the other hand, $\mathrm{Sp}_{k,n}$ is
a function of a sorting network. The $n\to\infty$ limit of
$\widehat{\mathrm{Sp}}_{k,n}$ is still connected to $\mathbf{T}_{\rm{FS}}(k)$
of Definition 1.4, but in a slightly different way.

######  Theorem 1.14.

Fix $k\in\mathbb{Z}_{\geq 1}$. We have the following convergence in
distribution:

$\lim_{n\to\infty}\frac{\widehat{\mathrm{Sp}}_{k,n}}{n^{\frac{3}{2}}}=\frac{1}{2}\widehat{Z}_{k},$

where $\widehat{Z}_{k}$ is a positive random variable of density
$\widehat{g}_{k}(x)$, $x>0$, given by

(1.3)
$\widehat{g}_{k}(x)=\frac{\sqrt{\pi}}{2}\frac{(2k-2)!!}{(2k-1)!!}\,\frac{\partial^{2}}{\partial
x^{2}}\bigl{(}\mathbb{P}\left[\mathbf{T}_{\rm{FS}}(k)>x\right]\bigr{)}.$

######  Remark 1.15.

In $k=1$ case $\mathbf{T}_{\rm{FS}}(k)$ becomes the absolute value of Gaussian
random variable $N(0,\tfrac{1}{2})$. Hence, $\widehat{Z}_{1}$ has density
$2xe^{-x^{2}}$, $x>0$, which matches the result of [R].

In addition, we present an alternative expression for $\widehat{Z}_{k}$ by
identifying it with a marginal of a certain two-dimensional determinantal
point process; we refer to [B] and references therein for general discussion
of the determinantal point processes.

######  Definition 1.16.

Let $\pazocal{X}_{k}^{{}^{\prime}}$ be a determinantal point process on
$\mathbb{Z}_{\geq 0}\times\mathbb{R}_{\geq 0}$, such that with respect to the
product of the counting measure on the first coordinate and the Lebesgue
measure on the second coordinate, the correlation kernel is

$\displaystyle
K^{{}^{\prime}}_{k}(x_{1},u_{1};\,x_{2},u_{2})=\mathbf{1}_{\\{u_{2}<u_{1},\,x_{2}<x_{1}\\}}\,\frac{(u_{2}-u_{1})^{x_{1}-x_{2}-1}}{(x_{1}-x_{2}-1)!}$
$\displaystyle+\frac{1}{(2\pi\mathbf{i})^{2}}\oint\limits_{C_{z}[0,\infty)}dz\oint\limits_{C_{w}[0,x_{1})}dw\,\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{\Gamma(z+x_{2}+1)}{\Gamma(x_{1}-w)}\frac{\Gamma(-\frac{z+x_{2}}{2})}{\Gamma(\frac{-x_{1}+w+1}{2})}\times$
$\displaystyle\frac{z+x_{2}-2k}{z+x_{2}-2k+1}\frac{x_{1}-w-2k}{x_{1}-w-2k-1}\frac{u_{1}^{w}u_{2}^{z}}{w+z+x_{2}-x_{1}+1}.$

The contours $C_{z}[0,\infty),C_{w}[0,x_{1})$ are counterclockwise, enclose
the integers in $[0,\infty)$ and $[0,x_{1})$ respectively, but are uniformly
bounded away from them. These two contours are also arranged in a way that
$w+z+x_{2}-x_{1}+1$ is uniformly bounded away from 0. The value of
$K^{{}^{\prime}}_{k}$ when $u_{1}$ or $u_{2}$ equals 0 is to be understood as
the limit as $u_{1}$ or $u_{2}$ tends to 0.

The $l^{\text{th}}$ level of $\pazocal{X}^{{}^{\prime}}_{k}$ is the
restriction of the process on the line $\\{l\\}\times\mathbb{R}_{\geq 0}$.

######  Remark 1.17.

For $m\geq 0$, there are $m$ particles in the $2m^{\text{th}}$ and
$2m+1^{\text{th}}$ level of $\pazocal{X}^{{}^{\prime}}_{k}$, except that on
the $2k^{\text{th}}$ level there are only $k-1$ particles.

If we denote the $j^{\text{th}}$ particle(from top to bottom) on the
$l^{\text{th}}$ level by $x^{l}_{j}$ and set $x^{l}_{j}=0$ for $j>\\#$
particles in the $l^{\text{th}}$ level, then the particles of
$\pazocal{X}^{{}^{\prime}}_{k}$ interlace, i.e,

$x^{l+1}_{j+1}\leq x^{l}_{j}\leq x^{l+1}_{j}.$

While we do not prove this, we expect that compared to $\pazocal{X}$,
$\pazocal{X}^{{}^{\prime}}_{k}$ can be thought as the corner process of aGUE
conditioned on the event that the smallest positive eigenvalues of the
$2k^{\text{th}}$ corner is 0, see Remark 3.2.

######  Theorem 1.18.

Let $\widehat{Z}_{k}$ be as in Theorem 3.6 and let $Z^{\prime}_{l}$ denote the
coordinate of the closest to the origin particle on the $l^{\text{th}}$ level
in the point process $\pazocal{X}^{\prime}_{k}$.

  * •

If $k=1$, then the law of $\widehat{Z}_{1}$ coincides with that of
$Z^{\prime}_{3}$.

  * •

If $k\geq 2$, then the law of $\widehat{Z}_{k}$ coincides with that of
$\min\\{Z^{\prime}_{2k-1},Z^{\prime}_{2k+1}\\}$.

###  1.3. Methods

Here is a sketch of our proof of Theorem 1.5:

  * •

We first use the Edelman-Greene bijection [EG] to map the problem to the study
of the asymptotic distribution of the entries of uniformly random standard
Young tableau of staircase shape, see Section 4.2.

  * •

We replace standard Young tableau by a continuous version, which we call
Poissonized Young tableau. In Section 3.2 we show that the error of this
replacement is negligible in our limit regime.

  * •

We use the formula of [GR] for the correlation kernel of the determinantal
point process, describing the entries of a Poissonized Young tableau.

  * •

Asymptotic analysis of this formula reveals a match to the correlation kernel
of the determinantal point process describing positive eigenvalues of aGUE
matrix.

We remark that (in contrast to the results in the bulk, i.e., for $k$ of order
$\alpha n$, $0<\alpha<1$, as in [GR, ADHV]) we do not claim any results for
the joint asymptotic distribution of several swaps; Theorem 1.5 only deals
with one-dimensional marginal distribution. A technical reason is that the
Edelman-Green bijection does not have any straightforward continuous version
acting on the eigenvalues of aGUE of finite sizes: it seems that one needs to
deal simultaneously with aGUE of all sizes, which is not a particularly
transparent operation. In future, it would be interesting to find a way to
overcome this difficulty.

Theorems 1.11 and 1.14 are proven in Section 4 as corollaries of Theorem 1.5.
The central idea is to develop discrete versions of (1.2), (1.3) valid for
random sorting networks of finite sizes. In fact, these discrete statements
are valid for any periodic point processes and, thus, can be of independent
interest, see Proposition 4.2.

Finally, Theorem 1.18 is proven in Section 4 by repeating the arguments of
Theorem 1.5 for the sorting networks conditional on the event $\\{s_{1}=k\\}$.
This requires asymptotic analysis of entries of standard Young tableau of a
slightly different shape, which is still possible by our methods. Thus, we
arrive at the identities of Theorem 1.18 in an indirect way, by showing that
the two sides of the identity are $n\to\infty$ limits of the same discrete
distributions.

### Acknowledgements

The work of V.G. was partially supported by NSF Grants DMS-1664619,
DMS-1949820, and by the Office of the Vice Chancellor for Research and
Graduate Education at the University of Wisconsin–Madison with funding from
the Wisconsin Alumni Research Foundation.

[Some of the old materials are hidden under comment below. It might be moved
to later sections, if it is necessary for them.]

##  2\. Preliminaries

One of the key tools in studying sorting networks is the Edelman-Green
bijection (see Section 4.2 for the details), which maps them to Standard Young
Tableaux (SYT). In this and the next section we develop asymptotic results for
SYT, which will be ultimately used in Section 4 to prove the theorems about
the random sorting networks announced in the introduction.

In addition in the last subsection of this section we recall the properties of
the eigenvalues of aGUE, which will be useful later on.

###  2.1. Young diagrams and Young tableaux

A partition is a sequence of non-negative integers
$\lambda_{1}\geq\lambda_{2}\geq...\geq 0$ such that
$|\lambda|:=\sum_{i=1}^{\infty}\lambda_{i}<\infty$. The length of $\lambda$,
denoted by $l(\lambda)$, is the number of strictly positive $\lambda_{i}s$.

We identify a partition $\lambda$ with a collection of $N=|\lambda|$ boxes,
such that there are $\lambda_{i}$ boxes in row $i$. We use coordinate $(i,j)$
to denote the j-th box at row i. In other words, the coordinates of boxes are
$\\{(i,j)\mid 1\leq j\leq\lambda_{i},i=1,2...\\}$. Such a combinatorial object
is named a _Young diagram_ , still denoted as $\lambda$. In particular, we say
the Young diagram is of staircase shape and write $\lambda=\Delta_{n}$ for
some $n\in\mathbb{Z}_{\geq 1}$, if $\lambda_{i}=n-i$ for $i=1,2,...,n$. We
also use the diagram $\lambda=\Delta_{n}\setminus(n-k,k)$ for $1\leq k\leq
n-1$, which has $\lambda_{i}=n-i$ for $i\neq n-k$ and $\lambda_{n-k}=k-1$.

A _standard Young tableaux_ (SYT) of shape $\lambda$ is an insertion of
numbers 1,2,…,$|\lambda|$ into boxes of $\lambda$ without repetitions, such
that the numbers in each row and column strictly increase from left to right
and from top to bottom. The numbers within a SYT are its _entries_. Fixing the
Young diagram $\lambda$, we can get a uniformly random SYT of shape $\lambda$,
denoted as $\mathbf{T}_{\lambda}$ by considering a uniform measure on the
space of all SYT of shape $\lambda$. See Figure 2 for an example with
$\lambda=\Delta_{6}$.

A _Poissonized Young tableaux_ (PYT) of shape $\lambda$ is an insertion of
$|\lambda|$ real numbers in $[0,1]$ into boxes of $\lambda$, such that the
numbers strictly increase along each row and column. We use
$\mathrm{PYT}(\lambda)$ to denote the space of all PYT of shape $\lambda$.
Note that a given PYT of shape $\lambda$ can be transformed canonically to a
SYT of the same shape, by replacing the k-th smallest entry with k. In the
opposite direction, we can get a uniformly random PYT of shape $\lambda$,
denoted as $\pazocal{T}_{\lambda}$ by the following steps: first generate a
uniformly random SYT of shape $\lambda$, then independently sample $|\lambda|$
i.i.d random variables uniform in [0,1], and replace the entry $k$ with the
k-th smallest random variable.

Figure 2. An example of staircase shaped standard Young tableaux, with $n=6$
in two different coordinate systems: the right one is defined in Section 2.1
and left one is defined in Section 2.2.

###  2.2. Rotation of Young Tableaux

Let $T_{\Delta_{n}}$ denote a standard Young tableaux of shape $\Delta_{n}$.
We make a change of coordinates $(i,j)\to(l,m)$ for the entries in
$T_{\Delta_{n}}$ by letting

$\displaystyle l=n-i+j,\qquad
m=\begin{cases}\frac{n-i-j}{2}+1,&n-i-j\;\;\text{is even},\\\
\frac{n-i-j+1}{2},&n-i-j\;\;\text{is odd}.\end{cases}$

We let $T_{\Delta_{n}}(l,m)$ denote the entry of $T_{\Delta_{n}}$ on the
$i^{\text{th}}$ row and the $j^{\text{th}}$ column, where $i$ and $j$ are
reconstructed from $l$ and $m$ by inverting the above formulas. In details, we
have

$i=\begin{cases}n-\frac{l}{2}-m+1,&l\text{ is even},\\\
n-\frac{l}{2}-m+\frac{1}{2},&l\text{ is odd};\end{cases}\qquad
j=\begin{cases}\frac{l}{2}-m+1,&l\text{ is even},\\\
\frac{l}{2}-m+\frac{1}{2},&l\text{ is odd}.\end{cases}$

The allowed values for $(i,j)$ are: $i\geq 1$, $j\geq 1$, $i+j\leq n$. The
allowed values for $(l,m)$ are $l=2,3,4,\dots,2n-2$ and
$m=1,2,\dots,\lfloor\frac{\min(l,2n-l)}{2}\rfloor$. The transformation
$(i,j)\to(l,m)$ is essentially a clockwise rotation by 45 degrees, as can be
seen in Figure 2.

Similarly, for standard Young tableaux $T_{\Delta_{n}\setminus(n-k,k)}$ we
make the same change of coordinates as above and deal with
$T_{\Delta_{n}\setminus(n-k,k)}(l,m)$. Formally, the entry at $(l,m)=(2k,1)$
does not exist, because it corresponds to the removed box at $(n-k,k)$, but we
can also think about this entry as being $N={n\choose 2}$; in this way
$T_{\Delta_{n}\setminus(n-k,k)}$ becomes $T_{\Delta_{n}}$ with the additional
restriction $T_{\Delta_{n}}(2k,1)=N$.

For Poissonized Young tableaux (PYT) $\pazocal{T}_{\lambda}$ of shapes
$\lambda=\Delta_{n}$ or $\lambda=\Delta_{n}\setminus(n-k,k)$, we define the
change of coordinate in exactly the same way and use $(m,l)$ coordinate
system.

###  2.3. Point processes associated with PYT

Let $\pazocal{T}_{\lambda}$ be a $\mathrm{PYT}$ of shape $\lambda$. As in
Section 2.2, we will focus on the case $\lambda=\Delta_{n}$ or
$\lambda=\Delta_{n}\setminus(n-k,k)$, for which we induce a point process on
$\mathbb{Z}_{\geq 0}\times\mathbb{R}_{\geq 0}$ from its entries.

######  Definition 2.1.

The projection of a $\mathrm{PYT}$ $\pazocal{T}$ with shape ${\Delta_{n}}$ or
$\Delta_{n}\setminus(n-k,k)$ is a point configuration on $\mathbb{Z}_{\geq
0}\times[0,1]$, such that there’s a particle at $(l,u)$, if for some
$l\in\mathbb{Z}_{\geq 2}$ and $m\in\mathbb{Z}_{\geq 1}$,

$u=1-\pazocal{T}(l,m).$

By definition, $u=1-\pazocal{T}(l,m)$ is the $m^{\text{th}}$ lowest particle
on level $\\{l\\}\times\mathbb{R}_{\geq 0}$ (except for level $l=2k$ in
$\pazocal{T}_{\Delta_{n}\setminus(n-k,k)}$, where $\pazocal{T}(2k,1)$ is
missing), and the particles on neighboring levels interlace by the setting of
PYT, see Figure 3.

Figure 3. Producing a point configuration from a $\mathrm{PYT}$ $T$ of shape
$\Delta_{6}$ with entries given in the figure as in Section 2.2 and 2.3: first
rotate $\Delta_{6}$ clockwise by 45 degree, then project its entries, which
are always interlacing.

If we take $\pazocal{T}_{\lambda}$ to be uniformly random, then the projection
of $\pazocal{T}_{\lambda}$ becomes a point process on $\mathbb{Z}_{\geq
0}\times[0,1]$. Note that this random point configuration is simple almost
surely. We also would like to rescale the $u$–coordinate.

######  Definition 2.2.

The point process $\pazocal{X}^{{}^{\prime}}_{n}$ on $\mathbb{Z}_{\geq
0}\times\mathbb{R}_{\geq 0}$ is obtained by taking a uniformly random
$\mathrm{PYT}$ $\pazocal{T}_{\Delta_{n}}$ , projecting its entries on
$\mathbb{Z}_{\geq 0}\times[0,1]$, and then rescaling the second coordinate of
each particle by the map $(l,u)\mapsto(l,n^{\frac{1}{2}}\cdot u)$. Similarly,
$\pazocal{X}^{{}^{\prime}}_{k,n}$ is obtained from uniformly random PYT
$\pazocal{T}_{\Delta_{n}\setminus(n-k,k)}$ by the same procedure.

