#  The spectral form factor in the ‘t Hooft limit
Intermediacy versus universality

Ward L. Vleeshouwers1,2* and Vladimir Gritsev1,3
1 Institute for Theoretical Physics, Universiteit van Amsterdam,
Science Park 904, Postbus 94485, 1098 XH Amsterdam, The Netherlands
2 Institute for Theoretical Physics, Universiteit Utrecht,
Princetonplein 5, Postbus 80.089, 3584 CC Utrecht, The Netherlands
3 Russian Quantum Center, Skolkovo, Moscow, Russia
<EMAIL_ADDRESS>
###### Abstract

The Spectral Form Factor (SFF) is a convenient tool for the characterization
of eigenvalue statistics of systems with discrete spectra, and thus serves as
a proxy for quantum chaoticity. This work presents an analytical calculation
of the SFF of the Chern-Simons Matrix Model (CSMM), which was first introduced
to describe the intermediate level statistics of disordered electrons at the
mobility edge [1]. The CSMM is characterized by a parameter $0\leq q\leq 1$,
where the Circular Unitary Ensemble (CUE) is recovered for $q\to 0$. The CSMM
was later found as a matrix model description of $U(N)$ Chern-Simons theory on
$S^{3}$ [2], which is dual to a topological string theory characterized by
string coupling $g_{s}=-\log q$. The spectral form factor is proportional to a
colored HOMFLY invariant of a $(2n,2)$-torus link with its two components
carrying the fundamental and antifundamental representations, respectively. We
check explicitly that taking $N\to\infty$ whilst keeping $q<1$ reduces the
connected SFF to an exact linear ramp of unit slope, thereby confirming the
main result from [3] for the specific case of the CSMM. We then consider the
‘t Hooft limit, where $N\to\infty$ and $q\to 1^{-}$ such that $y=q^{N}$
remains finite. As we take $q\to 1^{-}$, this constitutes the opposite extreme
of the CUE limit. In the ‘t Hooft limit, the connected SFF turns into a
remarkable sequence of polynomials which, as far as the authors are aware,
have not appeared in the literature thus far. A gap opens in the spectrum and,
after unfolding by a constant rescaling, the connected SFF approximates a
linear ramp of unit slope for all $y$ except $y\approx 1$, where the connected
SFF goes to zero. We thus find that, although the CSMM was introduced to
describe intermediate statistics and the ‘t Hooft limit is the opposite limit
of the CUE, we still recover Wigner-Dyson universality for all $y$ except
$y\approx 1$.

##  1 Introduction

Random matrix theory provides a phenomenological description of a wide variety
of systems appearing in physics and beyond, starting with nuclear physics [4],
and later finding applications in quantum chaos [5], disordered electronic [6]
and mesoscopic [7] systems, chromodynamics [8], models of 2d quantum gravity
and string theory [9], economics [10], information theory [11], and number
theory [12]. One of the central concepts in the theory of random matrices is
the Wigner-Dyson (WD) universality that is exhibited by the eigenvalue
correlation properties of a wide variety of ensembles. For ensembles defined
by the probability density with the weight function $w(x)=\exp(-V(x))$ where
$V(H)\sim\text{tr}(H^{2})+\ldots$ for Hermitian $N\times N$ matrix $H$, the
density of eigenvalues $x$ has the famous, “universal” semi-circular form
$\rho(x)=\sqrt{2N-x^{2}}/\pi$. This does not depend on the precise form of
$V(x)$ for a broad range of potentials and the matrix size $N$. Rescaling the
eigenvalues such that $\bar{\rho}(u)=1$ near the origin, typically called
unfolding, leads to the two-level eigenvalue kernel function
$G(u,v)=\sin[\pi(u-v)]/\pi(u-v)$. As a consequence, this leads to the
universal form of the Spectral Form Factor (SFF), a Fourier transform of the
density-density correlation function, $K(\xi)=1-\int
G(x,0)^{2}\exp{(ix\xi)}dx$ resulting for $\xi>0$ in the linear ramp until the
Heisenberg “time" $\xi=T_{H}=2\pi$ followed by a plateau for $\xi>T_{H}$.
Integrable systems, on the other hand, display uncorrelated Poisson statistics
leading to a constant SFF for all times. For systems with statistics
intermediate between Poissonian and WD, the SFF has mixed features. Typically,
it is of a dip-ramp-plateau shape, where, for small $\xi$, the SFF dips until
it reaches a minimum at Thouless time, after which it transitions into a
linear ramp which saturates at a plateau at Heisenberg time. These features
make the SFF a convenient tool for characterizing level statistics which has
found extensive use in the RMT and quantum chaos community. We also mention
here that the SFF has recently became popular in the string theory community,
predominantly in the context of the AdS/CFT correspondence, as well as the SYK
model and JT-gravity [13], [14], [15], [16], [17], [18].

Certain systems which are somewhere in between chaotic and integrable, such as
disordered electrons at the mobility edge of Anderson localization [19], or
pseudo-integrable billiards [20], display so-called intermediate statistics.
Various random matrix ensembles have been introduced which exhibit these
statistics. To the best of our knowledge, the first such model was a solvable
random matrix ensemble interpolating between Poisson and Wigner-Dyson
statistics that was introduced by Gaudin [21]. The same model was rediscovered
later by Yukawa [22] in the context of a so-called Pechukas-Yukawa gas.
Another type of RME that was introduced to describe intermediate statistics
are the so-called banded RME’s [23], for which the entries of the random
matrices decay in a power-law fashion away from the main diagonal. Another
such is the Moshe-Neuberger-Shapiro (MNS) ensemble [24], where the unitary
invariance of the ensemble is explicitly broken by a potential term involving
a fixed matrix, typically chosen to be diagonal. Bogomolny et al. [25]
discovered a number of ensembles which are based on the Lax matrices of
various integrable systems. These ensembles exhibit intermediate statistics
and multifractal eigenfunctions [26]. Further, a generalization [27] of the
Rosenzweig-Porter model [28] was shown to have a region in parameter space
with multifractal behavior. Moreover, a standard $\beta$-ensembles, introduced
in [29] have intermediate statistics [30].

Another class of matrix models was proposed in [1], [31] to describe
intermediate level statistics. These ensembles are defined in terms of a
$U(N)$-invariant measure, as opposed to the aforementioned ensembles with
intermediate statistics. It was observed that WD-universality is lost for
sufficiently shallow confining potentials, which asymptotically behave as
$V(H)\sim g_{s}^{-1}\log^{2}H$ for $|H|\gg 1$. Such potentials are associated
with indeterminate moment problems, which is to say that the weight function
$w(x)$ is not uniquely determined by its moments $m_{j}=\int x^{j}w(x)dx$
[32], [33]. A typical signature of this intermediate class of statistics is a
lack of a simple translationally-invariant kernel of $\sin x/x$-form as
before, which now becomes dependent on some parameter $q$. This opens up the
possibility of non-unique (and more complicated) scaling limits, [34], [35].

The connection of the CSMM with intermediate statistics was seen to arise in
earlier works as follows. It was shown that taking $N\to\infty$ and then
$q=e^{-g_{s}}\to 1$ leads to a kernel that is of the form
$G(\theta,\varphi)=\frac{g_{s}}{2\pi}\frac{\sin[\pi(\theta-\varphi)]}{\sinh[g_{s}(\theta-\varphi)/2]}$
[1], [32]. This was found to be precisely the kernel [31] for the
aforementioned banded random matrix ensembles and the MNS-ensemble. The
relation of the ($U(N)$-invariant) CSMM to the (non-$U(N)$-invariant) banded
and MNS-ensembles has been argued to arise due to a spontaneous breaking of
$U(N)$-invariance [36]. It was found that the CSMM reproduces both the nearest
neighbor spacing and the level number variance at the same value of the
parameter $q$ [37]. From the level number variance, one can determine certain
properties of the eigenvector statistics as well, in particular their
multifractal dimension(s) [38] [39]. The SFF was previously calculated using
the limiting $\sinh$-kernel in [40]. The CSMM appears in unitary guise as a
one-parameter dependent generalization of the circular unitary ensemble (CUE),
see e.g. [41] for a map between the unitary and Hermitian ensembles. Denoting
the parameter as $q=e^{-g_{s}}$, the CSMM reduces to the CUE for $q\to 0$ and
produces Poissonian statistics for $q\to 1$ such that $q^{N}=1$. This paper
focuses on the unitary version of the CSMM. The CUE also exhibits WD-
universality, consisting of a linear ramp of unit slope that saturates at a
plateau [42].