We study the asymptotic behavior of $\pazocal{X}_{n}^{{}^{\prime}}$ and
$\pazocal{X}^{{}^{\prime}}_{k,n}$ in Section 3.3. Our analysis uses the
technique of determinantal point processes with correlation kernels given by
double contour integrals, which we present next.

###  2.4. Determinantal Point Process

Here’s a brief review of some standard definitions, for a thorough
introduction see [B, DVe]. Let $S$ be a locally compact Polish space, say
$\mathbb{Z}$ or $\mathbb{R}^{n}$. We consider the space of discrete subsets in
$S$, which consists of countable subsets $X$ of $S$ without accumulation
points, and identify each such point configuration with the measure
$\sum_{x\in X}\delta_{x}$. The topology of this space is given as the weak
topology of Radon measure on $S$. We further use Borel $\sigma-$algebra
corresponding to this topology.

A point process is a random discrete subset of $S$, and we assume that it is
simple a.s, i.e, all the points have distinct positions. A _determinantal
point process_ $\pazocal{X}$ on $S$ is a point process with correlation kernel
$K:S\times S\rightarrow\mathbb{R}$, and a reference measure $\mu$ on $S$, such
that for any $f:S^{k}\rightarrow\mathbb{R}$ with compact support,

(2.1)
$\mathbb{E}\,\Bigg{[}\sum_{\begin{subarray}{c}x_{1},\ldots,x_{k}\in\pazocal{X}\\\
\,x_{1},\ldots,x_{k}\;\text{are
distinct}\end{subarray}}f(x_{1},\ldots,x_{k})\Bigg{]}=\int\limits_{S^{k}}\det\left[K(x_{i},x_{j})\right]f(x_{1},\ldots,x_{k})\,\mu^{\otimes
k}(dx_{1},\ldots,dx_{k}).$

Expectations of the form given by the l.h.s. of (2.1) determine the law of
$\pazocal{X}$ under mild conditions on $K$, see [Le]. This will always be the
case in this text as the correlation kernels we consider will be continuous.

The determinantal point processes appearing in this paper live in space
$S=\mathbb{Z}\times[0,1]$ or $S=\mathbb{Z}\times\mathbb{R}_{\geq 0}$, with
reference measure being the product of counting measure and Lebesgue measure,
denoted by $\\#_{\mathbb{Z}}\times\pazocal{L}([0,1])$ or
$\\#_{\mathbb{Z}}\times\pazocal{L}(\mathbb{R}_{\geq 0})$ resp. We use the
following lemma, whose proof we omit.

######  Lemma 2.3.

Let $Y_{n}$ be a determinantal point process on
$\mathbb{Z}\times\mathbb{R}_{>0}$ with reference measure
$\\#_{Z}\times\pazocal{L}(\mathbb{R}_{>0})$ and correlation kernel $K_{n}$.
Then the sequence $Y_{n}$ converges weakly as $n\to\infty$ to a determinantal
point process $\pazocal{X}$ with correlation kernel $K$ on
$\mathbb{Z}\times\mathbb{R}_{\geq 0}$ (with reference measure
$\\#_{\mathbb{Z}}\times\mathbb{R}_{\geq 0}$) and with almost surely no
particles at $(i,0)$, $i\in\mathbb{Z}$, if

$K_{n}^{{}^{\prime}}\rightarrow K$ uniformly on compact subsets of
$\mathbb{Z}\times\mathbb{R}_{\geq 0}$.

###  2.5. Uniformly Random Projection

When $\mathrm{PYT}$ $\pazocal{T}_{\Delta_{n}}$ and
$\pazocal{T}_{\Delta_{n}\setminus(n-k,k)}$ are uniformly random, their
projections are point processes on $\mathbb{Z}_{\geq 0}\times[0,1]$, whose
correlation functions were computed in [GR]. We restate the result there for
our case in the following theorem, which is a stepping stone of the proofs of
our main results.

######  Theorem 2.4.

The processes $\pazocal{X}^{{}^{\prime}}_{n}$ and
$\pazocal{X}^{{}^{\prime}}_{k,n}$ of Definition 2.2 are determinantal point
process on $\mathbb{Z}_{\geq 0}\times\mathbb{R}_{>0}$ with correlation kernel
$K_{\lambda}(x_{1},u_{1};x_{2},u_{2})$ given for
$x_{1},x_{2}\in\mathbb{Z}_{\geq 0}$ and $u_{1},u_{2}\in\mathbb{R}_{>0}$, by

$\displaystyle
K_{\lambda}(x_{1},u_{1};x_{2};u_{2})=\mathbf{1}_{\\{u_{2}<u_{1},x_{2}<x_{1}\\}}\frac{(u_{2}-u_{1})^{x_{1}-x_{2}-1}n^{-\frac{1}{2}(x_{1}-x_{2})}}{(x_{1}-x_{2}-1)!}$
$\displaystyle+\frac{1}{(2\pi
i)^{2}}\oint_{C_{z}[0,\lambda_{1}+n-x_{2})}dz\oint_{C_{w}[0,x_{1})}dw\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{G_{\lambda}(z+x_{2}-n)}{G_{\lambda}(x_{1}-n-1-w)}\cdot\frac{u_{2}^{z}\,u_{1}^{w}\,n^{-\frac{1}{2}(z+w+1)}}{w+z+x_{2}-x_{1}+1},$

where

$\displaystyle
G_{\lambda}(u)=\Gamma\left(u+1\right)\prod_{i=1}^{\infty}\frac{u+i}{u-\lambda_{i}+i}=\frac{\Gamma\left(u+1+n\right)}{\prod\limits_{i=1}^{n}(u-\lambda_{i}+i)},\quad
n\geq\ell(\lambda),$

with $\lambda=\Delta_{n}$ for $\pazocal{X}^{{}^{\prime}}_{n}$ and
$\lambda=\Delta_{n}\setminus(n-k,k)$ for $\pazocal{X}^{{}^{\prime}}_{k,n}$.
The contours $C_{z}[0,\lambda_{1}+n-x_{2})$ and $C_{w}[0,x_{1})$ are as shown
in Figure 4. Both are counter-clockwise, encloses only the integers in the
respective half open intervals $[0,\lambda_{1}+n-x_{2})$ and $[0,x_{1})$, and
are arranged so that $C_{z}$ and $x_{1}-x_{2}-1-C_{w}$ are disjoint, as in
Figure 4.

$z$-contour $C_{z}$$w$-contour $C_{w}$$0$$-1$$x_{1}-1$$\lambda_{1}+n-1-x_{2}$

Figure 4. The contours in the statement of Theorem 2.4, correspond to the case
$1\leq x_{1}\leq x_{2}$ (top) and $1\leq x_{2}<x_{1}$ (bottom). In the bottom
picture, $w$-contour splits into two components, in order to guarantee that
$C_{z}$ and $x_{1}-x_{2}-1-C_{w}$ are disjoint

###### Proof.

This is a corollary of [GR, Theorem 1.5]. In the notation of [GR], we choose:

$t_{1}=1-\frac{u_{1}}{n^{\frac{1}{2}}};\
t_{2}=1-\frac{u_{2}}{n^{\frac{1}{2}}};\ x_{1}^{[GR]}=x_{1}-n;\
x_{2}^{[GR]}=x_{2}-n;\qquad\lambda=\Delta_{n}\;\text{or}\;\Delta_{n}\setminus(n-k,k).$

Since our match of parameters involves rescaling of the real spatial
coordinate, we also need to use the following Lemma 2.8, whose proof we omit.
∎

######  Remark 2.5.

The arrangement of contours of $z,w$ inherits from [GR, Theorem 1.5], with an
extra shift due to our change of variables.

######  Remark 2.6.

By Definition 2.2, it actually does not hurt to consider only $x_{1},x_{2}\geq
2$.

######  Remark 2.7.

We can extend $\pazocal{X}_{n}^{{}^{\prime}}$ and
$\pazocal{X}_{k,n}^{{}^{\prime}}$ to point processes on $\mathbb{Z}_{\geq
0}\times\mathbb{R}_{\geq 0}$, in such a way that almost surely there is no
particle falling at $(x,0)$ for any $x\in\mathbb{Z}_{\geq 0}$. Moreover, the
kernel $K_{\lambda}$ can be extended to the same space based on its regular
behavior at $u_{1}$ or $u_{2}=0$. In order to see that the kernel is well-
behaved as $u_{1}\to 0$ or $u_{2}\to 0$, we shift the contour of $z$ and $w$
by $1$ to the right, collecting additional residue terms at $w=0$ and $z=0$ in
the deformation. Let us only detail the case
$\lambda=\Delta_{n}\setminus(n-k,k)$, for which we get

$\displaystyle
K_{\lambda}(x_{1},u_{1};x_{2},u_{2})=\mathbf{1}_{\\{u_{2}<u_{1},x_{2}<x_{1}\\}}\frac{(u_{2}-u_{1})^{x_{1}-x_{2}-1}n^{-\frac{1}{2}(x_{1}-x_{2})}}{(x_{1}-x_{2}-1)!}$
$\displaystyle+\frac{1}{(2\pi\mathbf{i})^{2}}\oint_{C_{z}[1,2n-1-x_{2})}dz\oint_{C_{w}[1,x_{1})}dw\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{G_{\lambda}(z+x_{2}-n)}{G_{\lambda}(x_{1}-n-1-w)}\cdot\frac{u_{2}^{z}\,u_{1}^{w}\,n^{-\frac{1}{2}(z+w+1)}}{w+z+x_{2}-x_{1}+1}$
$\displaystyle-\frac{1}{2\pi\mathbf{i}}\oint\limits_{C_{z}[1,2n-1-x_{2})}dz\Biggl{(}\prod_{i=1}^{n}\frac{x_{1}-2n+2i+\mathbf{1}_{\\{i=n-k\\}}-1}{x_{2}-2n+2i+\mathbf{1}_{\\{i=n-k\\}}+z}\Biggr{)}\cdot\frac{\Gamma(z+x_{2}+1)}{\Gamma(x_{1})\Gamma(z+1)}\cdot\frac{u_{2}^{z}\,n^{-\frac{1}{2}(z+1)}}{z+x_{2}-x_{1}+1}$
$\displaystyle\cdot$ $\displaystyle+\frac{\mathbf{1}_{\\{x_{2}\in
A_{k}\\}}}{2\pi\mathbf{i}}\Bigg{[}\oint\limits_{C_{w}[1,x_{1})}dw\Biggl{(}\prod_{\begin{smallmatrix}1\leq
i\leq n:\\\ \text{denominator}\neq
0\end{smallmatrix}}\frac{x_{1}-2n+2i+\mathbf{1}_{\\{i=n-k\\}}-1-w}{x_{2}-2n+2i+\mathbf{1}_{\\{i=n-k\\}}}\Biggr{)}\cdot\frac{\Gamma(x_{2}+1)\Gamma(-w)}{\Gamma(x_{1}-w)}\cdot\frac{u_{1}^{w}\,n^{-\frac{1}{2}(w+1)}}{w+x_{2}-x_{1}+1}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\mathbf{1}_{\\{w_{0}\geq
1\\}}\Biggl{(}\prod_{\begin{smallmatrix}1\leq i\leq n:\\\
\text{denominator}\neq
0\end{smallmatrix}}\frac{x_{1}-2n+2i+\mathbf{1}_{\\{i=n-k\\}}-1-w_{0}}{x_{2}-2n+2i+\mathbf{1}_{\\{i=n-k\\}}}\Biggr{)}\cdot\frac{\Gamma(x_{2}+1)(-1)^{1+w_{0}}}{\Gamma(x_{1}-w_{0})\Gamma(1+w_{0})}\cdot
u_{1}^{w_{0}}n^{-\frac{1}{2}(w_{0}+1)}\Bigg{]}$
$\displaystyle-\mathbf{1}_{\\{x_{2}-x_{1}+1\neq 0\\}}\mathbf{1}_{\\{x_{2}\in
A_{k}\\}}\Biggl{(}\prod_{\begin{smallmatrix}1\leq i\leq n:\\\
\text{denominator}\neq
0\end{smallmatrix}}\frac{x_{1}-2n+2i+\mathbf{1}_{\\{i=n-k\\}}-1}{x_{2}-2n+2i+\mathbf{1}_{\\{i=n-k\\}}}\Biggr{)}\cdot\frac{\Gamma(x_{2}+1)}{\Gamma(x_{1})}\cdot\frac{n^{-\frac{1}{2}}}{x_{2}-x_{1}+1}$
$\displaystyle-\mathbf{1}_{\\{x_{2}-x_{1}+1=0\\}}\mathbf{1}_{\\{x_{2}\notin
A_{k}\\}}\Biggl{(}\prod_{\begin{smallmatrix}1\leq i\leq
n\end{smallmatrix}}\frac{x_{1}-2n+2i+\mathbf{1}_{\\{i=n-k\\}}-1}{x_{2}-2n+2i+\mathbf{1}_{\\{i=n-k\\}}}\Biggr{)}\cdot\frac{\Gamma(x_{2}+1)}{\Gamma(x_{1})}\cdot
n^{-\frac{1}{2}},$

where
$A_{k}=\\{0,2,\dots,2k-2,2k-1,2k+2,\dots,2n-2\\}=\\{2n-2i-\mathbf{1}_{\\{i=n-k\\}}\mid
i=1,2,\dots,n\\}$, and $w_{0}=x_{1}-x_{2}-1$.

Jiaming: I add two lines in the above expression of the kernel: line 5 is the
residue at $w=x_{1}-x_{2}-1$, $z=0$, which should be taken(but not taken in
line 4) because $w+z+x_{2}-x_{1}+1$ becomes $z$ after first integrating w.r.t
$w$. Similarly the last line deals with the case $x_{2}-x_{1}+1=0$ where the
integrand is the same as line 3.

Each of the three contour integrals has zero limit as $u_{1}$ or
$u_{2}\rightarrow 0$ due to vanishing integrands and the last term does not
depend on $u_{1}$ and $u_{2}$ at all. There is a slight ambiguity for the
limit as $u_{1},u_{2}\to 0$ in the first term in the situation
$x_{1}=x_{2}+1$, however, we can resolve this ambiguity in an arbitrary
consistent way. This shows the kernel has a finite limit at $u_{1}$ or
$u_{2}=0$.

######  Lemma 2.8.

Let $Y_{n}$ be a determinantal point process on $\mathbb{Z}\times(0,1)$ with
reference measure $\\#_{Z}\times\pazocal{L}((0,1))$ and correlation kernel
$K_{n}$. For $c_{n}\in\mathbb{Z}$, $k\in\mathbb{R}$, we define the rescaled
and shifted point process

$Y^{{}^{\prime}}_{n}=\\{(x-c_{n},n^{k}u)\mid(x,u)\in Y_{n}\\}.$

Then, the correlation kernel of $Y^{{}^{\prime}}_{n}$ with reference measure
$\\#_{Z}\times\pazocal{L}(\mathbb{R}_{>0})$ is

$K^{{}^{\prime}}_{n}(x_{1},u_{1};\,x_{2},u_{2})=n^{-k}K_{n}\Bigg{(}x_{1}+c_{n},\frac{u_{1}}{n^{k}};\,x_{2}+c_{n},\frac{u_{2}}{n^{k}}\Bigg{)}.$

###  2.6. Anti-symmetric GUE

Let us recall a statement from [FN, Theorems 2.9 and 3.3]:

######  Proposition 2.9.

For $M$ in Definition 1.2 and $l=1,2,\dots$ we have:

  1. (a)

The $l^{\text{th}}$ corner of $M$ has $l$ real eigenvalues.

  2. (b)

The eigenvalues come in pairs, i.e, $\lambda$ is an eigenvalue of the
$l^{\text{th}}$ corner if and only if $-\lambda$ is an eigenvalue of the
$l^{\text{th}}$ corner.

  3. (c)

The eigenvalues of the $l^{\text{th}}$ corner interlace with the eigenvalues
of the $(l+1)^{\text{st}}$ corner a.s, which means that

$\lambda^{l+1}_{j+1}\leq\lambda^{l}_{j}\leq\lambda^{l+1}_{j},$

where $j=1,2,...,l$ and $\lambda^{l}_{j}$ denotes the $j^{\text{th}}$ largest
eigenvalue of the $l^{\text{th}}$ corner.

  4. (d)

Conditional on the positive eigenvalues of the $l^{\text{th}}$ corner, the
positive eigenvalues of the $i^{\text{th}}$ corners, $i=2,...,l-1$, are
distributed uniformly subject to the interlacing inequalities.

  5. (e)

The joint density of positive eigenvalues
$\bigl{(}\lambda^{l}_{1}\geq\dots\geq\lambda^{l}_{\lfloor\frac{l}{2}\rfloor}\bigr{)}$
of the $l^{\text{th}}$ corner is

$\displaystyle
p(\lambda^{l}_{1},...,\lambda^{l}_{\lfloor\frac{l}{2}\rfloor})=\begin{dcases}C_{l}\prod_{1\leq
i<j\leq\frac{l}{2}}\left((\lambda^{l}_{i})^{2}-(\lambda^{l}_{j})^{2}\right)\,\prod_{j=1}^{\frac{l}{2}}e^{-(\lambda^{l}_{j})^{2}},&\text{l}\;\;\text{is}\;\;\text{even};\\\
\quad\\\ C_{l}\prod_{1\leq
i<j\leq\frac{l-1}{2}}\left((\lambda^{l}_{i})^{2}-(\lambda^{l}_{j})^{2}\right)\,\prod_{j=1}^{\frac{l-1}{2}}\left[\lambda^{l}_{j}\,e^{-(\lambda^{l}_{j})^{2}}\right],&\text{l}\;\;\text{is}\;\;\text{odd},\end{dcases}$

where $C_{l}$ are normalization constants.

Note that the above five properties uniquely characterize the joint
distribution of $\\{\lambda^{k}_{j}\\}_{1\leq j\leq k<\infty}$. An alternative
characterization is through the correlation functions of the corresponding
point process $\pazocal{X}$:

######  Theorem 2.10 ([FN, Proposition 4.3]).

The aGUE corners process $\pazocal{X}$ of Definition 1.2 is a determinantal
point process with correlation kernel $K(s,x;t,y)$ (with respect to the
reference measure $\\#_{\mathbb{Z}}\otimes\pazocal{L}(\mathbb{R}_{\geq 0})$),
such that for $s\geq t$,

(2.2)
$K(s,x;t,y)=e^{-x^{2}}\sum_{l=1}^{[t/2]}\frac{H_{s-2l}(x)H_{t-2l}(y)}{N_{t-2l}},$

and for $s<t$,

(2.3)
$K(s,x;t,y)=-e^{-x^{2}}\sum_{l=-\infty}^{0}\frac{H_{s-2l}(x)H_{t-2l}(y)}{N_{t-2l}}.$

Here $H_{j}(x)$ is the $j^{\text{th}}$ “physicist’s Hermite polynomial” with
leading coefficient $2^{j}$, so that

$\int_{\mathbb{R}}H_{i}(x)H_{j}(x)e^{-x^{2}}dx=\delta_{ij}N_{j},\qquad
N_{j}=j!2^{j}\sqrt{\pi}.$

######  Remark 2.11.

The $s=t$ case of (2.2) differs from the kernel of (1.1) by the factor
$e^{-\tfrac{1}{2}x^{2}+\tfrac{1}{2}y^{2}}$, which cancels out when we compute
determinants.

##  3\. Local Limit of Standard Staircase Shaped Tableaux

The aim of this section is to investigate the scaling limit near the bottom-
left corner for the uniformly random standard Young tableaux of shapes
$\Delta_{n}$ and $\Delta_{n}\setminus(n-k,k)$ as $n\to\infty$.

###  3.1. Statement of the result

We use the $(l,m)$ coordinate system for SYT, as in Section 2.2. We recall
that $\pazocal{X}$ is the point process corresponding to the corners of aGUE
as in Definition 1.2. Let $\pazocal{X}(l,m)$ denote the $m^{\text{th}}$
particle of $\pazocal{X}$ in the level $l$, sorted in the increasing order; in
other words, $\pazocal{X}(l,m)$ is the $m^{\text{th}}$ positive eigenvalue of
$l\times l$ corner of the infinite aGUE matrix $M$ (with $m=1$ corresponding
to the smallest positive and $m=\lfloor\tfrac{l}{2}\rfloor$ to the largest
positive eigenvalue).

Further, we set $\pazocal{X}^{{}^{\prime}}_{k}$ to be the point process of
Definition 1.16 with an extra particle inserted at $(2k,0)$. We let
$\pazocal{X}^{{}^{\prime}}_{k}(l,m)$ denote the $m^{\text{th}}$ particle of
$\pazocal{X}^{{}^{\prime}}_{k}$ in the level $x=l$, sorted in the increasing
order.

######  Theorem 3.1.

For each $n\in\mathbb{Z}_{\geq 1}$, let $T_{\Delta_{n}}(l,m)$ and
$T_{\Delta_{n}\setminus(n-k,k)}(l,m)$ be uniformly random standard Young
tableaux of shapes $\Delta_{n}$ and ${\Delta_{n}\setminus(n-k,k)}$,
respectively, in the coordinate system of Section 2.2. Recall $N={n\choose
2}$. Then for each fixed $k$, as $n\to\infty$ we have

(3.1)
$\left\\{n^{\frac{1}{2}}\left(1-\frac{T_{\Delta_{n}}(l,m)}{N}\right)\right\\}_{l\geq
2,\,\,1\leq
m\leq\lfloor\frac{l}{2}\rfloor}\longrightarrow\bigl{\\{}\pazocal{X}(l,m)\bigr{\\}}_{l\geq
2,\,\,1\leq m\leq\lfloor\frac{l}{2}\rfloor}\;\qquad\text{ and }$ (3.2)
$\left\\{n^{\frac{1}{2}}\left(1-\frac{T_{\Delta_{n}\setminus(n-k,k)}(l,m)}{N}\right)\right\\}_{l\geq
2,\,\,1\leq
m\leq\lfloor\frac{l}{2}\rfloor}\longrightarrow\bigl{\\{}\pazocal{X}^{{}^{\prime}}_{k}(l,m)\bigr{\\}}_{l\geq
2,\,\,1\leq m\leq\lfloor\frac{l}{2}\rfloor},$

in the sense of convergence of finite-dimensional distributions.

######  Remark 3.2.

The SYT $T_{\Delta_{n}\setminus(n-k,k)}$ can be thought as $T_{\Delta_{n}}$
conditioned on its largest entry being at $(n-k,k)$. Hence, Theorem 3.1
suggest a conjecture: $\pazocal{X}^{{}^{\prime}}_{k}$ should have the same law
as $\pazocal{X}$ conditioned on having a particle at $(2k,0)$, i.e. on
$\pazocal{X}(2k,1)=0$. Note that additional efforts are needed to prove this,
because the topology of the convergence in Theorem 3.1 is not strong enough to
make conclusions about the conditional distributions.

The rest of this section is devoted to the proof of Theorem 3.1. We first
couple the two types of SYT with the uniformly random PYT of the same shape,
and induce two point processes
$\pazocal{X}_{n}^{{}^{\prime}},\pazocal{X}_{k,n}^{{}^{\prime}}$ on
$\mathbb{Z}_{\geq 0}\times\mathbb{R}_{\geq 0}$ respectively from these two
PYT, as in Definition 2.2. Then we prove that
$\pazocal{X}_{n}^{{}^{\prime}}\rightarrow\pazocal{X}^{{}^{\prime}}$ and
$\pazocal{X}_{k,n}^{{}^{\prime}}\rightarrow\pazocal{X}^{{}^{\prime}}_{k}$ as
$n\to\infty$, where $\pazocal{X}^{{}^{\prime}}$ and
$\pazocal{X}^{{}^{\prime}}_{k}$ are determinantal point processes on
$\mathbb{Z}_{\geq 0}\times\mathbb{R}_{\geq 0}$ with explicit correlation
kernels given by double contour integrals. Next, we identify
$\pazocal{X}^{{}^{\prime}}$ with $\pazocal{X}$, the corners process of aGUE of
Definition 1.2.

###  3.2. Coupling With PYT

The first step of the proof is to introduce a coupling between SYT with PYT of
the same shape and to show that the under this coupling the difference between
random SYT and PYT is negligible in the asymptotic regime of our interest.

Let $T_{\Delta_{n}}$ and $\pazocal{T}_{\Delta_{n}}$ be uniformly random SYT
and PYT, respectively, of shape $\Delta_{n}$. We couple these two tableaux in
the following way: for $N={n\choose 2}$, let $(P_{1}\ <\ P_{2}\ <\cdots<\
P_{N})$ be a uniformly random point on the simplex
$\\{(x_{1},\cdots,x_{N})\mid x_{i}\geq 0,\,1\leq i\leq
N,\,\sum_{i=1}^{N}x_{i}=1\\}$. Given a uniformly random SYT $T_{\Delta_{n}}$,
we replace entry $k$ in $T_{\Delta_{n}}$ by $P_{k}$ for $k=1,2,\cdots,N$; it
is straightforward to check that the result is a uniformly random PYT
$\pazocal{T}_{\Delta_{n}}$. In the opposite direction, $T_{\Delta_{n}}$ is
reconstructed by $\pazocal{T}_{\Delta_{n}}$ by replacing the $k$th largest
entry in $\pazocal{T}_{\Delta_{n}}$ by $k$ for each $k=1,2,\dots,N$.

######  Lemma 3.3.

Fix any pair of integers $(i,j)$ with $i\geq 2$ and $1\leq
j\leq\lfloor\tfrac{i}{2}\rfloor$. Using the coupling described above, we have
for each $\delta>0$

$n^{1-\delta}\left|\pazocal{T}_{\Delta_{n}}(i,j)-\frac{T_{\Delta_{n}}(i,j)}{N}\right|\rightarrow
0$

in probability, as $n\rightarrow\infty$.

###### Proof.

Let $T_{\Delta_{n}}(i,j)=\xi\in\\{1,2,\dots,N\\}$, where $\xi$ is a random
variable. Then, by the construction of the coupling,
$\pazocal{T}_{\Delta_{n}}(i,j)=P_{\xi}$. By Chebyshev’s inequality,

(3.3)
$\begin{split}&\quad\mathbb{P}\left[n^{1-\delta}\left|P_{\xi}-\frac{\xi}{N}\right|>\epsilon\right]\\\
&\leq\left(\frac{n^{1-\delta}}{\epsilon}\right)^{2}\mathbb{E}\left[\left(P_{\xi}-\frac{\xi}{N}\right)^{2}\right]\\\
&=\frac{n^{2-2\delta}}{\epsilon^{2}}\sum_{j=1}^{N}\mathbb{E}\left[\left(P_{\xi}-\frac{\xi}{N}\right)^{2}\,\Big{|}\,\xi=j\right]\cdot\mathbb{P}\left[\xi=j\right]\end{split}$

For a fixed $j$, the random variable $P_{j}$ is distributed according to Beta
distribution with parameters $j$ and $N+1-j$, which has mean $\frac{j}{N+1}$
and variance

(3.4)
$\mathbb{E}\left[\left(P_{j}-\frac{j}{N+1}\right)^{2}\right]=\frac{j(N+1-j)}{(N+1)^{2}(N+2)}\leq\frac{\left(\frac{N+1}{2}\right)^{2}}{(N+1)^{2}(N+2)}=\frac{1}{4(N+2)}.$

Thus,

(3.5)
$\mathbb{E}\left[\left(P_{j}-\frac{j}{N}\right)^{2}\right]=\mathbb{E}\left[\left(P_{j}-\frac{j}{N+1}\right)^{2}\right]+\left(\frac{j}{N(N+1)}\right)^{2}\leq\frac{1}{4(N+2)}+\frac{1}{(N+1)^{2}}\leq\frac{1}{N}.$

Since $N=\frac{n(n-1)}{2}$, (3.3) and (3.5) imply that as $n\rightarrow\infty$

$\mathbb{P}\left[n^{1-\delta}\left|P_{\xi}-\frac{\xi}{N}\right|>\epsilon\right]\leq\frac{n^{2-2\delta}}{\epsilon^{2}N}\rightarrow
0.\qed$

In a similar way, we couple uniformly random SYT
$T_{\Delta_{n}\setminus(n-k,k)}$ and PYT
$\pazocal{T}_{\Delta_{n}\setminus(n-k,k)}$ of shape
$\Delta_{n}\setminus(n-k,k)$: we let $(P_{1}\ <\ P_{2}\ <\cdots<\ P_{N-1})$ be
a uniformly random point on the simplex $\\{(x_{1},\cdots,x_{N-1})\mid
x_{i}\geq 0,\,1\leq i\leq N-1,\,\sum_{i=1}^{N-1}x_{i}=1\\}$. Given a uniformly
random SYT $T_{\Delta_{n}\setminus(n-k,k)}$, we replace entry $k$ in
$T_{\Delta_{n}\setminus(n-k,k)}$ by $P_{k}$ for $k=1,2,\cdots,N-1$; it is
straightforward to check that the result is a uniformly random PYT
$\pazocal{T}_{\Delta_{n}\setminus(n-k,k)}$. Repeating the proof of Lemma 3.3
we arrive at:

######  Lemma 3.4.

Fix any pair of integers $(i,j)$ with $i\geq 2$ and $1\leq
j\leq\lfloor\tfrac{i}{2}\rfloor$. Using the coupling described above, we have
for each $\delta>0$

$n^{1-\delta}\left|\pazocal{T}_{\Delta_{n}\setminus(n-k,k)}(i,j)-\frac{T_{\Delta_{n}\setminus(n-k,k)}(i,j)}{N}\right|\rightarrow
0$

in probability, as $n\rightarrow\infty$.

###  3.3. Limiting Processes

In this section we analyze asymptotic behavior of the point processes
associated to random PYT $\pazocal{T}_{\Delta_{n}}$ and
$\pazocal{T}_{\Delta_{n}\setminus(n-k,k)}$ by the procedure of Section 2.3.

######  Theorem 3.5.

The point process $\pazocal{X}^{{}^{\prime}}_{n}$ (corresponding to random PYT
$\pazocal{T}_{\Delta_{n}}$ via Definition 2.2) converges weakly as
$n\to\infty$ to a point process $\pazocal{X}^{{}^{\prime}}$ on
$\mathbb{Z}_{\geq 0}\times\mathbb{R}_{\geq 0}$. $\pazocal{X}^{{}^{\prime}}$ is
a determinantal point process (with respect to reference measure
$\\#_{\mathbb{Z}_{\geq 0}}\otimes\pazocal{L}(\mathbb{R}_{\geq 0})$) with
correlation kernel

$\displaystyle K^{{}^{\prime}}$
$\displaystyle(x_{1},u_{1};x_{2},u_{2})=I_{\\{u_{2}<u_{1},x_{2}<x_{1}\\}}\frac{(u_{2}-u_{1})^{x_{1}-x_{2}-1}}{(x_{1}-x_{2}-1)!}$
$\displaystyle+\frac{1}{(2\pi
i)^{2}}\oint_{C_{z}[0,\infty)}dz\oint_{C_{w}[0,x_{1})}dw\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{\Gamma(z+x_{2}+1)}{\Gamma(x_{1}-w)}\frac{\Gamma(-\frac{z+x_{2}}{2})}{\Gamma(\frac{-x_{1}+w+1}{2})}\frac{u_{1}^{w}u_{2}^{z}}{w+z+x_{2}-x_{1}+1}.$

Here $u_{1},u_{2}\in\mathbb{R}_{>0}$ and $x_{1},x_{2}\in\mathbb{Z}_{\geq 0}$;
when $u_{1}$ or $u_{2}$ is equal to $0$, the kernel is defined as the limit as
$u_{1}$ or $u_{2}$ tends to 0. $C_{z}[0,\infty),C_{w}[0,x_{1})$ are
counterclockwise–oriented contours which enclose the integers in $[0,\infty)$
and $[0,x_{1})$, respectively, but are uniformly bounded away from them. The
contours are arranged so that $w+z+x_{2}-x_{1}+1$ is uniformly bounded away
from 0, and there’s no residue given by $w+z+x_{2}-x_{1}+1$ when first
integrating $w$, as shown in Figure 5.

Figure 5. The contours in the statements of Theorem 3.5 and 3.6, for the case
$1\leq x_{1}\leq x_{2}$ and $1\leq x_{2}<x_{1}$ respectively.