The CSMM was later found by Mariño as a matrix model representation of type
$A$ topological open string theory on the cotangent space of $S^{3}$ [2],
which reduces to $U(N)$ Chern-Simons theory on $S^{3}$ [43]. The orthogonal
polynomials associated to this ensemble are the Stieltjes-Wigert or Rogers-
Szegö polynomials in Hermitian and unitary description, respectively [33],
[44]. Witten famously showed that Wilson line expectation values in $SU(2)$
Chern-Simons theory equal topological (Jones) invariants of knots and links
[45], which was later generalized to $U(N)$ for general $N$. Indeed, the SFF,
$\left\langle\lvert\text{tr}U^{n}\rvert^{2}\right\rangle$, is proportional to
the HOMFLY invariant of a $(2n,2)$-torus link with one Wilson line in the
fundamental and the other in the antifundamental representation.

In the string theory literature on the CSMM, the ’t Hooft limit has been
considered, where one takes $N\to\infty$ and simultaneously $q\to 1$ such that
$y=q^{N}$ remains finite. This idea goes back to pioneering work by ‘t Hooft
in the context of $U(N)$ gauge theories at large $N$. In the type $A$
topological open string theory on $T^{*}S^{3}$ described by the $U(N)$ Chern-
Simons theory, certain large $N$ dualities appear. In particular, it has been
argued that the topological $A$-type open string theory on $T^{*}S^{3}$
undergoes a conifold transition to a _closed_ type $A$ topological string
theory on the resolved conifold [46]. The magnitude of the $B$-field on the
$S^{2}$ blowup of the conifold is given by $t=Ng_{s}$, the ‘t Hooft parameter.
The mirror dual of the conifold geometry can be seen to arise from the
resolvent of the matrix model in the large $N$ limit [47], [48], see also
[49], [50]. These dualities and related results have important applications in
enumerative geometry and intersection theory.

As far as the authors are aware, the ‘t Hooft limit has heretofore not been
explicitly considered for the CSMM in the RMT literature. In [51], a closely
related limit was considered for a Hermitian version of the $q$-deformed
ensemble considered here, where the weak disorder (GUE) limit corresponds to
$q=e^{-\gamma}\to 1$ and $N\to\infty$ such that $\gamma N\to 0$, while the
strong disorder limit involves $\gamma N=\text{constant}$. In the latter
limit, which is essentially the ’t Hooft limit, an approximate expression for
the parametric density correlation function was found in [51]. Further, a
similar limit was considered for another, closely related, $q$-deformed
circular unitary ensemble in [52], see also [32]. It was found that deviations
from the CUE level density only persist in the infinite $N$ limit if one
simultaneously scales $q$ such that $(1-q)N$ remains finite, which is
essentially the ‘t Hooft limit.

In [3], we calculated the spectral form factor (SFF) for $N\to\infty$
invariant unitary matrix models satisfying the assumptions of Szegö’s theorem.
In this limit, we found that for _all_ such ensembles, the SFF is of a
surprisingly simple form, where the connected SFF always consists of an exact
linear ramp and plateau, and the disconnected part constitutes a dip which
consists of a squared simple power sum polynomial with of variables that can
be read off from the weight function. In the present work, we extend the
calculation of the SFF for the CSMM to finite $N$ and explore first how our
previous results are recovered in the $N\to\infty$. The present calculation
proceeds from the expansion of the SFF in terms of Toeplitz minors, which are
proportional to the product of two hook-shaped Schur polynomials of different
specialization. The simplification that occurs for $N\to\infty$ [53] is not
present here, which precludes us from generalizing these results to other
matrix models as we did for infinite $N$ [3].

We then proceed to take the ’t Hooft limit, where $N\to\infty$ and $q\to 1$.
Since the CSMM was introduced to describe intermediate statistics and reduces
to the CUE for $q\to 0$, one would naturally expect the CSMM to exhibit
deviation from WD-universality in the opposite limit, i.e. $q\to 1$. This is
our main physical motivation for studying the ‘t Hooft limit. We find that the
connected SFF, which we denote by $F(n)_{c}$, turns into a remarkable sequence
of polynomials of degree $2n-1$ in $y=q^{N}$, which we computed for
$n=1,\dots,11$. These polynomials do not seem to have appeared in the
literature thus far. Their somewhat complicated form belies the fact that they
are very close to linear ramps, with slope decreasing from 1 to 0 as we
increase $y$ from $0$ to $1$. In fact, we find that the $F(n)_{c}$ is an exact
linear ramp of slope $1/2$ for $y=1/2$, with the SFF’s for $y$ and $1-y$
exactly adding up to a linear ramp of unit slope. Although the form of low-
lying expansion coefficients can be found, the general structure of these
polynomials (beyond those which calculated explicitly) remains elusive.
Further, a gap opens in the eigenphase density, which we will simply refer to
as level density, for any $y>0$. We unfold by rescaling the spectrum so that
the level density has support on an interval of length $2\pi$. After
calculating the connected SFF using the unfolded eigenphases, we find that it
is very close to a linear ramp of unit slope for all $y$ other than $y$ close
to 1, with precision increasing with $n$. Indeed, we recover an exact linear
ramp of _unit_ slope for $y=1/2$, in spite of the fact that our unfolding
procedure involves only a rescaling, and the unfolded level density is not
flat at all (in fact it closely resembles a semicircle). We thus recover WD-
universality for $y$ sufficiently far from 1, in spite of the fact that the ‘t
Hooft limit involves $q\to 1$, which is the opposite limit of the CUE limit.
This the main result of the present work. Lastly, we consider the non-
commutativity of the limits $q\to 1$ and $N\to\infty$, which was already noted
for the partition function and trace averages
$\left\langle\text{tr}U^{n}\right\rangle$ in [54].

The outline of the paper is as follows. In section 2, we review the
calculation of $U(N)$ integrals “twisted” by the insertion of $U(N)$
characters (Schur polynomials), and we explicitly consider a few examples
relevant for the calculation of the SFF. In section 3, we present the
calculation of the SFF. In particular, in section 3.1, we calculate the SFF
for general $N$ and $q$, and show how the connected SFF reduces to a linear
ramp for $y=q^{N}\to 0$, thereby confirming one of the main results of [3]. We
further demonstrate explicitly that the linear ramp emerges from $U(N)$
averages over characters corresponding to identical (hook-shaped)
representations, which confirms a result derived in [3] for $N\to\infty$ with
$q<1$. In section 3.2, we demonstrate how a plateau emerges sufficiently far
from the origin and explain its arising from the properties of Schur
polynomials $s_{\lambda}$, in particular from the simple fact that
$s_{\lambda}(x_{1},\dots,x_{N})=0$ if the number of non-empty rows in
partition $\lambda$ exceeds the number of variables $N$. In section 3.3, we
consider the ‘t Hooft limit and compute the SFF and level density. We unfold
by a linear rescaling and demonstrate how WD-universality is recovered as a
result. Then, in section 3.4, we explore the non-commutativity of the limits
$N\to\infty$ and $q\to 1$ and calculate the SFF for small ‘t Hooft parameter.
We finish this work by presenting our outlook and conclusions.