######  Theorem 3.6.

The point process $\pazocal{X}^{{}^{\prime}}_{k,n}$ (corresponding to random
PYT $\pazocal{T}_{\Delta_{n}\setminus(n-k,k)}$ via Definition 2.2) converges
weakly to a proint process $\pazocal{X}^{{}^{\prime}}_{k}$ on
$\mathbb{Z}_{\geq 0}\times\mathbb{R}_{\geq 0}$.
$\pazocal{X}^{{}^{\prime}}_{k}$ is a determinantal point process (with respect
to reference measure $\\#_{\mathbb{Z}_{\geq
0}}\otimes\pazocal{L}(\mathbb{R}_{\geq 0})$) with correlation kernel given by

$\displaystyle K^{{}^{\prime}}_{k}$
$\displaystyle(x_{1},u_{1};x_{2},u_{2})=I_{\\{u_{2}<u_{1},x_{2}<x_{1}\\}}\frac{(u_{2}-u_{1})^{x_{1}-x_{2}-1}}{(x_{1}-x_{2}-1)!}$
$\displaystyle+\frac{1}{(2\pi
i)^{2}}\oint_{C_{z}[0,\infty)}dz\oint_{C_{w}[0,x_{1})}dw\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{\Gamma(z+x_{2}+1)}{\Gamma(x_{1}-w)}\frac{\Gamma(-\frac{z+x_{2}}{2})}{\Gamma(\frac{-x_{1}+w+1}{2})}$
$\displaystyle\quad\times\frac{z+x_{2}-2k}{z+x_{2}-2k+1}\frac{x_{1}-w-2k}{x_{1}-w-2k-1}\frac{u_{1}^{w}u_{2}^{z}}{w+z+x_{2}-x_{1}+1}.$

Here $u_{1},u_{2}\in\mathbb{R}_{>0}$ and $x_{1},x_{2}\in\mathbb{Z}_{\geq 0}$;
the value of $K^{{}^{\prime}}_{k}$ when $u_{1}$ or $u_{2}$ equals 0 is to be
understood as the limit as $u_{1}$ or $u_{2}$ tends to 0. The contours
$C_{z}[0,\infty)$ and $C_{w}[0,x_{1})$ are the same as in Theorem 3.5.

We are going to use two well-known properties of the Gamma function:

  1. (1)

$\displaystyle\frac{\Gamma(y+m)}{(m-1)!}=m^{y}(1+O(m^{-1}))$, where $m\geq 1$
and the big O term is uniform in $m$ as long as $y$ is bounded and bounded
away from negative integers.

  2. (2)

For $y\notin\\{0,-1,-2...\\}$, we have
$\displaystyle\Gamma(1-y)\Gamma(y)=\frac{\pi}{\sin(\pi y)}.$

###### Proof of Theorem 3.5.

We show the convergence in distribution by verifying conditions of Lemma 2.3,
namely, we show that the correlation kernel
$K^{{}^{\prime}}_{n}(x_{1},u_{1};x_{2},u_{2})$ of
$\pazocal{X}^{{}^{\prime}}_{n}$ converges to $K^{{}^{\prime}}$ uniformly on
compact subsets of $x_{1},x_{2}\in\mathbb{Z}_{\geq 0}$ and
$u_{1},u_{2}\in\mathbb{R}_{\geq 0}$.

By Theorem 2.4, the rescaled process $\pazocal{X}^{{}^{\prime}}_{n}$ is a
determinantal point process on $\mathbb{Z}_{\geq 0}\times\mathbb{R}_{>0}$, and
the correlation kernel of $\pazocal{X}^{{}^{\prime}}_{n}$ with reference
measure $\\#_{\mathbb{Z}_{\geq 0}}\otimes\pazocal{L}(\mathbb{R}_{>0})$,
denoted by $K^{{}^{\prime}}_{n}$ is

$\displaystyle
K^{{}^{\prime}}_{n}(x_{1},u_{1};x_{2},u_{2})=\mathbf{1}_{\\{u_{2}<u_{1},x_{2}<x_{1}\\}}\frac{(u_{2}-u_{1})^{x_{1}-x_{2}-1}n^{-\frac{1}{2}(x_{1}-x_{2})}}{(x_{1}-x_{2}-1)!}$
$\displaystyle+\frac{1}{(2\pi
i)^{2}}\oint_{C_{z}[0,2n-1-x_{2})}dz\oint_{C_{w}[0,x_{1})}dw\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{G_{\lambda}(z+x_{2}-n)}{G_{\lambda}(x_{1}-n-1-w)}\times\frac{u_{2}^{z}u_{1}^{w}}{w+z+x_{2}-x_{1}+1}n^{-\frac{1}{2}(z+w+1)}$
$\displaystyle\text{where}\;\;G_{\lambda}(u)=\frac{\Gamma(u+1+n)}{\prod_{i=1}^{n}(u-\lambda_{i}+i)},\quad
n\geq\ell(\lambda).$

In this case $\lambda=\Delta_{n}$, $\lambda_{i}=n-i(i=1,2,...,n-1)$. We will
show that the integrand converges as $n\rightarrow\infty$.

First we do some calculation:

$\displaystyle\quad\prod_{i=1}^{n}(z+x_{2}-2n+2i)=(-1)^{n}\prod_{i=1}^{n}(-z-x_{2}+2n-2i)$
$\displaystyle=(-1)^{n}2^{n}\prod_{i=1}^{n}(-\frac{z=x_{2}}{2}+n-i)=(-1)^{n}2^{n}\frac{\Gamma(-\frac{z+x_{2}}{2}+n)}{\Gamma(-\frac{z+x_{2}}{2})}$
$\displaystyle\sim{(-1)^{n}}2^{n}\frac{(n-1)!n^{-\frac{z+x_{2}}{2}}}{\Gamma(-\frac{z+x_{2}}{2})}.$

The last step is by Stirling’s approximation of the Gamma function.

Similarly

$\prod_{i=1}^{n}(x_{1}-w-2n+2i-1)\sim(-1)^{n}2^{n}\frac{(n-1)!n^{\frac{-x_{1}+w+1}{2}}}{\Gamma(\frac{-x_{1}+w+1}{2})}.$

Plugging the value of $\lambda$ into the definition of $G_{\lambda}$, we have

$\displaystyle
G_{\lambda}(z+x_{2}-n)=\frac{\Gamma(z+x_{2}-n+1+n)}{\prod_{i=1}^{n}(z+x_{2}-2n+2i)}=\frac{\Gamma(z+x_{2}+1)}{\prod_{i=1}^{n}(z+x_{2}-2n+2i)},\;\;\text{and}$
$\displaystyle
G_{\lambda}(x_{1}-n-1-w)=\frac{\Gamma(x_{1}-n-1-w+1+n)}{\prod_{i=1}^{n}(x_{1}-n-1-w-n+2i)}=\frac{\Gamma(x_{1}-w)}{\prod_{i=1}^{n}(x_{1}-w-2n+2i-1)}.$

Then using the results that we calculate out above, as $n\rightarrow\infty$,

$\frac{G_{\lambda}(z+x_{2}-n)}{G_{\lambda}(x_{1}-n-1-w)}\sim\frac{\Gamma(z+x_{2}+1)}{\Gamma(x_{1}-w)}\frac{\Gamma(-\frac{z+x_{2}}{2})}{\Gamma(\frac{-x_{1}+w+1}{2})}n^{\frac{-x_{1}+w+1}{2}}n^{\frac{z+x_{2}}{2}}.$

Multiplying $K^{{}^{\prime}}_{n}$ by $n^{\frac{x_{1}-x_{2}}{2}}$ does not
change $det[K^{{}^{\prime}}_{n}(x_{i},u_{i};x_{j},u_{j})]$, hence gives the
same point process. Then by the calculation above, when $n\rightarrow\infty$,
the integrand of the double contour integral in the kernel converges to

$\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{\Gamma(z+x_{2}+1)}{\Gamma(x_{1}-w)}\frac{\Gamma(-\frac{z+x_{2}}{2})}{\Gamma(\frac{-x_{1}+w+1}{2})}\frac{u_{1}^{w}u_{2}^{z}}{w+z+x_{2}-x_{1}+1}.$

Assuming the conditions of dominated convergence theorem holds(which will be
verified later). Then as $n\rightarrow\infty$,

$\displaystyle n^{\frac{x_{1}-x_{2}}{2}}K^{{}^{\prime}}_{n}\longrightarrow$
$\displaystyle\mathbf{1}_{\\{u_{2}<u_{1},x_{2}<x_{1}\\}}\frac{(u_{2}-u_{1})^{x_{1}-x_{2}-1}}{(x_{1}-x_{2}-1)!}$
$\displaystyle\;+\frac{1}{(2\pi
i)^{2}}\oint_{C_{z}[0,\infty)}dz\oint_{C_{w}[0,x_{1})}dw\
\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{\Gamma(z+x_{2}+1)}{\Gamma(x_{1}-w)}\frac{\Gamma(-\frac{z+x_{2}}{2})}{\Gamma(\frac{-x_{1}+w+1}{2})}\frac{u_{1}^{w}u_{2}^{z}}{w+z+x_{2}-x_{1}+1}.$

∎

###### Proof of Theorem 3.6.

The proof follows the same idea as proving Theorem 3.5, with only some slight
difference in calculation. We still start from the rescaled jumps of PYT
$\pazocal{X}^{{}^{\prime}}_{k,n}$, and its correlation kernel with reference
measure $\\#_{\mathbb{Z}_{\geq 0}}\otimes\pazocal{L}(R_{>0})$ is of the same
form:

$\displaystyle
K^{{}^{\prime}}_{k,n}(x_{1},u_{1};x_{2},u_{2})=\mathbf{1}_{\\{u_{2}<u_{1},x_{2}<x_{1}\\}}\frac{(u_{2}-u_{1})^{x_{1}-x_{2}-1}n^{-\frac{1}{2}(x_{1}-x_{2})}}{(x_{1}-x_{2}-1)!}$
$\displaystyle\quad\frac{1}{(2\pi
i)^{2}}\oint_{C_{z}[0,2n-1-x_{2})}dz\oint_{C_{w}[0,x_{1})}dw\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{G_{\lambda}(z+x_{2}-n)}{G_{\lambda}(x_{1}-n-1-w)}\times\frac{u_{2}^{z}u_{1}^{w}}{w+z+x_{2}-x_{1}+1}n^{-\frac{1}{2}(z+w+1)},$

where $G_{\lambda}$ is defined in the same way as above.

Depending on the value of $k$, here $\lambda$ is a Young diagram such that

$\displaystyle\lambda_{i}=\begin{cases}n-i&i=1,2,\cdots,n-1\;\;\text{except}\;\;n-k;\\\
\\\ k-1&i=n-k.\end{cases}$

For this $\lambda$,

$\displaystyle\quad\quad G_{\lambda}(z+x_{2}-n)$
$\displaystyle=\frac{\Gamma(z+x_{2}-n+1+n)}{\prod_{i=1}^{n}(z+x_{2}-n-n+2i)}\cdot\frac{z+x_{2}-n-k+n-k}{z+x_{2}-n-k+n-k+1}$
$\displaystyle=\frac{\Gamma(z+x_{2}+1)}{\prod_{i=1}^{n}(z+x_{2}-2n+2i)}\cdot\frac{z+x_{2}-2k}{z+x_{2}-2k+1}.$

Similarly

$\displaystyle G_{\lambda}(x_{1}-n-1-w)$
$\displaystyle=\frac{\Gamma(x_{1}-n-1-w+1+n)}{\prod_{i=1}^{n}(x_{1}-n-1-w-n+2i)}\cdot\frac{x_{1}-n-1-w-k+n-k}{x_{1}-n-1-w-k+k-k+1}$
$\displaystyle=\frac{\Gamma(x_{1}-w)}{\prod_{i=1}^{n}(x_{1}-w-2n+2i-1)}\cdot\frac{x_{1}-w-2k-1}{x_{1}-w-2k}.$

Again multiplying the kernel by $n^{\frac{x_{1}-x_{2}}{2}}$. By the
calculation above, when $n\rightarrow\infty$, the integrand of the double
contour integral in the kernel converges to

$\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{\Gamma(z+x_{2}+1)}{\Gamma(x_{1}-w)}\frac{\Gamma(-\frac{z+x_{2}}{2})}{\Gamma(\frac{-x_{1}+w+1}{2})}\frac{z+x_{2}-2k}{z+x_{2}-2k+1}\frac{x_{1}-w-2k}{x_{1}-w-2k-1}\frac{u_{1}^{w}u_{2}^{z}}{w+z+x_{2}-x_{1}+1}.$

To finish the proof of 3.5 and 3.6, now we do all the checking of technical
details which are necessary for applying Lemma 2.3. The checking for
$\lambda=\Delta_{n}$ and $\lambda=\Delta_{n}\setminus(n-k,k)$ follow exactly
the same idea, so without loss of generality we use
$\lambda=\Delta_{n}\setminus(n-k,k)$ here.

After showing the convergence of the integrand describing the kernel
$K^{{}^{\prime}}_{k,n}$, one still need to check that the convergence of
integrand indeed implies the convergence of the integral, and the convergence
of integral is uniform on compact sets.

To apply dominated convergence theorem, we want to get a uniform upper bound
of the integrand, such that the upper bound is integrable over the limiting
contour(the contours are enlarging so more and more residues are counted).
Recall that the integrand of $n^{\frac{x_{1}-x_{2}}{2}}\
K^{{}^{\prime}}_{k,n}$ is

$\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{G_{\lambda}(z+x_{2}-n)}{G_{\lambda}(x_{1}-n-1-w)}\cdot\frac{z+x_{2}-2k}{z+x_{2}-2k+1}\cdot\frac{x_{1}-w-2k}{x_{1}-w-2k-1}\cdot\frac{(\frac{u_{2}}{n^{\frac{1}{2}}})^{z}(\frac{u_{1}}{n^{\frac{1}{2}}})^{w}}{w+z+x_{2}-x_{1}+1}n^{-\frac{1}{2}}n^{\frac{x_{1}-x_{2}}{2}}.$

Using the definition of $G_{\lambda}$, this can be written as

$\displaystyle\frac{\Gamma(-w)}{\Gamma(z+1)}$
$\displaystyle\frac{\Gamma(z+x_{2}+1)}{\Gamma(x_{1}-w)}\frac{\Gamma(-\frac{z+x_{2}}{2})}{\Gamma(\frac{-x_{1}+w+1}{2})}\frac{\Gamma(\frac{-x_{1}+w+1}{2}+n)}{\Gamma(-\frac{z+x_{2}}{2}+n)}$
$\displaystyle\cdot$
$\displaystyle\frac{z+x_{2}-2k}{z+x_{2}-2k+1}\cdot\frac{x_{1}-w-2k}{x_{1}-w-2k-1}\cdot\frac{u_{1}^{w}u_{2}^{z}}{w+z+x_{2}-x_{1}+1}n^{-\frac{w+z+1}{2}}n^{\frac{x_{1}-x_{2}}{2}}.$

Note that as already mentioned in the statements, we can naturally set some
restrictions on the limiting contours $C_{z}[0,\infty),C_{w}[0,x_{1})$, such
that the contours are uniformly bounded away from all the integers. By these
restrictions,

$\frac{1}{w+z+x_{2}-x_{1}+1}\cdot\frac{z+x_{2}-2k}{z+x_{2}-2k+1}\cdot\frac{x_{1}-w-2k}{x_{1}-w-2k-1}$

is bounded uniformly over the contours and over $n$.

Then we use Stirling’s approximation and the restrictions on the contours to
estimate the Gamma functions, and find that

$\frac{\Gamma(z+x_{2}+1)}{\Gamma(z+1)}\cdot\frac{\Gamma(-w)}{\Gamma(x_{1}-w)}\quad\text{and}\quad\frac{\Gamma(\frac{-x_{1}+w+1}{2}+n)}{\Gamma(-\frac{z+x_{2}}{2}+n)}n^{\frac{x_{1}-x_{2}}{2}}n^{-\frac{w+z+1}{2}}$

are also bounded uniformly over the contours and over $n$. Then what remains
is

$u_{1}^{w}u_{2}^{z}\frac{\Gamma(-\frac{z+x_{2}}{2})}{\Gamma(\frac{-x_{1}+w+1}{2})}.$

Using the identity (2) of Gamma function listed above, this equals to

$-u_{1}^{w}u_{2}^{z}\frac{1}{\Gamma(\frac{-x_{1}+w+1}{2})}\frac{\pi}{sin(\pi\frac{z+x_{2}}{2})}\frac{1}{\Gamma(1+\frac{z+x_{2}}{2})}.$

Using Stirling approximation again,

$\frac{|u_{2}^{z}|}{|\Gamma(1+\frac{z+x_{2})}{2}|}=exp\Bigg{\\{}-\frac{Re(z)}{2}\
log(Re(z))+Re(z)(log(u)+O(1))\Bigg{\\}},$

and

$\frac{\pi}{sin(\pi\frac{z+x_{2}}{2})}$

is bounded uniformly over $C_{z}[0,\infty)$.

We conclude that the integrand are bounded uniformly over $n$ by something
finite, and hence the integral w.r.t $w$ is finite since the limiting contour
of $C_{w}$ is finite. And the integral w.r.t $z$ is also finite since the
integrand decays exponentially with $z$.

The convergence is uniform over $x_{i}$ in compact subsets of $\mathbb{Z}$,
and $u_{i}$ in compact subsets of $\mathbb{R}_{>0}$, since when $u_{i}$ is
restricted in compact sets of $\mathbb{R}_{>0}$, $|u_{1}^{w}|$ is always
bounded, $|u_{2}^{z}|$ is bounded for $|z|$ bounded, and $|u_{2}^{z}|\leq
M^{z}$ for some $M\geq 0$ when $|z|$ is large.

It remains to show that the convergence is also uniform for $u$ in
$[0,t](0<t<\infty)$. For this part we only need to shift the contour of $z$
and $w$ in the same way as Remark 2.7. Then the double contour integral splits
into three contour integrals and a term independent of $u_{1},u_{2}$. The same
bounds as above apply to integrands of the three contour integrals, for
$u_{1},u_{2}\in[0,t]$, hence the three integrals all converge uniformly.

The above arguments show that
$\pazocal{X}^{{}^{\prime\prime}}_{n}\rightarrow\pazocal{X}^{{}^{\prime}}$,
$\pazocal{X}^{{}^{\prime\prime}}_{k,n}\rightarrow\pazocal{X}^{{}^{\prime}}_{k}$.
∎

Using Theorem 3.5 and 3.6, we can express the kernels of
$\pazocal{X}^{{}^{\prime}}$ and $\pazocal{X}^{{}^{\prime}}_{k}$ more
explicitly.

######  Theorem 3.7.

Let

$\displaystyle f(t)$
$\displaystyle=I_{\\{u_{2}<u_{1},x_{2}<x_{1}\\}}\frac{(tu_{2}-tu_{1})^{x_{1}-x_{2}-1}}{(x_{1}-x_{2}-1)!}$
$\displaystyle+\frac{1}{(2\pi
i)^{2}}\oint_{C_{z}[0,\infty)}dz\oint_{C_{w}[0,x_{1})}dw\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{\Gamma(z+x_{2}+1)}{\Gamma(x_{1}-w)}\frac{\Gamma(-\frac{z+x_{2}}{2})}{\Gamma(\frac{-x_{1}+w+1}{2})}\frac{(tu_{1})^{w}(tu_{2})^{z}}{w+z+x_{2}-x_{1}+1}.$

Then $K^{{}^{\prime}}(x_{1},u_{1};x_{2},u_{2})=f(1)$. And $f(t)$ satiesfies
such an equality:

(3.6)
$\begin{split}&\quad(x_{2}-x_{1}+1)f(t)+tf^{{}^{\prime}}(t)=[t^{x_{2}-x_{1}+1}f(t)]^{{}^{\prime}}\frac{1}{t^{x_{2}-x_{1}}}\\\
&=\sum_{j=0}^{a}\Bigg{[}(-1)^{2j+\tilde{x_{1}}}\frac{(tu_{1})^{2j+\tilde{x_{1}}}}{(2j+x_{1}-\tilde{x_{1}}-1)!\
\Gamma(j+\frac{1+\tilde{x_{1}}-x_{1}}{2})}\frac{1}{(2j+\tilde{x_{1}})!}\Bigg{]}\\\
&(\tilde{x_{1}}=\mathbf{1}_{\\{x_{1}\ odd\\}},\
a=\frac{x_{1}-2-\tilde{x_{1}}}{2})\\\
&\times\begin{cases}2\sum_{j=0}^{\infty}(-1)^{j+1+\frac{x_{2}}{2}}\frac{(tu_{2})^{2j}\Gamma(2j+x_{2}+1)}{(2j)!\Gamma(j+\frac{x_{2}}{2}+1)},&x_{2}\;\;\text{even};\\\
\\\
2\sum_{j=0}^{\infty}(-1)^{j+\frac{3}{2}+\frac{x_{2}}{2}}\frac{(tu_{2})^{2j+1}\Gamma(2j+1+x_{2}+1)}{(2j+1)!\Gamma(j+\frac{x_{2}+1}{2}+1)},&x_{2}\;\;\text{odd}.\end{cases}\end{split}$

###### Proof.

By doing a little bit of calculus, we see that

$\displaystyle(x_{2}-x_{1}+1)f(t)+tf^{{}^{\prime}}(t)=[t^{x_{2}-x_{1}+1}f(t)]^{{}^{\prime}}\frac{1}{t^{x_{2}-x_{1}}}$
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi
i)^{2}}\oint_{C_{z}[0,\infty)}dz\oint_{C_{w}[0,x_{1})}dw\
\frac{\Gamma(-w)}{\Gamma(z+1)}\frac{\Gamma(z+x_{2}+1)}{\Gamma(x_{1}-w)}\frac{\Gamma(-\frac{z+x_{2}}{2})}{\Gamma(\frac{-x_{1}+w+1}{2})}(tu_{1})^{w}(tu_{2})^{z}.$

We do some modification on the integrand. Applying the identity in Lemma
LABEL:lemma:gamma(b) to $\Gamma(-w),\ \Gamma(-\frac{z+x_{2}}{2})$ and
$\Gamma(\frac{-x_{1}+w+1}{2})$, the integrand can be written as

$\Bigg{[}(tu_{1})^{w}\frac{\Gamma(\frac{x_{1}-w+1}{2})}{\Gamma(x_{1}-w)}\frac{1}{\Gamma(w+1)}\frac{sin(\pi\frac{x_{1}-w+1}{2})}{sin(-\pi
w)}\Bigg{]}\times\Bigg{[}(tu_{2})^{z}\frac{\Gamma(z+x_{2}+1)}{\Gamma(\frac{z+x_{2}}{2}+1)}\frac{1}{\Gamma(z+1)}\frac{\pi}{sin(-\pi\frac{z+x_{2}}{2})}\Bigg{]}.$

Then the expression in the theorem follows directly from residue theorem. ∎

######  Theorem 3.8.

Let

$\displaystyle
f(t)=I_{\\{u_{2}<u_{1},x_{2}<x_{1}\\}}\frac{(tu_{2}-tu_{1})^{x_{1}-x_{2}-1}}{(x_{1}-x_{2}-1)!}$
$\displaystyle+\frac{1}{(2\pi
i)^{2}}\oint_{C_{z}[0,\infty)}dz\oint_{C_{w}[0,x_{1})}dw\frac{\Gamma(-w)}{\Gamma(z+1)}$
$\displaystyle\frac{\Gamma(z+x_{2}+1)}{\Gamma(x_{1}-w)}\frac{\Gamma(-\frac{z+x_{2}}{2})}{\Gamma(\frac{-x_{1}+w+1}{2})}$
$\displaystyle\frac{z+x_{2}-2k}{z+x_{2}-2k+1}\frac{x_{1}-w-2k}{x_{1}-w-2k-1}\frac{(tu_{1})^{w}(tu_{2})^{z}}{w+z+x_{2}-x_{1}+1}$

Then $K^{{}^{\prime}}_{k}(x_{1},u_{1};x_{2},u_{2})=f(1)$. And $f(t)$
satiesfies such an equality:

$\displaystyle(x_{2}-x_{1}+1)f(t)+tf^{{}^{\prime}}(t)$
$\displaystyle=[t^{x_{2}-x_{1}+1}f(t)]^{{}^{\prime}}\frac{1}{t^{x_{2}-x_{1}}}$
$\displaystyle=\frac{1}{(2\pi
i)^{2}}\oint_{C_{z}[0,\infty)}dz\oint_{C_{w}[0,x_{1})}dw$
$\displaystyle\Bigg{[}(tu_{1})^{w}\frac{\Gamma(\frac{x_{1}-w+1}{2})}{\Gamma(x_{1}-w)}\frac{1}{\Gamma(w+1)}\frac{x_{1}-w-2k}{x_{1}-w-2k-1}\frac{sin(\pi\frac{x_{1}-w+1}{2})}{sin(-\pi
w)}\Bigg{]}$
$\displaystyle\times\Bigg{[}(tu_{2})^{z}\frac{\Gamma(z+x_{2}+1)}{\Gamma(\frac{z+x_{2}}{2}+1)}\frac{1}{\Gamma(z+1)}\frac{z+x_{2}-2k}{z+x_{2}-2k+1}\frac{\pi}{sin(-\pi\frac{z+x_{2}}{2})}\Bigg{]}.$

###### Proof.

The result follows directly from the definition of $f(t)$ and a little bit of
calculus. ∎

######  Remark 3.9.

In the above two theorems, the unbounded contour $C_{z}[0,\infty)$ is obtained
as a limit of $C_{z}[0,n)$ as $n\rightarrow\infty$, and the integration over
it is really an infinite sum of residues inside the contour.

######  Corollary 3.10.

By Theorem 3.7, when $x_{1}=x_{2}=2k$: the first term of $f(t)$ vanishes and
$f(t)$ is given purely by a double contour integral. By restricting the
contours $C_{z}$ and $C_{w}$ so that $Re(w+z+1)>0$, we see that in this
particular case

$\lim_{t\rightarrow 0}tf(t)=0.$

Hence
$K^{{}^{\prime}}(2k,u_{1};2k,u_{2})=f(1)=\int_{0}^{1}[tf(t)]^{{}^{\prime}}dt$.
And (LABEL:eq_ft) simplifies to

$\displaystyle[tf(t)]^{{}^{\prime}}=f(t)+tf(t)$ $\displaystyle=$
$\displaystyle\Bigg{[}\sum_{j=0}^{k-1}(-1)^{k+sj+1}\frac{(tu_{1})^{2j}}{(2k-2j-1)!(2j)!}\frac{1}{\Gamma(j-k+\frac{1}{2})}\Bigg{]}\times\Bigg{[}2\times\sum_{i=0}^{\infty}(-1)^{i+1}\frac{(tu_{2})^{2i}(2i+2k)!}{(2i)!(i+k)!}\Bigg{]}.$

Then integrating w.r.t $t$, we get (for $u_{1},u_{2}\in\mathbb{R}_{\geq 0}$)

$\displaystyle
K^{{}^{\prime}}(2k,u_{1};2k,u_{2})=2^{k+1}\sum_{i=0}^{\infty}\sum_{j=0}^{k-1}$
$\displaystyle(-1)^{i+k}\frac{u_{1}^{2j}}{(2k-2j-1)!\Gamma(j+\frac{1}{2}-k)}\cdot\frac{1}{(2j)!}\cdot\frac{u_{2}^{2i}}{i!}\cdot\frac{1}{2j+2i+1}$
$\displaystyle\times$ $\displaystyle(2i+1)(2i+3)...(2i+2k-1).$

We write out the explicit expression for two simple cases, that $k=1$ and
$k=2$.

When $k=1$:

$K^{{}^{\prime}}(2,u_{1};2,u_{2})=\frac{2}{\sqrt{\pi}}e^{-u_{2}^{2}}$. There’s
only one particle on level $\\{2\\}\times\mathbb{R}_{\geq 0}$ which
corresponds to the entry of the SYT at lower corner, with coordinate
$(n-1,1)$.

When $k=2$:

$K^{{}^{\prime}}(4,u_{1};4,u_{2})=\frac{1}{\sqrt{\pi}}[(1-2u_{1}^{2})(1-2u_{2}^{2})+2]e^{-u_{2}^{2}}$.
There are two particles on level $\\{4\\}\times\mathbb{R}_{\geq 0}$ which
correspond to the two entries of the SYT with coordinates $(n-3,1),(n-2,2)$.

Proof of Theorem 3.1: Theorem 3.5, 3.6 show that the point process
$\pazocal{X}^{{}^{\prime}}_{n}$, $\pazocal{X}^{{}^{\prime}}_{k,n}$ induced
from $n^{\frac{1}{2}}(1-\pazocal{T}(p,m))$, the rescaled entries of PYT of
shape $\Delta_{n}$, $\Delta_{n}\setminus(n-k,k)$ converge weakly to
$\pazocal{X}^{{}^{\prime}}$, $\pazocal{X}^{{}^{\prime}}_{k}$ respectively. In
particular, treating the entries of the PYT of these two shapes as a infinite
random vector, it converges in the sense of finite dimensional distribution to
the corresponding particles in $\pazocal{X}^{{}^{\prime}}_{n}$ or
$\pazocal{X}^{{}^{\prime}}_{k,n}$.

On the other hand, by Lemma 3.3, there’s a coupling of PYT $\pazocal{T}$ and
SYT $T$ that, for any $(p,m)$,

$n^{\frac{1}{2}}[(1-\frac{T(p,m)}{N+1})-(1-\pazocal{T}(p,m))]$

converges to 0 in probability. This implies the convergence in probability of
any finite dimensional sub-vector. Combining the above two results, we obtain
the weak convergence of finite dimensional sub-vector of

$\\{n^{\frac{1}{2}}(1-\frac{T(p,m)}{N+1})\\}$

to the same limit. For the case $\lambda=\Delta_{n}\setminus(n-k,k)$, choosing
the denominator to be $N$ instead of $N+1$ does not affect the limiting
behavior.

(b) is already proven. We will check in Section 3.4 that
$\pazocal{X}^{{}^{\prime}}=\pazocal{X}$, which finishes the proof of (a). ∎

###  3.4. Connection to Anti-symmetric GUE

Now we prove that the distribution of the limiting process of staircase shaped
SYT coinsides with the corner process of aGUE, by matching the correlation
kernel of these two determinantal point process using some explicit
calculation.

######  Theorem 3.11.

The two point process on $\mathbb{Z}_{\geq 0}\times\mathbb{R}_{\geq 0}$, given
by $\pazocal{X}^{{}^{\prime}}$, the limiting process in Theorem 3.5 and
$\pazocal{X}$, the corner process of a $\infty\times\infty$ aGUE, are equal in
distribution.

###### Proof.

Since the two point processes are on the same space, it suffices to check that
the two kernels are equal.

(1). Instead of checking the equivalence of the two kernels in there most
general form, we start by proving the special case, when
$s=t=2k(k\in\mathbb{Z}_{\geq 1})$:

$K^{{}^{\prime}}(2k,u_{1};2k,u_{2})=K(2k,u_{2};2k,u_{1}).$

By Corollary 3.10, when $s=t=2k$, the correlation kernel of limiting rescaled
jumps $\pazocal{X}^{{}^{\prime}}$ is

$\displaystyle
K^{{}^{\prime}}(2k,u_{1};2k,u_{2})=2^{k+1}\sum_{i=0}^{\infty}\sum_{j=0}^{k-1}$
$\displaystyle(-1)^{i+k}\frac{u_{1}^{2j}}{(2k-2j-1)!\Gamma(j+\frac{1}{2}-k)}\frac{1}{(2j)!}\frac{u_{2}^{2i}}{i!}\frac{1}{2j+2i+1}$
$\displaystyle\quad(2i+1)(2i+3)\cdots(2i+2k-1).$

Recall the kernel of aGUE given in Theorem 2.10($s\geq t$):

$K(s,u_{2};t,u_{1})=e^{-u_{2}^{2}}\sum_{l=1}^{t/2}\frac{H_{s-2l}(u_{2})H_{t-2l}(u_{1})}{N_{t-2l}}.$

Step 1:

To show these two are equal, it suffices to show that as polynomials of
$u_{1}^{2}$, their coefficients of each term are equal. Let’s consider the
$u_{1}^{2j}(j=0,1,2,...)$ term. For $K^{{}^{\prime}}$, it has coefficient

(3.7)
$\sum_{i=0}^{\infty}\frac{(-u_{2}^{2})^{i}}{i!}(2i+1)(2i+3)\cdots(2i+2k-1)\frac{1}{2i+2j+1}(-1)^{k}2^{k}\\\
\times\frac{1}{(2k-2j-1)!\Gamma(j+\frac{1}{2}-k)}\frac{2}{(2j)!}.$

As for $K$, we use the explicit expression of Hermite polynomial:

$H_{t-2l}(u_{1})=2^{t-2l}\sum_{r=0}^{t/2-l}(-1)^{r}\frac{(t-2l)!}{2^{2r}r!(t-2l-2r)!}u_{1}^{t-2l-2r}$

Using this expression, when $s=t=2k$, we have that the coefficient of
$u_{1}^{2j}$ of $K$ is

(3.8)
$\frac{2}{(2j)!}e^{-u_{2}^{2}}(-1)^{k-j}2^{2j-2k}\frac{1}{\sqrt{\pi}}\sum_{l=1}^{k-j}(-1)^{l}2^{2l}\frac{1}{(k-l-j)!}H_{2k-2l}(u_{2}).$

It suffices to show (3.7) and (3.8) are equal.