##  2 Twisted $U(N)$ integrals

Consider the weight function of some ensemble, expressed here as

$\displaystyle f(z)$
$\displaystyle=\sum_{k\in\mathds{Z}}d_{k}z^{k}=\prod_{j=1}^{\infty}(1+x_{j}z)(1+x_{j}z^{-1})=E(x;z)E(x;z^{-1})\leavevmode\nobreak\
,$ (1)

where $x=(x_{1},x_{2},\dots)$ are the variables of $E(x;z)$, the generating
function of elementary symmetric polynomials. For $U\in U(N)$ with eigenvalues
$e^{i\phi_{j}}$, we write

$\displaystyle\tilde{f}(U)=\prod_{j=1}^{N}f(e^{i\phi_{j}})\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
s_{\lambda}(U)=s_{\lambda}(e^{i\phi_{j}})\leavevmode\nobreak\ .$ (2)

In the limit $N\to\infty$, the twisted $U(N)$ integral goes to [55]

$\displaystyle\left\langle
s_{\lambda}(U^{-1})s_{\mu}(U)\right\rangle\coloneqq\frac{\int_{U(N)}\tilde{f}(U)s_{\lambda}(U^{-1})s_{\mu}(U)dU}{\int_{U(N)}\tilde{f}(U)dU}=\sum_{\nu}s_{(\lambda/\nu)^{t}}(x)s_{(\mu/\nu)^{t}}(x)\leavevmode\nobreak\
,$ (3)

where the superscript t denotes transposition and where the sum is over all
partitions $\nu$ such that $\mu\supseteq\nu\subseteq\lambda$. Further,
$x=(x_{1},x_{2},\dots)$, the set of variables appearing in (1). If we have a
finite number of distinct non-zero $x_{j}$, (3) can be valid even for finite
$N$ [3]. We denote by $\lvert x\rvert$ the number of distinct, non-zero
$x_{j}$. Then, (3) remains valid as long as both $\ell(\lambda)$ and
$\ell(\mu)$ are greater than $N-\lvert x\rvert$. However, we are most
interested in the CSMM, for which $f(z)=\Theta_{3}(z)$. Using the Jacobi
triple product expansion, we see that $x_{j}=q^{j-1/2}$ in equation (1) for
all $j\in\mathds{Z}^{+}$, so that we cannot apply (3). In particular, the
partition function of the CSMM can be written as

$\displaystyle Z=\int
dU\prod_{j=1}^{\infty}\det\left(\mathds{1}+q^{j-1/2}U\right)\det\left(\mathds{1}+q^{j-1/2}U^{-1}\right)\leavevmode\nobreak\
.$ (4)

All expectation values $\left\langle\dots\right\rangle$ that we write from now
on are with respect to the above partition function. To calculate the SFF of
the CSMM, then, we will consider twisted $U(N)$ integrals for general $N$ and
weight function. These can be expressed as a minors of a Toeplitz matrix of
symbol $f(z)=\sum_{k\in\mathds{Z}}d_{k}z^{k}$, i.e. a Toeplitz matrix with
$d_{k}$ on the $k^{\text{th}}$ diagonal [56],

$\displaystyle
D_{N-1}^{\lambda,\mu}(f)=\det(d_{\lambda_{j}-j-\mu_{k}+k})_{j,k=1}^{N}=\int_{U(N)}\tilde{f}(U)s_{\lambda}(U^{-1})s_{\mu}(U)dU\leavevmode\nobreak\
.$ (5)

The weight function under consideration in the present work is the third theta
function, which, for $0<\lvert q\rvert<1$, can be expressed in the triple
product expansion as

$\displaystyle\Theta_{3}(z)$
$\displaystyle=\sum_{k\in\mathds{Z}}q^{k^{2}/2}z^{k}=(q;q)_{\infty}\prod_{j=1}^{\infty}(1+q^{j-1/2}z)(1+q^{j-1/2}z^{-1})$
$\displaystyle=(q;q)_{\infty}E(q^{j-1/2};z)E(q^{j-1/2};z^{-1})\leavevmode\nobreak\
.$ (6)

Note that, with this definition, $d_{k}=q^{k^{2}/2}$ rather than
$d_{k}=q^{k^{2}}$, the latter being another common convention. We then have

$\displaystyle
D_{N-1}^{\lambda,\mu}(\Theta_{3})=\det\left(q^{(\lambda_{j}-j-\mu_{k}+k)^{2}/2}\right)_{j,k=1}^{N}\leavevmode\nobreak\
.$ (7)

First taking $\lambda=\emptyset=\mu$, we have

$\displaystyle
D_{N-1}(\Theta_{3})=\det\left(q^{(k-j)^{2}/2}\right)_{j,k=1}^{N}=\prod_{j<k}(1-q^{k-j})=\prod_{k=1}^{N-1}(1-q^{j})^{N-j}\leavevmode\nobreak\
.$ (8)

Taking only $\lambda=\emptyset$ gives

$\displaystyle
D_{N-1}^{\mu}(\Theta_{3})=\det(d_{k-\mu_{k}-j})_{j,k=1}^{N}=\int_{U(N)}\tilde{f}(U)s_{\mu}(U)dU=\det\left(q^{({k-\mu_{k}-j})^{2}/2}\right)_{j,k=1}^{N}\leavevmode\nobreak\
.$ (9)

We have (see e.g. the appendix of [54]),

$\displaystyle\det(q^{(\mu_{k}+k-j)^{2}/2})_{j,k=1}^{N}=q^{\sum_{k}\mu_{k}^{2}/2}\prod_{j>k}(1-q^{\mu_{j}-\mu_{k}+k-j})\leavevmode\nobreak\
.$ (10)

Then,

$\displaystyle W_{\lambda\mu}\coloneqq\left\langle
s_{\lambda}(U^{-1})s_{\mu}(U)\right\rangle=\frac{D^{\lambda\mu}_{N-1}}{D_{N-1}}\leavevmode\nobreak\
.$ (11)

Up to a simple framing factor, this equals the Chern-Simons average over a
pair of Wilson lines tied into a Hopf link, where one Wilson line carries a
$U(N)$ representation $\lambda$ and the other carries $\mu$. For
$\mu=\emptyset$, the Hopf link reduces to an unknot carrying rep $\lambda$,
and vice versa for $\lambda=\emptyset$. Using (10), we then have

$\displaystyle
W_{\mu}=\frac{\det\left(q^{({k-\mu_{k}-j})^{2}/2}\right)}{\det\left(q^{({k-j})^{2}/2}\right)}=q^{\sum_{j}\mu_{j}^{2}/2}\frac{\prod_{j<k}(1-q^{k-j-\mu_{k}+\mu_{j}})}{\prod_{j<k}(1-q^{k-j})}=q^{-n(\mu)+\sum_{j}\mu_{j}^{2}/2}s_{\mu}(1,q,\dots,q^{N-1})\leavevmode\nobreak\
,$ (12)

where $n(\mu)=\sum_{j=1}^{N}(j-1)\mu_{j}$. When we take $N\to\infty$, this
equals [53]

$\displaystyle
W^{\infty}_{\mu}=s_{\mu^{t}}(q^{j-1/2})=q^{\lvert\mu\rvert/2+n(\mu^{t})-n(\mu)}s_{\mu}(q^{j-1})\leavevmode\nobreak\
.$ (13)

The power appearing in the prefactor is given by the sum of the content,
$c(x)$, over the partition $\mu$. Specifically, $c(x)=j-i$ for
$x=(i,j)\in\mu$, and [57]

$\displaystyle\sum_{x\in\mu}c(x)=n(\mu^{t})-n(\mu)\leavevmode\nobreak\ ,$ (14)

For general $\lambda$ and $\mu$, finite $N$, and $\lvert q\rvert<1$, we have

$\displaystyle W_{\lambda\mu}$ $\displaystyle=\frac{1}{Z_{N}}\int
s_{\lambda}(U^{-1})s_{\mu}(U)f(U)dU$
$\displaystyle=q^{\sum_{j=1}^{N}(\lambda_{j}^{2}/2+\mu_{j}^{2}/2-(j-1)(\lambda_{j}+\mu_{j}))}s_{\mu}(q^{j-1})s_{\lambda}(q^{-\mu_{1}},q^{1-\mu_{2}},\dots,q^{N-1-\mu_{N}})$
$\displaystyle=q^{-n(\lambda)-n(\mu)+\sum_{j=1}^{N}(\lambda_{j}^{2}/2+\mu_{j}^{2}/2)}s_{\mu}(q^{j-1})s_{\lambda}(q^{-\mu_{1}},q^{1-\mu_{2}},\dots,q^{N-1-\mu_{N}})\leavevmode\nobreak\
,$ (15)

which, for $N\to\infty$, goes to [53]

$\displaystyle
W^{\infty}_{\lambda\mu}=\sum_{\nu}s_{(\lambda/\nu)^{t}}(q^{j-1/2})s_{(\mu/\nu)^{t}}(q^{j-1/2})\leavevmode\nobreak\
.$ (16)

Consider, for example, $\lambda=\Box=\mu$. Then, using [equation (5.9) in
[57]]

$\displaystyle
s_{\lambda}(x,y)=\sum_{\mu}s_{\lambda/\mu}(x)s_{\mu}(y)\leavevmode\nobreak\ ,$
(17)

and
$\lim_{N\to\infty}[N]_{q}=\lim_{N\to\infty}\frac{1-q^{N}}{1-q}=\frac{1}{1-q}$
for $\lvert q\rvert<1$, we have

$\displaystyle
W_{\Box\Box}=[N]+q^{2}[N][N-1]\overset{N\to\infty}{\to}\frac{1}{1-q}+\frac{q^{2}}{(1-q)^{2}}\leavevmode\nobreak\
.$ (18)

On the other hand,

$\displaystyle W^{\infty}_{\Box\Box}$ $\displaystyle=$
$\displaystyle\sum_{\nu}(s_{(\Box/\nu)}(q^{j-1/2}))^{2}=(s_{\Box}(q^{j-1/2}))^{2}+(s_{\emptyset}(q^{j-1/2}))^{2}$
(19) $\displaystyle=$ $\displaystyle
q[N]^{2}+1=\frac{q}{(1-q)^{2}}+1=\lim_{N\to\infty}W_{\Box\Box}\leavevmode\nobreak\
.$ (20)

As one can see, the fact that terms of the form $q^{N}$ go to zero as
$N\to\infty$ leads to the agreement between these expressions. Consider the
Schur polynomial appearing in the unknot and Hopf link, given by the $q$-hook
length formula,

$\displaystyle
s_{\lambda}(q^{j-1})=q^{n(\lambda)}\prod_{x\in\lambda}\frac{[N+c(x)]}{[h(x)]}=q^{n(\lambda)}\dim_{q}(\lambda)\leavevmode\nobreak\
,$ (21)

where $h(x)$ is the hook-length of $x\in\lambda$. The quantity
$\dim_{q}(\lambda)$ is known as the quantum dimension, or $q$-dimension. Its
expression in (21) is simply the usual hook length formula where numbers are
replaced by $q$-numbers $[N]=\frac{1-q^{N}}{1-q}$. Using the $q$-hook-length
formula, one finds that the hook-shaped Schur polynomial is given by

$s_{(a,1^{b})}(x_{i}=q^{i-1})=q^{b(b+1)/2}\frac{[N+a-1]!}{[N-b-1]![a-1]![b]![a+b]}\leavevmode\nobreak\
.$ (22)

For $c(x)\ll N,\leavevmode\nobreak\ \forall x\in\lambda$ and $N\to\infty$,
(21) gives

$\displaystyle
s_{\lambda}(q^{j-1})=\frac{q^{n(\lambda)}}{(1-q)^{\lvert\lambda\rvert}}\prod_{x\in\lambda}[h(x)]^{-1}=q^{n(\lambda)}\prod_{x\in\lambda}(1-q^{h(x)})^{-1}\leavevmode\nobreak\
.$ (23)

In particular, $s_{\lambda}(q^{j-1})$ depends only on $n(\lambda)$ and the
hook lengths, so that e.g. $\frac{[N+a-1]}{[N-b-1]}=(1-q)^{-(a+b)}$.
Therefore, for a partition $\lambda$ for which

$\displaystyle\sum_{x\in\lambda}c(x)=n(\lambda^{t})-n(\lambda)=0\leavevmode\nobreak\
,$ (24)

the unknot $W^{\infty}_{\lambda}$ is invariant under taking
$\lambda\to\lambda^{t}$. One can clearly see from equation (21) that this is
not the case for finite $N$. These examples illustrate the simplification
which occurs as $q^{N}\to 0$, which we will further comment on in the
following section.

###  2.1 The ‘t Hooft limit

The ‘t Hooft limit is given by the following double scaling,

$\displaystyle N\to\infty\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ g_{s}\to 0\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \text{such that }t\coloneqq
Ng_{s}=\text{finite}\leavevmode\nobreak\ .$ (25)

The ‘t Hooft limit of the CSMM has been considered in the context of
topological string theory in e.g. [46], [48], [49]. In this case, one has
$y=q^{N}=e^{-t}\neq 0$. In this limit, $q$ taken to a finite power will simply
give 1, whereas $q$ to the power of multiples of $N$ will give powers of $y$,
which we need to keep track off to calculate the SFF. In the ‘t Hooft limit,
the hook-shaped Schur polynomial in (22) goes to

$\displaystyle\lim_{q\to
1^{-}}\frac{1}{(a-1)!b!(a+b)}\left(\frac{1-y}{1-q}\right)^{a+b}\leavevmode\nobreak\
,$ (26)

which we write as a limit as it is a divergent quantity. However, we will find
that the connected SFF is in fact not divergent for the explicit examples we
calculated. For the connected SFF not to be divergent, a precise cancellation
between various powers of $(1-q)^{-1}$ has to take place, which means that we
cannot simply use (22) in this calculation. Instead, we will write,

$\displaystyle
s_{(a,1^{b})}(x_{i}=q^{i-1})=\underbrace{\frac{q^{b(b+1)/2}}{[a-1]![b]![a+b]}}_{=A_{a,b}}\frac{[N+a-1]!}{[N-b-1]!}$
$\displaystyle=A_{a,b}\frac{\prod_{k=0}^{a+b-1}(1-yq^{a-1-k})}{(1-q)^{a+b}}!,$
$\displaystyle=\frac{A_{a,b}}{(1-q)^{a+b}}(yq^{a-1};q^{-1})_{a+b},$ (27)

where one should keep in mind that we take the limit $q\to 1^{-}$. In
particular, for $m$ finite, one can write

$\displaystyle\begin{bmatrix}N+m\\\
k\end{bmatrix}=\frac{(yq^{m};q^{-1})_{k}}{[k]!(1-q)^{k}}\leavevmode\nobreak\
.$ (28)

as a convenient way to extract factors of $y$.

##  3 Spectral form factor

We proceed to calculate the SFF, which is given by

$\displaystyle
K(n)\coloneqq\frac{1}{N}\left\langle\lvert\text{tr}U^{n}\rvert^{2}\right\rangle=\frac{1}{N}\sum_{r,s=0}^{n-1}(-1)^{r+s}\left\langle
s_{(n-r,1^{r})}s_{(n-s,1^{s})}\right\rangle\leavevmode\nobreak\ ,$ (29)

where we applied the expansion of the power sum polynomial,

$\displaystyle\text{tr}U^{n}=\sum_{j=1}^{N}e^{in\phi_{j}}=\sum_{r=0}^{n-1}(-1)^{r}s_{(n-r,1^{r})}(e^{i\phi_{j}})\leavevmode\nobreak\
.$ (30)

Plugging equation (29) into (2), we have

$\displaystyle
K(n)=\frac{q^{n^{2}}}{N}\sum_{r,s=0}^{n-1}(-1)^{r+s}q^{-n(r+s)}s_{(n-s,1^{s})}(q^{j-1})s_{(n-r,1^{r})}(q^{-(n-s)},1,\dots,q^{s-1},q^{s+1},\dots,q^{N-1})\leavevmode\nobreak\
.$ (31)

The first Schur polynomial appearing in the sum,
$s_{(n-s,1^{s})}(q^{j-1})=s_{(n-s,1^{s})}(1,\dots,q^{N-1})$, is given in (22).
The second Schur polynomial on the right hand side of (31), of the form
$s_{\lambda}(q^{j-\mu_{j}-1})$ for hook-shaped $\lambda$ and $\mu$, is more
complicated. We write $\lambda=(a,1^{b})$ and $\mu=(c,1^{d})$. Defining the
sets of variables $x=q^{-c}$, $y=q^{d+1},\dots,q^{N-1}$, $z=1,\dots,q^{d-1}$,
we use the following expression [equation (5.10) in [57]],

$\displaystyle
s_{\lambda}(x,y,z)=\sum_{\rho,\nu}s_{\lambda/\rho}(x)s_{\rho/\nu}(y)s_{\nu}(z)\leavevmode\nobreak\
,$ (32)

where the sum runs over all partitions satisfying
$\nu\subset\rho\subset\lambda$. For $\lambda=(a,1^{b})$ and $x,y,z$ as defined
above, we get non-zero contributions only when $\lambda/\rho$ is a horizontal
strip, as $x=q^{-c}$ consists only of a single variable. We then have to
carefully distinguish between two types of partitions $\rho$.

  1. 1.

For $\rho=(e,1^{b})$, we again have $\lambda/\rho=(a-e)$ which is obviously a
horizontal strip. Then,

$\displaystyle s_{\lambda/\rho}(q^{-c})=q^{-c(a-e)}\leavevmode\nobreak\ .$
(33)

The requirement that $\nu\subset\rho$ then gives $\nu=(f,1^{g})$ so that
$\rho/\nu=(e-f)\otimes(1^{b-g})$ for $\nu\neq\emptyset$ and $\rho/\nu=\mu$ for
$\nu=\emptyset$.

  2. 2.