Step 2:

Cancel the term $\frac{2}{(2j)!}$ in both $(\ref{eq_coefficient_1})$ and
$(\ref{eq_coefficient_2})$. First consider coefficient in $K^{{}^{\prime}}$.
Note that $0\leq j\leq k-1$, which implies $\frac{1}{2i+2j+1}$ cancels with
one of the terms among $(2i+1)(2i+3)\cdots(2i+2k-1)$.

Moreover, $(2i+1)\cdots(2i+2k-1)/(2i+2j+1)$ is a polynomial of $i$ and could
be uniquely spanned by $i$, $i(i-1)$,
$i(i-1)(i-2),\cdots,i(i-1)\cdots(i-k+2)$. Canceling these basis with the $i!$
in denominator and then using the Talor series of $e^{-u_{2}^{2}}$,

$\sum_{i=0}^{\infty}\frac{(-u_{2}^{2})^{i}}{i!}(2i+1)\cdots(2i+2k-1)\cdot\frac{1}{2i+2j+1}$

can be written as

$P^{j}_{k-1}(-u_{2}^{2})e^{-u_{2}^{2}},$

s.t

$P_{k-1}^{j}(-u_{2}^{2})=\sum_{m=0}^{k-1}a_{m}^{j}(-u_{2}^{2})^{m}$

is a polynomial of $-u_{2}^{2}$ ($a_{m}^{j}$ are the coefficients depending on
$k,m,j$).

Let $C_{1}(k,j)$ denote

$(-1)^{k}2^{k}\frac{1}{(2k-2j-1)!\Gamma(j+\frac{1}{2}-k)},$

the remaining part of coefficient of $K^{{}^{\prime}}$ of term $u_{1}^{2j}$.
Hence we can write the coefficient $(\ref{eq_coefficient_1})$ as

$P^{j}_{k-1}(-u_{2}^{2})C_{1}(k,j)e^{-u_{2}^{2}}.$

Similarly, we can write the coefficient $(\ref{eq_coefficient_2})$ of $K$ of
term $u_{1}^{2j}$ as

$Q_{k,j}(u_{2})C_{2}(k,j)e^{-u_{2}^{2}},$

s.t

$Q_{k,j}(u_{2})=\sum_{l=1}^{k-j}(-1)^{l}2^{2l}\frac{1}{(k-l-j)!}H_{2k-2l}(u_{2})$

is of the form $\sum_{m=0}^{k-1}b_{m}^{j}u_{2}^{2m}$ ($b_{m}^{j}$ are
coefficients depending on $k,m,j$). And

$C_{2}(k,j)=(-1)^{k-j}2^{2j-2k}\frac{1}{\sqrt{\pi}}$

denotes the remaining part.

In order to show the two coefficients are equal, i.e,

$P^{j}_{k-1}(-u_{2}^{2})C_{1}(k,j)e^{-u_{2}^{2}}=Q_{k,j}(u_{2})C_{2}(k,j)e^{-u_{2}^{2}}$

it suffices to show for any $j=0,1,...,k-1$, the two statements below hold:

I) $a_{m}^{j}=$ (some constant independent of $m$) $\cdot\ b_{m}^{j}(-1)^{m}$
($m=0,1,\cdots,k-1$)

II) The highest degree term of the two coefficients are equal.

We will show I) in step 3, and II) in step 4.

Step 3:

Again we can use the concrete expression of Hermite polynomial:

$H_{2k-2l}(u_{2})=2^{2k-2l}\sum_{r=0}^{k-l}(-1)^{r}\frac{(2k-2l)!}{2^{2r}r!(2k-2l-2r)!}u_{2}^{2k-2l-2r}.$

By this expression, and the definition of $Q_{k,j}$, for $0\leq j\leq k-1$

$\displaystyle(-1)^{m}b_{m}^{j}$ $\displaystyle=$
$\displaystyle(-1)^{m}\sum_{l=1}^{(k-j)\wedge(k-m)}(-1)^{l}2^{2l}\frac{1}{(k-l-j)!}2^{2k-2l}(-1)^{k-m-l}\frac{(2k-2l)!}{2^{2k-2m-2l}(k-m-l)!(2m)!}$
$\displaystyle=$
$\displaystyle(-1)^{k}\frac{2^{2m}}{(2m)!}\sum_{l=1}^{(k-j)\wedge(k-m)}\frac{2^{2l}(2k-2l)!}{(k-l-j)!(k-l-m)!}$

Now let’s turn to study $a_{m}^{j}$. Recall that for a fixed $j\geq 0,k\geq
1$, $a_{m}^{j}$ are given by a group of unique coefficients that satisfy

$(2i+1)(2i+3)\cdots(2i+2k-1)\frac{1}{2i+2j+1}=\sum_{m=0}^{k-1}a_{m}^{j}[i(i-1)\cdots(i-m+1)],$

i.e, represent a polynomial by a group of certain basis. To show that
$a_{m}^{j}=(-1)^{m}b_{m}^{j}$ up to some constant, it suffices to show roots
of

$(2i+1)(2i+3)\cdots(2i+2k-1)\frac{1}{2i+2j+1}$

all are roots of

$\sum_{m=0}^{k-1}(-1)^{m}b_{m}^{j}[i(i-1)\cdots(i-m+1)].$

Roots of $(2i+1)(2i+3)\cdots(2i+2k-1)\frac{1}{2i+2j+1}$ are easy to see:
$i=-l-\frac{1}{2}$ for $l=0,1,...,k-1$ except $l=j$.

Denote

$\sum_{m=0}^{k-1}(-1)^{m}b_{m}^{j}[i(i-1)\cdots(i-m+1)]$

by $F(k,j)(i)$, and recall that $0\leq j\leq k-1$. We have

$\displaystyle
F(k,j)(i)=(-1)^{k}\Bigg{[}\sum_{m=0}^{j-1}\sum_{l=1}^{k-j}A_{m}B_{l,m}^{k,j}+\sum_{m=j}^{k-1}\sum_{l=1}^{k-m}A_{m}B_{l,m}^{k,j}\Bigg{]},\;\;\text{such
that}$ $\displaystyle
A_{m}=i(i-1)\cdots(i-m+1)\frac{2^{2m}}{(2m)!},B_{l,m}^{k,j}=\frac{2^{2l}(2k-2l)!}{(k-l-j)!(k-l-m)!}.$

We start with the base case $k=j+1$, in which $i=-l-\frac{1}{2}$,
$l=0,1,...,j-1$.

######  Lemma 3.12.

Let $j\in\mathbb{Z}_{\geq 1}$.
$\sum_{m=0}^{j}i(i-1)\cdots(i-m+1)\frac{2^{2m}}{(2m)!}\frac{1}{(j-m)!}=0$, for
all $i=-l-\frac{1}{2}$, $l=0,1,...,j-1$.

###### Proof.

$\displaystyle\sum_{m=0}^{j}(-l-\frac{1}{2})(-l-\frac{3}{2})\cdots(-l+\frac{1}{2}-m)\frac{2^{2m}}{(2m)!}\frac{1}{(j-m)!}$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{j}(-1)^{m}\frac{(2l+1)(2l+3)\cdots(2l+2m-1)}{2^{m}}\frac{2^{2m}}{(2m)!}\frac{1}{(j-m)!}$
$\displaystyle=$ $\displaystyle
C(l)\times\sum_{m=0}^{j}\frac{1}{(j-m)!}\frac{1}{m!}(-1)^{m}(2m+1)\cdots(2m+2l-1)$
$\displaystyle\quad(C(l)\;\;\text{is a constant depending on}\;\;l).$

Again expanding $(2m+1)\cdots(2m+2k-1)$ by basis $m$,
$m(m-1),\cdots,m(m-1)\cdots(m-j+1)$ but for $l=0,1,...,j-1$, each basis cancel
with $\frac{1}{m!}$, and then we have linear span of a group of binomial
expansion of $(1+(-1))^{n}$($n$ is some positive integer), hence it is 0. On
the other hand, for $l=j$, one of the expansion is $(1+(-1))^{0}$, hence it is
not 0. ∎

For general $k\geq j+1$, we have the following induction result.

######  Lemma 3.13.

For $i=-l-\frac{1}{2}$, $l=0,1,...,k-1$,

$F(k,j)(i)=0\;\;\text{implies}\;\;F(k+1,j)(i)=0.$

###### Proof.

By definition we have for any $i$,

$B_{l+1,m}^{k+1,j}=4B_{l,m}^{k,j}.$

Hence,

$\displaystyle F(k+1,j)$ $\displaystyle=$ $\displaystyle(-1)\cdot
4F(k,j)+(-1)\cdot\sum_{m=0}^{k}A_{m}B_{1,m}^{k+1,j}$ $\displaystyle=$
$\displaystyle(-1)\cdot
4F(k,j)+(-1)\sum_{m=0}^{k}i(i-1)\cdots(i-m+1)\frac{2^{2m}}{(2m)!}\frac{1}{(k-m)!}\cdot\frac{2^{2}(2k)!}{(k-j)!}.$

The second term in the last line equals to 0, by Lemma 3.12. ∎

By the two lemma above, $F(k,j)(i)=0$ for $i=-l-\frac{1}{2}$, $l=0,1,...,j-1$,
$0\leq j\leq k-1$. If $j+1\leq l\leq k-1$(which implies $k\geq j+2)$, it
suffices to consider the case $i=-(k-1)-\frac{1}{2}$, i.e, to verify

$F(k,j)(-k+\frac{1}{2})=0$

for $k\geq j+2$. The general cases again follow from Lemma 3.12.

By definition,

$\displaystyle F(k,j)(i)$ $\displaystyle=$
$\displaystyle(-1)^{k}\Bigg{[}\sum_{m=0}^{j-1}\sum_{l=1}^{k-j}A_{m}B_{l,m}^{k,j}+\sum_{m=j}^{k-1}\sum_{l=1}^{k-m}A_{m}B_{l,m}^{k,j}\Bigg{]}$
$\displaystyle=$
$\displaystyle(-1)^{k}\sum_{l=1}^{k-j}\sum_{m=1}^{k-l}A_{m}B_{l,m}^{k,j}.$

Let $N=k-l$, omit the constant term $(-1)^{k}$ and plug in $i=-k+\frac{1}{2}$,
the above expression is written as

$\displaystyle\sum_{N=j}^{k-1}\Bigg{[}\sum_{m=0}^{N}(-\frac{1}{2})^{m}(2k-1)(2k+1)\cdots(2k+2m-3)\frac{2^{2m}}{(2m)!}\frac{1}{(N-m)!}\Bigg{]}\frac{2^{2(k-N)}(2N)!}{(N-j)!}$
$\displaystyle=$ $\displaystyle
2^{2k}\sum_{N=j}^{k-1}\Bigg{[}\sum_{m=0}^{N}(-1)^{m}(2k-1)(2k+1)\cdots(2k+2m-3)\frac{2^{2m}}{(2m)!}\frac{1}{(N-m)!}\Bigg{]}\frac{(2N)!}{(N-j)!\cdot
2^{2N}}$ $\displaystyle=$ $\displaystyle
2^{2k}\sum_{N=j}^{k-1}\Bigg{[}\sum_{m=0}^{N}(-1)^{m}\frac{(2k-1)(2k+1)\cdots(2k+2m-3)}{1\cdot
3\cdot
5\cdots(2m-1)}\frac{2^{m}}{m!}\frac{1}{(N-m)!}\Bigg{]}\frac{(2N)!}{(N-j)!\cdot
2^{2N}}.$

Recall we are considering the case $0\leq j\leq k-2$. Hence by another change
of variable $k-1\rightarrow k$, and removing the constant term
$\frac{2^{2k}}{(2k-1)!!}$, the above expression becomes

$\displaystyle\sum_{N=j}^{k}\Bigg{[}\sum_{m=0}^{N}(-1)^{m}(2m+1)(2m+3)\cdots(2m+2k-3)(2m+2k-1)\frac{1}{m!(N-m)!}\Bigg{]}\frac{(2N)!}{(N-j)!\cdot
2^{2N}},$

for $0\leq j\leq k-1$.

Once again expand $(2m+1)(2m+3)\cdots(2m+2k-1)$ by the basis as above.

$\sum_{m=0}^{N}(-1)^{m}\frac{1}{m!(N-m)!}m(m-1)\cdots(m-l+1)=0$

for $0\leq l\leq N-1$, since this is again a binomial sum of 1 and -1. As for
$N+1\leq l\leq k$,

$(-1)^{m}\frac{1}{m!(N-m)!}m(m-1)\cdots(m-l+1)=0$

for any $m=0,1,...,N$. Therefore, it finally remains to show the following
result.

######  Lemma 3.14.

Fix $j,k\in\mathbb{Z}$, $0\leq j\leq k-1$. Treat $(2m+1)(2m+3)\cdots(2m+2k+1)$
as a polynomial of $m$ and expand it by basis $m$, $m(m-1)$, $\cdots$,
$m(m-1)\cdots(m-k+1)$. Denote the coefficient of $m(m-1)\cdots(m-N+1)$ as
$C^{N}_{k}$. Then

$\sum_{N=j}^{k}(-1)^{N}C^{N}_{k}\frac{(2N)!}{(N-j)!\cdot 2^{2N}}=0.$

To prove the lemma, we first provide two inductive results of $C^{N}_{k}$
which follow directly from the definition.

######  Proposition 3.15.

For $N=k$,

$C^{k}_{k}=2^{k}.$

For $0\leq N\leq k-1$,

$\displaystyle C^{N}_{k}$ $\displaystyle=2^{N}\cdot e_{k-N}(1,3,...,2k-1)$
$\displaystyle+C^{k}_{k}\cdot e_{k-N}(0,1,...,k-1)(-1)^{k-N-1}$
$\displaystyle+C^{k}_{k-1}\cdot e_{k-N-1}(0,1,...,k-2)(-1)^{k-N-2}$
$\displaystyle+...+C^{N+2}_{k}\cdot e_{2}(0,1,...,N+1)(-1)$
$\displaystyle+C^{N+1}_{k}\cdot e_{1}(0,1,...,N)(-1)^{0},$

where $e_{j}(x_{1},...,x_{m})$ denotes the $j^{\text{th}}$ elementary
symmetric polynomial of variables $x_{1},...,x_{m}$.

######  Proposition 3.16.

Define $C_{k}^{N}$ to be 0 for $N<0$. Then

$C^{N}_{k+1}=(2N+2k+1)C^{N}_{k}+2C^{N-1}_{k}.$

###### Proof.

By definition,

$\sum_{N=0}^{k}C^{N}_{k}m(m-1)\cdots(m-N+1)=(2m+1)\cdots(2m+2k-1).$
$\sum_{N=0}^{k+1}C^{N}_{k+1}m(m-1)\cdots(m-N+1)=(2m+1)\cdots(2m+2k-1)(2m+2k+1).$

Multiplying the first equality by $(2m+2k+1)$, we have

$\displaystyle(2m+1)(2m+3)\cdots(2m+2k-1)(2m+2k+1)$ $\displaystyle=$
$\displaystyle\sum_{N=0}^{k}2C^{N}_{k}m(m-1)\cdots(m-N+1)(m-N)$
$\displaystyle+\sum_{N=0}^{k}(2N+2K+1)C^{N}_{k}m(m-1)\cdots(m-N+1).$

Comparing the coefficients of the last two equality gives the statement. ∎

Proof of Lemma 3.14:

For $0\leq j\leq k-1$, use $(k,j)$ to denote the authentically of the lemma
for a pair of fixed parameter $k$ and $j$. One can use Proposition 3.15 to
check $(j+1,j)$ and $(j+2,j)$ for any $j\geq 0$. Then we can assume that
$k\geq j+2$, and we claim that $(k+1,j)$ is implied by $(k,j+1)$, $(k,j)$ and
$(k,j-1)$(ignore $(k,j-1)$ if $j=0$). This inductive claim clearly covers all
possible $(k,j)$.