For $\rho=(e,1^{b-1})$, we have $\lambda/\rho=\Box\otimes(a-e)$ so that

$\displaystyle s_{\lambda/\rho}(q^{-c})=q^{-c(a-e+1)}\leavevmode\nobreak\ .$
(34)

Then, $\nu=(f,1^{g})$ so that $\rho/\nu=(e-f)\otimes(1^{b-g-1})$ and
$\rho/\nu=\rho$ for $\nu=\emptyset$. This situation of course does not occur
for $b=0$, in which case $\lambda=(a)$.

Note that $s_{(a)}(x)=h_{a}(x)$ and $s_{(1^{a})}(x)=e_{a}(x)$, the complete
homogeneous and elementary symmetric polynomials of degree $a$, respectively.
We then have, for $\rho/\nu=(e-f)\otimes(1^{b-g})$

$\displaystyle s_{\rho/\nu}(y)=h_{e-f}(y)e_{b-g}(y)\leavevmode\nobreak\ .$
(35)

We illustrate these two choices for $\rho$ in equations (33) and (34) for
$\lambda=(4,1^{2})$. As a specific example, we set $e=2$, so that the first
choice of $\rho=(e,1^{b})=(2,1^{2})$. This gives
$\lambda/\rho=(4,1^{2})/(2,1^{2})=(2)$. $\lambda/\rho$ is represented in terms
of Young diagrams below.

4,1,1 / 2,1,1  =  2 .

Consider now the second choice, $\rho=(e,1^{b-1})=(2,1)$, so that
$\lambda/\rho=(4,1^{2})/(2,1)=(2)\times(1)$, represented in Young diagrams as
follows.

4,1,1 / 2,1 =  1 $\times$ 2

The result is again a horizontal strip, consisting in this case of two
disconnected components. We consider these two choices for $\mu$ and sum over
all partitions satisfying $\nu\subset\rho\subset\lambda$ to calculate
$s_{(n-r,1^{r})}(x,y,z)$ 111In particular, we sum over $g$ from 0 to
min$(b,d-1)$ or to min$(b-1,d-1)$, corresponding to $\rho=(e,1^{b})$ or
$\rho=(e,1^{b-1})$, respectively. We then sum over $f$ from $1$ to $e$ and
lastly over $e$ from $0$ to $a$. and, with that, the full SFF. We discuss the
results of this calculation below.

###  3.1 Appearance of the linear ramp for $n<N$

For finite $N>n$, we consider how the linear ramp appears for the connected
SFF, which is found by subtracting the disconnected contribution,
$\left\langle\text{tr}U^{n}\right\rangle^{2}$, from the full SFF. The
disconnected contribution is thus given by the square of

$\displaystyle\left\langle\text{tr}U^{n}\right\rangle=q^{n^{2}/2}\sum_{r=0}^{n-1}(-1)^{r}q^{-nr}s_{(n-r,1^{r})}(q^{j-1})\leavevmode\nobreak\
.$ (36)

For $n=1,2,3$, this equals

$\displaystyle\left\langle\text{tr}U\right\rangle$
$\displaystyle=q^{1/2}[N]=\frac{q^{1/2}(1-q^{N})}{1-q}$
$\displaystyle\left\langle\text{tr}U^{2}\right\rangle$
$\displaystyle=\frac{(1-q^{N})(-q+q^{N}+q^{N+1}+q^{N+2})}{1-q^{2}}$
$\displaystyle\left\langle\text{tr}U^{3}\right\rangle$
$\displaystyle=\frac{q^{3}-q^{N+1}(1+q+q^{2})^{2}+q^{2N}(1+q^{2})(1+q+q^{2})^{2}-q^{3N}(1+q^{2})(1+q+q^{2}+q^{3}+q^{4})}{q^{3/2}(1-q^{3})}$
(37)

We write,

$\displaystyle F(n)=NK(n)\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
F(n)_{c}=NK(n)_{c}\leavevmode\nobreak\ .$ (38)

The connected SFF for small $n$ is then given by,

$\displaystyle F(1)_{c}=$ $\displaystyle q\leavevmode\nobreak\
s_{(1)}(q^{j-1})s_{(1)}(q^{-1},q,q^{2},\dots)-q(s_{(1)}(1,q,q^{2}\dots))^{2}$
$\displaystyle=$ $\displaystyle q[N](1-q)=1-q^{N}$ $\displaystyle F(2)_{c}=$
$\displaystyle\frac{(1-q^{N})(2q-q^{N}+q^{2N}-q^{2+N}+2q^{1+2N}+q^{2+2N})}{q}\leavevmode\nobreak\
,$ $\displaystyle F(3)_{c}=$
$\displaystyle-\frac{1}{q^{4}}(-3q^{4}+q^{3N}-2q^{N}+q^{5N}+q^{2+N}+2q^{3+N}+3q^{4+N}++2q^{5+N}+q^{6+N}+$
$\displaystyle-2q^{1+2N}-4q^{2+2N}-8q^{3+2N}-8q^{4+2N}+-8q^{5+2N}-4q^{6+2N}-2q^{7+2N}+$
$\displaystyle+6q^{1+3N}+10q^{2+3N}+16q^{3+3N}+18q^{4+3N}+16q^{5+3N}++10q^{6+3N}+6q^{7+3N}+$
$\displaystyle+q^{8+3N}-6q^{1+4N}-12q^{2+4N}-16q^{3+4N}-18q^{4+4N}-16q^{5+4N}-12q^{6+4N}+$
$\displaystyle-6q^{7+4N}-2q^{8+4N}+2q^{1+5N}+5q^{2+5N}+6q^{3+5N}+8q^{4+5N}+6q^{5+5N}+5q^{6+5N}+$
$\displaystyle+2q^{7+5N}+q^{8+5N})\leavevmode\nobreak\ .$ (39)

Examples of the SFF for higher $n$ are too long to print here. One thing one
can see from (3.1), which persists for higher $n$, is that the SFF is of the
form,

$\displaystyle F(n)_{c}=n+\mathcal{O}(q^{A})\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ A=N+\dots\leavevmode\nobreak\ .$
(40)

Therefore, for $N\to\infty$ and $q<1$ fixed, $q^{N}\to 0$ and $F(n)_{c}\to n$.
This reproduces the exact linear ramp that was found for all RME’s satisfying
the assumptions of Szegö’s theorem in our previous work [3], thus recovering
our main result for the case of the CSMM via a different computation.

####  3.1.1 Linear ramp from Schur bilinears

As mentioned above, as we take $N$ to infinity for $q<1$, the connected SFF
for $n/N<1$ is a linear ramp of unit slope [3]. In this limit, in the
expansion of the form factor in terms of averages of bilinears of hook-shaped
Schur polynomials,

$\displaystyle F(n)=\sum_{r,s=0}^{n-1}(-1)^{r+s}\left\langle
s_{(n-r,1^{r})}(U^{-1})s_{(n-s,1^{s})}(U)\right\rangle\leavevmode\nobreak\ ,$
(41)

we get a contribution equal to 1 when considering two identical hook-shaped
partitions $(n-r,1^{r})=(n-s,1^{s})$. Since there are $n$ hook-shaped
partitions containing $n$ boxes, we get a contribution equal to $n$, which is
the linear ramp. More details can be found in [3].

The above consideration leads us to conclude that, for finite $N$ and for
$r=s\leq N-1$, we should have that the summand of the SFF in (31) is of the
following form

$\displaystyle A(N,n,q,r,r)$ $\displaystyle\coloneqq\left\langle
s_{(n-r,1^{r})}s_{(n-r,1^{r})}\right\rangle$
$\displaystyle=q^{n^{2}-2nr}s_{(n-r,1^{r})}(q^{j-1})s_{(n-r,1^{r})}(q^{-(n-r)},1,\dots,q^{r-1},q^{r+1},\dots,q^{N-1})$
$\displaystyle=1+\mathcal{O}(q)\leavevmode\nobreak\ .$ (42)

Further, we should have

$\displaystyle A(N,n,q,r,s)=\mathcal{O}(q)\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ r\neq s\leavevmode\nobreak\ .$ (43)

There are two types of terms of $\mathcal{O}(q)$ in the above expressions.
First of all, there are terms of the form $q^{N+\dots}$, which go to zero as
we take $N\to\infty$ for fixed $q<1$. Secondly, there are powers of $q$ not
containing factors of $N$, which do not go to zero as $N\to\infty$. Therefore,
to recover the linear ramp as $N\to\infty$, all the lower powers of $q$ should
mutually cancel out between the various terms in the sum in (31). It should be
clear from equation (31) that the fact that such a cancellation occurs is a
priori far from obvious.

We start by verifying (3.1.1). Note that $q$-numbers and products thereof
(such as $q$-factorials and $q$-binomials) are themselves of $\mathcal{O}(1)$.
For example,

$\displaystyle[N]_{q}=\frac{1-q^{N}}{1-q}=(1-q^{N})\sum_{k=0}^{\infty}q^{k}=1+q+q^{2}+\dots-q^{N}-q^{N+1}+\dots\leavevmode\nobreak\
.$ (44)

Let us consider $A(N,n,q,0,0)$. Plugging

$\displaystyle
s_{(n)}(q^{-n},q,\dots,q^{N-1})=q^{-n^{2}}+\mathcal{O}(q^{-n^{2}+1})\leavevmode\nobreak\
.$ (45)

into (3.1.1) leads to

$\displaystyle
q^{n^{2}}s_{(n)}(q^{j-1})s_{(n)}(q^{-n},q,\dots,q^{N-1})=q^{n^{2}}\begin{bmatrix}N+n-1\\\
n\end{bmatrix}\left(q^{-n^{2}}+\mathcal{O}(q^{-n^{2}+1})\right)=1+\mathcal{O}(q)\leavevmode\nobreak\
,$ (46)

where we use the aforementioned fact that $q$-binomials are of the form
$1+\mathcal{O}(q)$. When $r=s\neq 0$, the calculation is slightly more
involved. First, we read off from (21) that

$\displaystyle
s_{(n-r,1^{r})}(q^{j-1})=q^{r(r+1)/2}\left(1+\mathcal{O}(q)\right)\leavevmode\nobreak\
.$ (47)

We then determine the lowest power of $q$ appearing in

$\displaystyle
s_{(n-r,1^{r})}(q^{-(n-r)},1,\dots,q^{r-1},q^{r+1},\dots,q^{N-1})$ (48)
$\displaystyle=$
$\displaystyle\sum_{\mu,\nu}s_{(n-r,1^{r})/\mu}(q^{-(n-r)})s_{\mu/\nu}(1,\dots,q^{r-1})s_{\nu}(q^{r+1},\dots,q^{N-1})\leavevmode\nobreak\
,$ (49)

where we used (32) on the right hand side. Consider the case where
$\mu=(1^{r})$ and $\nu=\emptyset$. This gives $\lambda/\mu=(n-r-1)\times(1)$,
so that $s_{(n-r,1^{r})/\mu}(q^{-(n-r)})=q^{-(n-r)^{2}}$. Further, we have
$s_{\mu}(q,\dots,q^{r-1})=q^{r(r-1)/2}$, and $s_{\nu}=s_{\emptyset}=1$.
Therefore,

$\displaystyle
s_{(n-r,1^{r})}(q^{-(n-r)},1,\dots,q^{r-1},q^{r+1},\dots,q^{N-1})=q^{-(n-r)^{2}}q^{r(r-1)/2}(1+\mathcal{O}(q))\leavevmode\nobreak\
.$ (50)

Plugging this into (3.1.1) gives

$\displaystyle
A(N,n,q,r,r)=q^{n^{2}-2nr}q^{r(r+1)/2}q^{-(n-r)^{2}}q^{r(r-1)/2}(1+\mathcal{O}(q))=1+\mathcal{O}(q)\leavevmode\nobreak\
.$ (51)

One can readily check that any other choice of $\mu$ and $\nu$ leads to higher
powers of $q$. For example, choosing $\nu=(1)$ increases the power of $q$ by
2, and choosing a different partition for $\mu$ either increases the power of
$q$ by $n-r$ or gives zero (when $\ell((n-r,1^{r})/\mu)>1$). This demonstrates
equation (3.1.1).

Let us now consider the case where $r\neq s$, to derive equation (43). We take
$r<s$ without loss of generality. Taking first $r=0$, we have

$\displaystyle A(N,n,0,s,q)$
$\displaystyle=q^{n^{2}-ns}s_{(n-s,1^{s})}(q^{j-1})s_{(n)}(q^{-(n-s)},y,z)$
$\displaystyle=q^{n^{2}-ns}q^{s(s+1)/2}q^{-n(n-s)}(1+\mathcal{O}(q))=q^{s(s+1)/2}(1+\mathcal{O}(q))=\mathcal{O}(q)\leavevmode\nobreak\
.$ (52)

where $y=(1,\dots,q^{s-1})$ and $z=(q^{s+1},\dots,q^{N-1})$, as before.
Lastly, we check the case where $0\neq s\neq r\neq 0$, choosing again $r<s$
without loss of generality. Following the same procedure that lead to (50), we
find

$\displaystyle
s_{(n-r,1^{r})}(q^{-(n-s)},1,\dots,q^{s-1},q^{s+1},\dots,q^{N-1})=q^{-(n-r)^{2}}q^{r(r-1)/2}(1+\mathcal{O}(q))\leavevmode\nobreak\
,$ (53)

so that this term, too, appears with a positive power of $q$,

$\displaystyle
A(N,n,r,s,q)=q^{((r-s)^{2}+r+s)/2}(1+\mathcal{O}(q))\leavevmode\nobreak\ .$
(54)

We have thus shown that terms with $r=s$ contribute $1+\mathcal{O}(q)$,
whereas Schur bilinears with $r\neq s$ contribute terms of $\mathcal{O}(q)$.
As mentioned above, powers of $q$ which do not contain a factor $N$ cancel out
in the sum over $r$ and $s$, leaving only a linear ramp plus terms of the form
$q^{(N+\dots)}$, as can be seen in (3.1).

The calculations described here have a simple knot-theoretical interpretation.
Remember that $A(N,n,r,s,q)$ is proportional to the HOMFLY invariant of a Hopf
link, where the two components of the Hopf link carry $U(N)$ respresentations
corresponding to partitions $(n-r,1^{r})$ and $(n-s,1^{s})$, respectively. The
above results entail that Hopf links of Wilson lines carrying hook-shaped
representations only give a contribution of (order) unity for two identical
representations. Therefore, in the limit $q\to 0$, where the CSMM reduces to
the CUE, all Hopf link invariants with different $n$-box hook-shaped
partitions go to zero. On the other hand, those with identical $n$-box hook-
shaped partitions go to one. This means, for example, that an unknot carrying
$(a,1^{b})$ with $b\neq 0$ has invariant equal to zero, but if we tie two of
these unknots together to form a Hopf link the resulting invariant equals one.
On the other hand, an unknot carrying representation $(a)$ has invariant equal
to one, but if we tie it to an unknot carrying $(a-b,1^{b})$ with $b\neq 0$ to
form a Hopf link, the result is again zero.

###  3.2 Relaxing the assumption that $n<N$

If we relax the assumption that $n<N$, the SFF will eventually reach a plateau
for large enough $n$. A well-known, heuristic way to see that such a plateau
is eventually reached is as follows. If we consider a diagonal matrix
$V=\text{ diag}(e^{i\phi_{1}},e^{i\phi_{2}},\dots,e^{i\phi_{N}})$ with all
$\phi_{j}$ taking random values in $[0,2\pi)$, and we average over $\phi_{j}$,
we get

$\displaystyle\left\langle\lvert\text{tr}V^{n}\rvert^{2}\right\rangle=\left\langle\sum_{k,l=1}^{N}e^{in(\phi_{k}-\phi_{m})}\right\rangle=N\leavevmode\nobreak\
.$ (55)

Here, we use the fact that $e^{in(\phi_{k}-\phi_{m})}$ equals 1 for $k=m$,
whereas for $k\neq m$ it is a random variable on ($2n$ copies of) the complex
unit circle, which averages to zero. A system with eigenphases distributed
randomly across the unit circle therefore has a constant SFF, as was mentioned
in the introduction. On the other hand, random unitary matrices $U$ display
level repulsion with overwhelming probability, so that their eigenvalues tend
to distribute more evenly across the complex unit circle. Therefore,
$\left\langle\lvert\text{tr}U^{n}\rvert^{2}\right\rangle$ is much lower than
$N$ for small values of $n$. However, for $n$ close to $N$, we have that $n$
becomes of the order of the average spacing $\phi_{k+1}-\phi_{k}$. In that
case, $e^{in(\phi_{k}-\phi_{l})}$ is an approximately random element of the
complex unit circle for all $k\neq l$, so that these again average to zero and
only the constant contribution $N$ coming from $k=l$ remains. This explains
the origin of the plateau for $n\gtrsim N$.