Now suppose we get $(k,j+1)$, $(k,j)$ and $(k,j-1)$. Use Proposition 3.16, and
write $2N+2k+1$ as $2(N-j)+(2j+2k+1)$,

$\displaystyle\sum_{N=j}^{k+1}(-1)^{N}C^{N}_{k+1}\frac{(2N)!}{(N-j)!\cdot
2^{2N}}$ $\displaystyle=$ $\displaystyle
2\sum_{N=j+1}^{k}(-1)^{N}C^{N}_{k}\frac{(2N)!}{(N-(j+1))!\cdot 2^{2N}}$
$\displaystyle+$
$\displaystyle(2j+2k+1)\sum_{N=j}^{k}(-1)^{N}C^{N}_{k}\frac{(2N)!}{(N-j)!\cdot
2^{2N}}$ $\displaystyle+$ $\displaystyle
2\sum_{N=j}^{k+1}(-1)^{N}C^{N-1}_{k}\frac{(2N)!}{(N-j)!\cdot 2^{2N}}.$

The first and second line on the right side of last equality are 0 by (k,j+1)
and (k,j). Then it remains to show

(3.9) $\sum_{N=j}^{k+1}(-1)^{N}C^{N-1}_{k}\frac{(2N)!}{(N-j)!\cdot 2^{2N}}=0$

for $k\geq j+2$.

$\displaystyle(\ref{eq_lastcheck})=-\frac{1}{2}\sum_{N-1=j-1}^{k}(-1)^{N-1}C^{N-1}_{k}\frac{(2N-2)!}{(N-j)!2^{2N-2}}(2N-1)N.$

Write $(2N-1)N$ as $2(N-j)(N-j-1)+(4j+1)(N-j)+j(6j+3)$. Then

$\displaystyle\cdot(\ref{eq_lastcheck})=\ $ $\displaystyle(-\frac{1}{2})\cdot
2\sum_{N-1=j+1}^{k}(-1)^{N-1}C^{N-1}_{k}\frac{(2N-2)!}{[(N-1)-(j+1)]!\cdot
2^{2N-2}}$ $\displaystyle+$
$\displaystyle(-\frac{1}{2})(4j+1)\sum_{N-1=j}^{k}(-1)^{N-1}C^{N-1}_{k}\frac{(2N-2)!}{[(N-1)-j]!\cdot
2^{2N-2}}$ $\displaystyle+$
$\displaystyle(-\frac{1}{2})j(6j+3)\sum_{N-1=j-1}^{k}(-1)^{N-1}C^{N-1}_{k}\frac{(2N-2)!}{[(N-1)-(j-1)]!\cdot
2^{2N-2}}.$

The three lines on the right side of the last equality are 0 by $(k,j+1)$,
$(k,j)$ and $(k,j-1)$ respectively(when $j=0$ the coefficient $j(6j+3)$ of the
last line is 0).This finishes the proof of Lemma 3.14.∎

Step 4:

The final thing that we need to show is that the highest degree terms of
$P^{j}_{k-1}(-u_{2}^{2})C_{1}(k,j)$ and $Q_{k,j}(u_{2})C_{2}(k,j)$ are equal,
i.e,

$a_{k-1}^{j}C_{1}(k,j)=(-1)^{k-1}b_{k-1}^{j}C_{2}(k,j)(\forall\ 0\leq j\leq
k-1).$

By definition of $a_{m}^{j}$, one can see that $a_{k-1}^{j}=2^{k-1}$. Hence

$\text{LHS}=2^{k-1}(-1)^{k}2^{k}\frac{1}{(2k-2j-1)!\Gamma(j+\frac{1}{2}-k)}.$

On the other hand,

$\text{RHS}=(-1)^{k-1}\Bigg{[}(-1)\frac{2^{2}\cdot
2^{2k-2}}{(k-1-j)!}\Bigg{]}\cdot\Bigg{[}(-1)^{k-j}2^{2j-2k}\frac{1}{\sqrt{\pi}}\Bigg{]}.$

One can check directly that LHS=RHS. Done.

(2). After showing the equality of this special case(which corresponds to the
equilibrium in law of these two point processes on each level
$\\{2k\\}\times\mathbb{R}_{\geq 0}(k=1,2,...)$, the equality in law in general
on $\mathbb{Z}_{\geq 0}\times\mathbb{R}_{\geq 0}$ should follow from the
conditional uniformity of PYT and aGUE (under interlacing restriction). The
conditional uniformity of aGUE is stated in Proposition 2.9 (d), and the same
property holds for the rescaled PYT, proven in Lemma 3.17.

######  Lemma 3.17.

As a point process on $\mathbb{Z}_{\geq 0}\times\mathbb{R}_{\geq 0}$,
$\pazocal{X}^{{}^{\prime}}$ satisfies the conditional uniformity under
interlacing restriction, which is the same property as stated in Proposition
2.9.

###### Proof.

This can be seen directly from the construction of
$\pazocal{X}^{{}^{\prime}}$. We obtain $\pazocal{X}_{n}^{{}^{\prime}}$ by
rotating and rescaling the entries of a staircase shape PYT, such that the
rotated PYT has the conditional uniformity under interlacing restriction by
definition of PYT, and the rescaling preserves this property since it has the
same action on all the entries.

$\pazocal{X}^{{}^{\prime}}$ is the limiting process of
$\pazocal{X}_{n}^{{}^{\prime}}$, satisfying the same interlacing condition. It
remains to show $\pazocal{X}^{{}^{\prime}}$ inherits the conditional
uniformity of $\pazocal{X}^{{}^{\prime}}_{n}$. Consider the joint density
$P_{n}(y,z)$ of some random vector $(y,z)$ such that
$y\in\mathbb{R}^{k},z\in\mathbb{R}^{m}$. If $P_{n}(y|z)$ is uniform on some
connected open set $E\in\mathbb{R}^{k}$(one should think $E$ as the
interlacing region) and $P_{n}(y,z)\rightarrow P(y,z)$ uniformly on each
compact sets, then in particular $P(y|z)$ should still be uniform on $E$.

The joint density of particles of $\pazocal{X}_{n}^{{}^{\prime}}$ on the first
$M$ levels($M$ is any positive finite integer) is given by a determinant of
its correlation kernel, which converges uniformly on compact sets by the proof
of Theorem 3.5. Hence by the above general argument, the conditional
uniformity of $\pazocal{X}_{n}^{{}^{\prime}}$ passes through its limit. ∎

This finishes the proof of Theorem 3.11. ∎

##  4\. Results of Spacings

We begin this section by introducing a point process on circle, whose
distribution is invariant under rotation. We give two definitions of spacing,
the definition of waiting time of this point process, which generalize spacing
of RSN defined in Section 1, and prove that their distribution functions
satisfy certain explicit relation, which are discrete versions of (1.2) and
(1.3). After that we show that we can induce a point process of this kind from
finite RSN, and therefore the same relation also hold for the spacing of RSN,
which leads to Theorem 1.11 and 1.14 after a limit transition. Finally, we
recall the well-known Edelman-Greene bijection between SYT and sorting
network, and use it as a bridge to reach Theorem 1.5 and 1.18, based on the
results of Theorem 3.1.

###  4.1. Relation With Waiting Time

Consider a random point process on a discrete circle of length
$K\in\mathbb{Z}_{\geq 2}$. The holes where the particles could locate are
distributed homogeneously on the circle, all of which have distance 1 to their
neighbors on the two sides. We index the holes by 1,2,…,$K$ in the clockwise
order. Assume that the distribution of the point process is invariant under
rotation.

Figure 6. Two configurations of point process on discrete circle of length
$K=8$, described in Example 4.1. The black disk means a hole with a particle,
and the white disk means a hole without particle. The number near the disk
gives the index of the hole.

Now we define the spacing of this point process, which are random variables
with positive integer values. We always start at hole $K$. Define the waiting
time $W\in\\{1,2,...,K\\}$ to be the smallest index $l$ where there’s a
particle at hole $l$. Let $\mathrm{Sp}_{1}\in\\{1,2,...,K\\}$ to be the
distance(the length of the arc crossing hole $K$) of the two closest particles
on the circle, one on the right of hole K and the other on or on the left of
hole $K$. Next we define $\mathrm{Sp}_{2}\in\\{1,2,...,K\\}$ to be the
clockwise distance from hole $K$ to the next hole with a particle,
conditioning on the event that there’s a particle at hole $K$. We assume that
this event has positive probability to avoid dealing with the trivial case.

Denote $f_{1}(L)$ as $\mathbb{P}\left[\mathrm{Sp}_{1}=L\right]$, and
$f_{2}(L),g(L)$ as the distribution function of $\mathrm{Sp}_{2}$ and the
waiting time respectively in the same way. Let $\Delta$ stand for the
difference operator: $\Delta g(L):=g(L+1)-g(L)$. Finally, let

$M=\mathbb{P}\left[there\ is\ a\ particle\ at\ hole\ K\right].$

######  Example 4.1.

Figure 8 gives two possible configurations of a point process on discrete
circle of length $K=8$. On the first configuration, $\mathrm{Sp}_{1}=4$, $W=1$
and $\mathrm{Sp}_{2}$ is not defined, since there’s no particle at hole $8$.
On the second configuration, $\mathrm{Sp}_{1}=W=\mathrm{Sp}_{2}=3$.

Since one of them can be obtained from another by rotation, these two
configurations have the same probability to occur in the point process.

######  Proposition 4.2.

Define $f_{1}(L),f_{2}(L),g(L)$ and constant $M$ as above. Then we have the
following relations:

(4.1) $-\Delta g(L)\cdot L=f_{1}(L).$

(4.2) $-\Delta g(L)=M\cdot f_{2}(L).$

We first prove a key lemma of this point process.

######  Lemma 4.3.

$-\Delta g(L)=\mathbb{P}\left[there^{\prime}s\ a\ particle\ at\ hole\ K,\
following\ spacing=L\right]$.

###### Proof.

Conditioning on the configuration of particles on the whole circle modulo
rotation, $g(L)$ counts the probability of the event that a particle falls on
hole $L$, and for this to happen, there should be no particle at hole
$1,2,...,L-1$ and therefore the distance of the next particle on the left of
hole $L$ to hole $L$ must be $\geq L$. If we make a rotation clockwise, such
that the next particle on the left of hole $L$ is switched to hole K, what we
get is a configuration that there’s a particle at hole K, and the following
spacing $\geq L$.

Since the point process is invariant under rotation, the configurations before
and after rotation discussed above have same probability. This shows that

$g(L)=\mathbb{P}\left[there^{\prime}s\ a\ particle\ at\ hole\ K,\ following\
spacing\geq L\right].$

Then the lemma follows by taking $-\Delta g(L)=g(L)-g(L+1)$.

∎

Proof of proposition 4.2:

(4.2) is a direct corollary of Lemma 4.3: just recall
$f_{2}(L)=\mathbb{P}\left[\mathrm{Sp}_{2}=L|there^{\prime}s\ a\ particle\ at\
hole\ K\right]$ and use the Bayesian law.

To prove (4.1), note that by definition, conditioning on the configuration on
the whole circle(modulo rotation), $f_{1}(L)$ counts the probability that hole
K falls between two neighboring particles with distance $L$.

By the invariance under rotation, this conditional probability is equal to

$\displaystyle\frac{total\ arc\ length\ of\ length\ L\ spacing}{K}$
$\displaystyle=$ $\displaystyle\frac{(\\#spacing\ with\ length\ L)\cdot
L}{K}.$

Again by rotational invariance, conditioning on certain configuration(modulo
rotation),

$\displaystyle\frac{\\#spacing\ with\ length\ L}{K}$ $\displaystyle=$
$\displaystyle\frac{\\#particles\ with\ following\ spacing=L}{K}$
$\displaystyle=$ $\displaystyle\mathbb{P}\left[there^{\prime}s\ a\ particle\
at\ hole\ K,\ following\ spacing=L\right].$

Therefore by removing the conditioning on both sides, we have

$f_{1}(L)=\mathbb{P}\left[there^{\prime}s\ a\ particle\ at\ hole\ K,\
following\ spacing=L\right]\cdot L.$

Identity (4.1) then follows from Lemma 4.3.∎

######  Remark 4.4.

The above proposition might be generalized to continuous setting. Consider a
point process on real line that is translation invariant (one may assume that
the correlation function $\\{\rho_{k}\\}_{k\geq 1}$ exists, to make this
assumption precise). Pick an arbitrary fixed position on the real line, and
define $\mathrm{Sp}_{1},\mathrm{Sp}_{2}$ and $W$ in a similar way as in the
discrete setting. Let $f_{1}(x),f_{2}(x)$ and $g(x)$ denote the density of the
above three random variables respectively, then we believe that under some
extra technical assumption, or if the continuous point process is the proper
rescaled limit of some discrete process that we studied above, one should be
able to get the identity of the following kind:

$\displaystyle-xg^{{}^{\prime}}(x)=f_{1}(x).$
$\displaystyle-g^{{}^{\prime}}(x)=M\cdot f_{2}(x).$

where $M$ is equal to the first correlation function $\rho_{1}$, which is a
constant by the invariance of the point process under rotation.

For the special case where the continuous process is the rescaled limit of the
discrete process induced from finite RSN, the limiting relations are stated in
Theorem 1.11 and 1.14 and proven below.

Now we specify the above result to uniformly random sorting network. Consider
the space of all finite sorting networks of a fixed rank $n$, denoted as
$\Omega_{n}$. Proposition 1.6 defines an automorphism in $\Omega_{n}$:

$\omega=(s_{1},...,s_{N})\mapsto(s_{2},...,s_{N},n-s_{1}).$

Then $\Omega_{n}$ can be classified into finite equivalence classes, such that
two sorting networks are in the same class iff one can obtain one from another
by translation. Pick an arbitrary element $\omega=(s_{1},s_{2},...,s_{N})$ in
each equivalence class, we can write that equivalence class as a periodic
sequence of the form

$[...,n-s_{N-1},n-s_{N},s_{1},s_{2},...s_{N},n-s_{1},n-s_{2},...n-s_{N},s_{1},s_{2},...]$

or

$[s_{1},s_{2},...s_{N},n-s_{1},n-s_{2},...n-s_{N}]$

in short with length $2N$. We still denote the elements $s_{i}$ in the
periodic sequence as _swaps_. Then each sorting network is denoted as a length
$N$ segment $(s_{1},s_{2},...,s_{N})$ in this periodic sequence, and we use
$s_{l}=k$ to denote that the $l^{\text{th}}$ swap of the corresponding sorting
network is $k$. Note that some segments might appear in more than one position
in a periodic sequence, but the multiplicities of the segments in the same
periodic sequence must be the same.

######  Example 4.5.

When $n=4$, $\Omega_{4}$ can be classified into 2 periodic sequences:

$\displaystyle[1\ 2\ 3\ 2\ 1\ 2\ 3\ 2\ 1\ 2\ 3\ 2]$ $\displaystyle and\quad$
$\displaystyle[1\ 3\ 2\ 1\ 3\ 2\ 3\ 1\ 2\ 3\ 1\ 2],$

containing 4 and 12 sorting networks, and the multiplicities of each sorting
network in the periodic sequences are 3 and 1 respectively.

Figure 7. One of the two periodic sequences in $\Omega_{4}$, as shown in
Example 4.5. The black dots highlight the swaps on row 1, such that all the
spacing are of length 4.

In this text we are studying the distance between two neighboring swaps on the
same fixed row $k$, which we define as spacing of RSN in Definition 1.9 and
1.12. Once we fix the row $k$, we induce a point process on discrete circle of
length $K=2N$ from the sorting network of size $n$, by taking the periodic
sequence $[s_{1},s_{2},...s_{N},n-s_{1},n-s_{2},...n-s_{N}]$ where the sorting
network $(s_{1},s_{2},...,s_{N})$ lives in, corresponding the $l^{\text{th}}$
swap in the periodic sequence with the hole $l$ on the circle, and placing a
particle at hole $l$ whenever the $l^{\text{th}}$ swap is equal to $k$.