Alternatively, the emergence of the plateau can be understood to arise from
the fact that $s_{\lambda}(x)=0$ for $\ell(\lambda)>\lvert x\rvert$. In
particular, the averages appearing on the right hand side of (29), are
weighted matrix integrals written in (5) over Schur polynomials of the form
$s_{(n-r,1^{r})}(U)=s_{(n-r,1^{r})}(e^{i\phi_{j}})$. If
$\ell(s_{(n-r,1^{r})})=r+1>N$, then $s_{(n-r,1^{r})}(U)=0$. Therefore, only
Schur bilinears of the form

$\displaystyle\left\langle
s_{(n-r,1^{r})}s_{(n-s,1^{s})}\right\rangle\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ r,s\leq N-1\leavevmode\nobreak\ ,$ (56)

give a non-zero contribution to (29). It was demonstrated in the previous
subsection that, for $r,s\leq N-1$,

$\displaystyle\left\langle
s_{(n-r,1^{r})}s_{(n-s,1^{s})}\right\rangle=\delta_{r,s}+\mathcal{O}(q)\leavevmode\nobreak\
,$ (57)

from which arises the linear ramp. From equation (57), it follows that the
linear ramp arising from $r=s\leq N-1$, saturates at a plateau for $n=N$.
However, one should note that this does not take into account terms of
$\mathcal{O}(q^{N})$, which will turn out to have a significant impact on the
shape of the SFF, delaying the onset of the plateau as one increases $q^{N}$.

To implement $n>N$ in the expression for the SFF derived at the start of this
section, one should take into account that

  1. 1.

$s_{(e,1^{b})}(y)=0$ for $b>N-d-2$

  2. 2.

$e_{b-g}(y)=0$ for $g>d+b+1-N$.

where $x=q^{-c}$, $y=q^{d+1},\dots,q^{N-1}$, and $z=1,\dots,q^{d-1}$, as
before. Note that the functions above both arise as $s_{\rho/\nu}(y)$ for
$\rho=(e,1^{b})$ in equation (32), where $e_{b-g}(y)$ is given in equation
(35), whereas $s_{\rho/\nu}(y)$ reduces to $s_{(e,1^{b})}(y)$ for
$\nu=\emptyset$. We plot $K(n)$ resulting from this calculation for $N=10$ and
$N=20$ and various choices of and $q$.

Figure 1: The SFF for $N=10$ and various values of $q$.

As one can see, the SFF is closest to a linear ramp for small values of $q$,
which is to be expected as the limit $q\to 0$ corresponds to the CUE. Further,
disconnected SFF becomes large for small $n$ as we increase $q$, leading to
large, oscillating deviations close to the origin. Comparing figures 1 and 2
reveals that, for fixed $q$, deviation from a linear ramp decreases as we
increase $N$. This suggests that a combination of $q$ and $N$ controls the
approximate slope away from the origin.

The aforementioned observations that, as we increase $q$, a dip emerges and
the slope of the SFF decreases, are not unrelated. In particular, we find that
$\sum_{n=1}^{k}F(n)$ for large enough $k\geq N$ is almost independent of $q$.
That is, as we increase $q$, we get positive contributions to $A(k)$ arising
from the disconnected SFF which are compensated by a decrease in the slope of
$F(n)$. We define, for $k>N$, the logarithm of the difference between the sum
over the SFF of the CSMM and the CUE ($q\to 0$) SFF,

$\displaystyle
A(k):=\log\left[\sum_{n=1}^{k}F(n)-N^{2}/2-N(N-k)\right]\leavevmode\nobreak\
.$ (58)

We plot the results for $N=10$ and $k=10,\dots,20$ below. It is clear that the
difference decreases quite rapidly with $k$ until it stabilizes around some
small value. Further, we see that the difference decreases more slowly and
acquires a larger minimum value as we increase $q$.

Figure 2: The SFF for $N=20$ and various values of $q$.

Figure 3: The logarithm of the difference between the sum over the CSMM SFF
for $N=10$ and the CUE ($q\to 0$) SFF, plotted for various values of $q$. We
see that the difference is very small and decreases quite rapidly with $k$
but, conversely, increases with $q$.

Figure 4: The numerically computed connected unfolded SFF’s for $N=10$ and
$N=20$, which can be distinguished by the fact that they arrive at a plateau
at $n=10$ and $n=20$, respectively. Note that we plot only the connected SFF
here, as opposed to figures 1 and 2, to simplify comparison with Wigner-Dyson
universality. It is clear that Wigner-Dyson universality is recovered to high
precision for both $N=10$ and $N=20$ and all $q$ considered here.

The SFF’s plotted above were found without unfolding. To see the effect of
unfolding, the connected SFF was computed numerically for $N=10$ and $N=20$
numerically, using the Metropolis algorithm to generate the spectra. The
unfolding is done via Gaussian kernel density estimation using the Silverman
rule for bandwidth selection. The data sets for $N=10$ and $N=20$ contain at
least 10.000 and 5.000 samples, respectively, such that at least 100.000
levels are involved in the calculation of both SFF’s. These are plotted below,
where, to distinguish them from the analytically calculated (and non-unfolded)
SFF’s, we denote the numerically computed SFF’s by $L(n)$ 222The data for
these plots was kindly provided by Wouter Buijsman. We emphasize once more
that these are the connected SFF’s, where we omit the disconnected part to
enable easier comparison with Wigner-Dyson universality. The SFF’s for $N=10$
and $N=20$ are distinguished by the fact that they arrive at a plateau at
$n=10$ and $n=20$, respectively. As we can see, for both $N=10$ and $N=20$ and
all values of $q$ we consider, the connected SFF’s exhibit Wigner-Dyson
universality rather than intermediate statistics. We will arrive at similar
conclusions for the ‘t Hooft limit below, albeit via different methods.

###  3.3 The SFF in the ‘t Hooft-limit

Taking the ‘t Hooft limit, $N\to\infty$ and $g_{s}\to 0$ such that
$t=Ng_{s}=\text{finite}$, leads to $q\to 1$ and $q^{N}=y$ with $0\leq y<1$. In
this limit, the SFF turns into a remarkable sequence of polynomials of degree
$2n-1$ in $y$. We will first consider these polynomials and their properties,
before turning to the question of unfolding. We have calculated the connected
SFF for $n=1,\dots,11$, resulting in the expressions below.