Recall that as shown in Example 4.5, multiplicities of sorting network in
different periodic sequences might be different, but multiplicities of SNs in
the same periodic sequence are always the same. Therefore, conditioning on
certain periodic sequence, picking a SN uniformly, is the same as uniformly
rotating the periodic sequence and picking the segment formed by the first $N$
swaps. In particular, this shows that the point process induced from RSN is
invariant under rotation. Moreover, by setting $a=0$, the spacing of RSN in
Definition 1.9 and 1.12 coincide with $\mathrm{Sp}_{1}$ and $\mathrm{Sp}_{2}$
defined at the beginning of this section respectively, and the first swapping
time of RSN is the same as the waiting time $W$.

For a uniformly random sorting network of size $n$, let
$f_{1}^{{}^{\prime}}(L)=\mathbb{P}\left[\mathrm{Sp}_{k,n}=L\right]$,
$f_{2}^{{}^{\prime}}(L)=\mathbb{P}\left[\widehat{\mathrm{Sp}}_{k,n}=L\right]$,
$g^{{}^{\prime}}(L)=\mathbb{P}\left[T_{FS,n}(k)=L\right]$, and let

$M=\mathbb{P}\left[s_{1}=k\right]=\frac{|\\{\omega\in\Omega_{n}:s_{1}=k\\}|}{|\Omega_{n}|}.$

Then by the above arguments, the spacings of RSN satisfy the following
proposition, which is a restatement of Proposition 4.2.

######  Proposition 4.6.

Define $f_{1}^{{}^{\prime}}(L),f_{2}^{{}^{\prime}}(L),g^{{}^{\prime}}(L)$ and
constant $M$ as above. Then we have the following relations:

(4.3) $-\Delta g^{{}^{\prime}}(L)\cdot L=f_{1}^{{}^{\prime}}(L).$ (4.4)
$-\Delta g^{{}^{\prime}}(L)=M\cdot f_{2}^{{}^{\prime}}(L).$

Now we give the proof of Theorem 1.11 and 1.14, which state the explicit
correspondence of the rescaled limiting spacing law and the first swapping
law.

Proof of Theorem 1.11:

We first consider a finite RSN of size $n$. Let
$\Lambda_{-}\in\mathbb{Z}_{\geq 0}$, $\Lambda_{+}\in\mathbb{Z}_{>0}$ denote
the distance between $X$, $Y$ and $a$ respectively, then
$\Lambda_{-},\Lambda_{+}\geq 0$, and
$\Lambda_{-}+\Lambda_{+}=\mathrm{Sp}_{k,n}$. Note that
$\Lambda_{+}\overset{d}{=}\mathbf{T}_{FS,n}(k)$.

By translation invariance of sorting networks,

$\mathbb{P}\left[\Lambda_{-}\geq a,\Lambda_{+}\geq
b\right]=\mathbb{P}\left[\Lambda_{-}\geq a^{{}^{\prime}},\Lambda_{+}\geq
b^{{}^{\prime}}\right]$

if $a,b,a^{{}^{\prime}},b^{{}^{\prime}}\in\mathbb{Z}_{\geq 0}$,
$a+b=a^{{}^{\prime}}+b^{{}^{\prime}}$.

For any $x,y\geq 0$, any $a\in\mathbb{Z}$,

$\displaystyle\mathbb{P}\left[2\frac{\Lambda_{-}}{n^{\frac{3}{2}}}>x,2\frac{\Lambda_{+}}{n^{\frac{3}{2}}}>y\right]$
$\displaystyle=$ $\displaystyle\mathbb{P}\left[\Lambda_{-}\geq\lceil
x\frac{n^{\frac{3}{2}}}{2}\rceil,\Lambda_{+}\geq\lceil
y\frac{n^{\frac{3}{2}}}{2}\rceil\right]$ $\displaystyle=$
$\displaystyle\mathbb{P}\left[\Lambda_{-}\geq 0,\Lambda_{+}\geq\lceil
x\frac{n^{\frac{3}{2}}}{2}\rceil+\lceil
y\frac{n^{\frac{3}{2}}}{2}\rceil\right]$ $\displaystyle=$
$\displaystyle\mathbb{P}\left[2\frac{\Lambda_{+}}{n^{\frac{3}{2}}}\geq\frac{\lceil
x\frac{n^{\frac{3}{2}}}{2}\rceil+\lceil
y\frac{n^{\frac{3}{2}}}{2}\rceil}{(\frac{n^{\frac{3}{2}}}{2})}\right].$

By translation invariance property and Theorem 1.5(which we will prove in
Section 4.4),

$\mathbb{P}\left[2\frac{\Lambda_{+}}{n^{\frac{3}{2}}}\geq\frac{\lceil
x\frac{n^{\frac{3}{2}}}{2}\rceil+\lceil
y\frac{n^{\frac{3}{2}}}{2}\rceil}{(\frac{n^{\frac{3}{2}}}{2})}\right]\rightarrow
1-F(x+y)$

as $n\rightarrow\infty$, where $F$ is the distribution function of
$\mathbf{T}_{FS}(k)$.

The result above show that
$(2\frac{\Lambda_{-}}{n^{\frac{3}{2}}},2\frac{\Lambda_{+}}{n^{\frac{3}{2}}})$
converges in law to some random vector $(Z_{1},Z_{2})$ with a joint density,
let’s denote it by $f(x,y)(x,y\geq 0)$. Again we use $g(x)$ to denote the
density of $Z_{1}+Z_{2}$, which is the limiting spacing. We already have
$G(a,b):=\mathbb{P}\left[Z_{1}>a,Z_{2}>b\right]=1-F(a+b)$(any $a,b\geq 0)$,
then

$\displaystyle\mathbb{P}\left[Z_{1}+Z_{2}>b\right]$
$\displaystyle=\int_{0}^{b}\int_{b-y}^{\infty}f(x,y)dxdy+\int_{b}^{\infty}\int_{0}^{\infty}f(x,y)dxdy$
$\displaystyle=\int_{0}^{b}-G_{y}(b-y,y)dy+\int_{0}^{\infty}-G_{x}(x,b)dx$
$\displaystyle=\int_{0}^{b}F^{{}^{\prime}}(b)dy+\int_{0}^{\infty}F^{{}^{\prime}}(x+b)dx$
$\displaystyle=bF^{{}^{\prime}}(b)+\int_{b}^{\infty}F^{{}^{\prime}}(x)dx\Rightarrow
g(b)=-bF^{{}^{\prime\prime}}(b).\qed$

Proof of Theorem 1.14:

Since the row $k$ of RSN is fixed, in the proof we denote the density function
of $\mathbf{T}_{FS}(k)$ by $g(x)$ and omit the subscript $k$.

Recall

$M=\frac{\\#SNs:s_{1}=k}{\\#SNs}.$

By Edelman-Greene bijection, it suffices to count the number of SYTs with the
corresponding shape. An explicit formula of the number of SYTs of a given
shape $\lambda$ is a standard fact, which can be found in, say, [AHRV, Theorem
7]. Then we have

$\displaystyle\frac{1}{M}=$ $\displaystyle\binom{n}{2}\frac{\prod_{1\leq
j<k}(2k-2j)}{\prod_{1\leq j<k}(2k-1-2j)}\frac{\prod_{k<i\leq
n}(2i-2k)}{\prod_{k<i\leq n}(2i-2k-1)}$ $\displaystyle=$
$\displaystyle\binom{n}{2}\frac{(2n-2-2k)!!}{(2n-1-2k)!!}\frac{(2k-2)!!}{(2k-1)!!}\sim\frac{\sqrt{\pi}}{4}n^{\frac{3}{2}}\frac{(2k-2)!!}{(2k-1)!!}.$

Consider $0\leq a<b\leq\infty$. By Proposition 4.6 (ii),

$-[g^{{}^{\prime}}(L+1)-g^{{}^{\prime}}(L)]=M\cdot f_{2}^{{}^{\prime}}(L).$

Take the summation on both sides over $L\in[\lfloor
a\frac{n^{\frac{3}{2}}}{2}\rfloor,\lfloor b\frac{n^{\frac{3}{2}}}{2}\rfloor]$,
we get

(4.5) $\sum_{\lfloor a\frac{n^{\frac{3}{2}}}{2}\rfloor}^{\lfloor
b\frac{n^{\frac{3}{2}}}{2}\rfloor}f_{2}^{{}^{\prime}}(L)=-\frac{1}{M}[g^{{}^{\prime}}(\lfloor
b\frac{n^{\frac{3}{2}}}{2}\rfloor)-g^{{}^{\prime}}(\lfloor
a\frac{n^{\frac{3}{2}}}{2}\rfloor)].$

By Theorem 1.5, when $n$ is large,

$(\ref{eq_summation})\approx-\frac{1}{M}\frac{2}{n^{\frac{3}{2}}}[g(b)-g(a)],$

which converges to
$-\frac{\sqrt{\pi}}{2}\frac{(2k-2)!!}{(2k-1)!!}[g(b)-g(a)]$. This shows that
$\frac{2}{n^{\frac{3}{2}}}\widehat{\mathrm{Sp}}_{k,n}$ converges in
distribution as $n\rightarrow\infty$. Let $\widehat{Z}_{k}$ denote the random
variable with its limiting distribution, then

$\mathbb{P}\left[\widehat{Z}_{k}\in[a,b]\right]=-\frac{\sqrt{\pi}}{2}\frac{(2k-2)!!}{(2k-1)!!}[g(b)-g(a)].$

After taking derivative on both sides we get the result density. ∎

######  Remark 4.7.

Based on Theorem 1.5 and Proposition 4.6 (i), Theorem 1.11 can also be proven
using a similar argument.

######  Remark 4.8.

One might be able to obtain Theorem 1.14 by a direct check of results in
Theorem 1.5 and Theorem 1.18(which is proven in Section 4.4 in an alternative
way as a consequence of Theorem 3.6), but the calculation turns out to be
really tedious even for $k=2$. We choose to prove Theorem 1.14 in the present
way because we find this approach more inspiring.

###  4.2. Edelman-Greene Bijection

There is a well-known one-one correspondence between a sorting network of size
$n$, and a standard Young tableaux of shape $\Delta_{n}$. In particular, a
uniform probability measure on $\Omega_{n}$ can be directly transferred to a
uniform probability measure on the space of all SYT of shape $\Delta_{n}$.

Moreover, given any SYT of shape $\Delta_{n}$, denoted by $T$, there is an
algorithm to generate the sorting network corresponding to $T$ explicitly.
This result is established in [EG].

This bijection and the algorithm is one of the stepping stones in our limiting
theorem of spacing. Now we describe how the algorithm works.

1\. Given the SYT $T$, find the coordinate $(n-l,l)(l=1,2,...,n-1)$ of the
biggest entry $N=\frac{n(n-1)}{2}$.

2\. This shows the first swap $s_{1}$ of the corresponding sorting network is
$l$.

3\. Define the sliding path in the following way: the first box of the path is
$(n-l,l)$. Compare the entries at $(n-l-1,l)$ and $(n-l,l-1)$, and take the
box with bigger entry as the second box(if only one coordinate is in the Young
diagram then take that one). Repeat this procedure, until arriving box (1,1),
which is the last box of the sliding path.

4\. Slide the entries on the sliding path in a way that, the biggest entry $N$
is removed, and all the remaining entries on the path are moved to the
neighboring box on the path. Then increase each remaining entries by 1, and
fill the box $(1,1)$ with entry 1.

5\. Note that after step 1-4, $T$ becomes a new SYT of shape $\Delta_{n}$.
Repeat step 1-4, to get swap $s_{2}$,$s_{3}$,…,$s_{N}$ of the corresponding
sorting network.

######  Remark 4.9.

If SYT $T$ corresponds to sorting network $(s_{1},...,s_{N})$, then after
running step 1-4 by once, we get a new SYT corresponding to sorting network
$(s_{2},s_{3},...,s_{N},n-s_{1})$(the last entry is $n-s_{1}$ by Proposition
1.6). Moreover, still by translation invariance, we will get SYT $T$ back
after repeating step 1-4 for $2N$ times.

###  4.3. Correspondence With SYT

Using Edelman-Greene correspondence, we reduce the problem in RSN to the
problem on staircase shaped SYT.

First swapping time:

Recall that $T_{FS,n}(k)$ denotes the first swapping time of a random sorting
network of size n, $N=\binom{n}{2}$. According to the Edelman-Greene
bijection, this time corresponds to the box (n-k,k) of the corresponding
staircase shape standard Young tableaux of size $n$, such that

(4.6) $\begin{split}T_{FS,n}(k)&=N+1-EG(\omega)(n-k,k)\\\
&\stackrel{{\scriptstyle
d}}{{=}}(N+1)(1-T_{\Delta_{n}}^{rsc}(n-k,k))\end{split}$

where $\,T_{\Delta_{n}}^{rsc}(i,j):=\frac{T_{\Delta_{n}}(i,j)}{N+1}$,
$T_{\Delta_{n}}(i,j)$ is the entry $(i,j)$ of a uniformly random SYT of shape
$\Delta_{n}$.

Conditioning on $T(n-k,k)=N$:

By Definition 1.12 we only need to care about what happens after time $a$. By
Edelman-Greene bijection, the conditioned process starting at time $a$
correspond to the uniformly random SYT $T$ of shape $\Delta_{n}$ conditioned
that $T(n-k,k)=N$. Moreover, the waiting time is

(4.7)
$\widehat{\mathrm{Sp}}_{k,n}=\begin{cases}(N-T(n-k-1,k))\wedge(N-T(n-k,k-1)),&\text{if}\;\;k=2,3,...,n-2;\\\
N-T(n-2,1),&\text{if}\;\;k=1;\\\ N-T(1,n-2),&\text{if}\;\;k=n-1.\end{cases}$

######  Remark 4.10.

The conditioned SYT described above can also be thought as a uniformly random
SYT with shape $\Delta_{n}\setminus(n-k,k)$.

###  4.4. Spacing distribution

Now we are able to write out the spacing distribution using the limiting
theorems of SYT discussed in Section 3.

Completing the proof of Theorem 1.5 and 1.18.

First swapping time:

Theorem 3.1 (a) implies in particular that

$n^{\frac{1}{2}}[1-T^{rsc}_{\Delta_{n}}(n-k,k)]$

converges weakly to the coordinate of the closest to the origin particle of
$\pazocal{X}$ on level $x=2k$. By Definition 1.2, this random variable equals
in law to the smallest positive eigenvalues of a $2k\times 2k$ aGUE.

On the other hand, it was shown in Section 4.3 that $T_{FS,n}(k)$ equals in
distribution to

$(N+1)[1-T^{rsc}_{\Delta_{n}}(n-k,k)].$

Since $N\approx n^{2}/2$, we need to scale the first swapping time by
$n^{\frac{3}{2}}/2$.

The expression of $\mathbb{P}\left[\mathbf{T}_{FS}(k)>t\right]$ via Fredholm
determinant follows directly from definition of determinantal point process
and Theorem 2.10, for the case $s=t=2k$.

######  Remark 4.11.

When $k=1$, there’s only one particle on level $x=2$ and $\mathbf{T}_{FS}(1)$
equals in distribution to the coordinate of this particle. Then density of
this height is given directly by
$K(2,u;2,u)=\frac{2}{\sqrt{\pi}}e^{-u^{2}}(u\geq 0)$, as mentioned in
Corollary 3.10. This result is first obtained in [R] Theorem 1.7.

Spacing Under Definition 1.12:

Theorem 3.1 (b) in particular implies that

$n^{\frac{1}{2}}[1-T^{rsc}_{\Delta_{n}\setminus(n-k,k)}(n-k,k-1)],$

and

$n^{\frac{1}{2}}[1-T^{rsc}_{\Delta_{n}\setminus(n-k,k)}(n-k-1,k)]$

converge weekly to the coordinates of the closest to the origin particles of
$\pazocal{X}_{k}^{{}^{\prime}}$ on level $x=2k-1$, $x=2k+1$ respectively. By
(4.7) in Section 4.3, we still need to scale the spacing
$\widehat{\mathrm{Sp}}_{k,n}$ by $n^{\frac{3}{2}}/{2}$.

When $k=1$, $Z_{3}$ is the coordinate of the unique particle of
$\pazocal{X}^{{}^{\prime}}_{1}$ on level $x=3$. The density of this height is
given directly by $K^{{}^{\prime}}_{1}(3,u;3,u)=2ue^{-u^{2}}(u\geq 0)$.
Although without giving a formal definition, [R] used our second definition of
spacing implicitly in its proof, and obtained this $k=1$ result in its Theorem
1.7. ∎

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