$\displaystyle F(1)_{c}$ $\displaystyle=1-y\leavevmode\nobreak\ ,$
$\displaystyle F(2)_{c}$ $\displaystyle=2-4y+6y^{2}-4y^{3}\leavevmode\nobreak\
,$ $\displaystyle F(3)_{c}$
$\displaystyle=3-9y+36y^{2}-84y^{3}+90y^{4}-36y^{5}\leavevmode\nobreak\ ,$
$\displaystyle F(4)_{c}$
$\displaystyle=4-16y+120y^{2}-560y^{3}+1420y^{4}-1968y^{5}+1400y^{6}-400y^{7}\leavevmode\nobreak\
,$ $\displaystyle F(5)_{c}$
$\displaystyle=5-25y+300y^{2}-2300y^{3}+10150y^{4}-26880y^{5}+43400y^{6}-41800y^{7}+22050y^{8}-4900y^{9}\leavevmode\nobreak\
,$ $\displaystyle F(6)_{c}$
$\displaystyle=6-36y+630y^{2}-7140y^{3}+47880y^{4}-200592y^{5}+544824y^{6}-974160y^{7}+1137780y^{8}+$
$\displaystyle-834960y^{9}+349272y^{10}-63504y^{11}\leavevmode\nobreak\ ,$
$\displaystyle F(7)_{c}$
$\displaystyle=7-49y+1176y^{2}-18424y^{3}+173460y^{4}-1042524y^{5}+4187736y^{6}-11565624y^{7}+$
$\displaystyle+22246686y^{8}-29742020y^{9}+27087984y^{10}-16024176y^{11}+5549544y^{12}-853776y^{13}\leavevmode\nobreak\
,$ $\displaystyle F(8)_{c}$
$\displaystyle=8-64y+2016y^{2}-41664y^{3}+522480y^{4}-4237632y^{5}+23380896y^{6}-90830784y^{7}+$
$\displaystyle+253846296y^{8}-515838400y^{9}+762521760y^{10}-810927936y^{11}+604107504y^{12}+$
$\displaystyle-299065536y^{13}+88339680y^{14}-11778624y^{15}\leavevmode\nobreak\
,$ $\displaystyle F(9)_{c}$
$\displaystyle=9-81y+3240y^{2}-85320y^{3}+1372140y^{4}-14394996y^{5}+103900104y^{6}-535847400y^{7}+$
$\displaystyle+2026445850y^{8}-5713765200y^{9}+12118597920y^{10}-19364383584y^{11}+23165382240y^{12}+$
$\displaystyle-20414698920y^{13}+12853423440y^{14}-5468226192y^{15}+1407913650y^{16}-165636900y^{17}\leavevmode\nobreak\
,$ $\displaystyle F(10)_{c}$
$\displaystyle=10-100y+4950y^{2}-161700y^{3}+3240600y^{4}-42617520y^{5}+388588200y^{6}+$
$\displaystyle-2556668400y^{7}+12488661900y^{8}-46202499200y^{9}+131172321280y^{10}+$
$\displaystyle-287919216000y^{11}+489596250000y^{12}-642659556000y^{13}+644511582000y^{14}+$
$\displaystyle-484405727520y^{15}+263957736900y^{16}-98425126800y^{17}+22457091800y^{18}+$
$\displaystyle-2363904400y^{19}\leavevmode\nobreak\ ,$ $\displaystyle
F(11)_{c}$
$\displaystyle=11-121y+7260y^{2}-287980y^{3}+7031310y^{4}-113142744y^{5}+1269259992y^{6}+$
$\displaystyle-10345746840y^{7}+63147440070y^{8}-295025713840y^{9}+1071727584928y^{10}+$
$\displaystyle-3059501029728y^{11}+6907003486240y^{12}-12358366232520y^{13}+17490417413040y^{14}+$
$\displaystyle-19447530019632y^{15}+16771920490182y^{16}-10982054062980y^{17}+5272925154640y^{18}+$
$\displaystyle-1749762036880y^{19}+358415185128y^{20}-34134779536y^{21}\leavevmode\nobreak\
.$ (59)

To the best of the authors’ knowledge, the above polynomials have not appeared
in the literature before. Their complicated form belies the fact that the SFF
appears to be very close to a straight line for any $y$, with decreasing slope
for increasing $y$, see figure 5. We emphasize again that the SFF’s plotted in
figure 5 were found without applying any unfolding. We will consider the issue
of unfolding by rescaling the spectrum in section 3.3.2, see in particular
figure 7. In fact, there are three choices of $y$ for which the SFF is a
perfectly straight line. Writing $F(n;y)_{c}$ to indicate dependence on $y$,
we have

$\displaystyle F(n;0)_{c}$ $\displaystyle=n\leavevmode\nobreak\ ,$
$\displaystyle F(n;1/2)_{c}$ $\displaystyle=\frac{n}{2}\leavevmode\nobreak\ ,$
$\displaystyle F(n;1)_{c}$ $\displaystyle=0\leavevmode\nobreak\ .$ (60)

The fact that $F(n;0)_{c}=n$ was already mentioned in section 3.1 and is, in
essence, one of the results derived in [3]. The last equality, written as a
limit for $y$, can easily be seen to be generally true and will be further
commented on in section 3.4. The middle equality, on the other hand, is a
priori completely unexpected (at least to the authors).

Figure 5: The connected SFF in the ‘t Hooft limit, without unfolding, written
in equation (3.3). As one can see, the resulting SFF is very close to a
straight line for any choice of $y$.

One can see from the above plot that the SFF appears to be symmetric around
$F(n;1/2)_{c}=\frac{n}{2}$. Indeed, we find that

$\displaystyle F(n;y)_{c}+F(n;1-y)_{c}=n\leavevmode\nobreak\ .$ (61)

It can be seen that the polynomials $F(n)_{c}$ appearing in (3.3) can
factorized into the product of a factor $n(1-y)$ and polynomials $p_{n}(y)$ of
degree $2n-2$. The first few of them are given by

$\displaystyle p_{1}$ $\displaystyle=$ $\displaystyle 1,$ $\displaystyle
p_{2}$ $\displaystyle=$ $\displaystyle 1-y+2y^{2},$ $\displaystyle p_{3}$
$\displaystyle=$ $\displaystyle 1-2y+10y^{2}-18y^{3}+12y^{4},$ $\displaystyle
p_{4}$ $\displaystyle=$ $\displaystyle
1-3y+27y^{2}-113y^{3}+242y^{4}-250y^{5}+100y^{6},$ $\displaystyle p_{5}$
$\displaystyle=$ $\displaystyle
1-4y+56y^{2}-404y^{3}+1626y^{4}-3750y^{5}+4930y^{6}-3430y^{7}+980y^{8},$
$\displaystyle p_{6}$ $\displaystyle=$ $\displaystyle
1-5y+100y^{2}-1090y^{3}+6890y^{4}-26542y^{5}+64262y^{6}-98098y^{7}$
$\displaystyle+$ $\displaystyle 91532y^{8}-47628y^{9}+10584y^{10},$
$\displaystyle p_{7}$ $\displaystyle=$ $\displaystyle
1-6y+162y^{2}-2470y^{3}+22310y^{4}-126622y^{5}+471626y^{6}-1180606y^{7}$
$\displaystyle+$ $\displaystyle
1997492y^{8}-2251368y^{9}+1618344y^{10}-670824y^{11}+121968y^{12},$
$\displaystyle p_{8}$ $\displaystyle=$ $\displaystyle
1-7y+245y^{2}-4963y^{3}+60347y^{4}-469357y^{5}+2453255y^{6}-8900593y^{7}+22830194y^{8}$
(62) $\displaystyle-$ $\displaystyle
41649606y^{9}+53665614y^{10}-47700378y^{11}+27813060y^{12}-9570132y^{13}+1472328y^{14}.$

We were able to identify the coefficients for the following powers of $y$ as

$\displaystyle y^{0}:$ $\displaystyle 1$ (63) $\displaystyle y^{1}:$
$\displaystyle-n$ $\displaystyle y^{2}:$ $\displaystyle n^{2}(n+3)/2$
$\displaystyle y^{3}:$ $\displaystyle-n(n-1)(2+10n+6n^{2}+n^{3})/6$
$\displaystyle y^{4}:$ $\displaystyle
n(n-1)(-72-224n-28n^{2}+87n^{3}+40n^{4}+5n^{5})/144$ $\displaystyle y^{2n-1}:$
$\displaystyle-(C^{2n}_{n})^{2}(2n-1)/2(n+1)$ $\displaystyle y^{2n}:$
$\displaystyle(C^{2n}_{n})^{2}/(n+1),$

where we have the binomial coefficient $C^{2n}_{n}=(2n)!/(n!)^{2}$. It seems
that no further information about the expansion coefficients of $p_{n}(y)$ can
be obtained easily. This prevents us from generalizing the connected SFF
beyond the 11 terms written in (3.3).

We list here some further observations on these polynomials. We notice that
all $p_{n}(y)-1$ are divisible by $y(2y-1)$. Denoting
$s_{n}(y)=(p_{n}(y)-1)/(y(2y-1))$, we find that these can be expressed as
functions of $x$, which reduces the degree even further. Finally, considering
the polynomials $w_{n}(y)=s_{n+1}(y)-s_{n}(y)$ one can realize 333We are very
grateful to Dr. Denis Kurlov for pointing this fact out to us. that they have
the following remarkable properties:

  1. a)

all the roots of $w_{n}(y)$ are real;

  2. b)

they occupy the interval $[0,1]$;

  3. c)

the roots have the interlacing property, meaning that the roots of lower
degree polynomials are located in between the roots of higher degree
polynomials444Such interlacing has been observed in related contexts, see
[58].

This and other observations suggest that the polynomials $w_{n}(y)$ could form
a family of orthogonal polynomials. However, the polynomials in (3.3) are only
of even degree, while the odd-degree polynomials are missing. It would be very
interesting to extend the sequence of polynomials to terms of higher degree
and further explore some of the considerations described above.

####  3.3.1 Level density

We now consider the question of unfolding in the ‘t Hooft limit, where we have

$\displaystyle N^{-1}\left\langle\text{tr}U^{n}\right\rangle=\frac{1}{2n\log
y}\left[P_{n}(2y-1)-P_{n-1}(2y-1)\right]\leavevmode\nobreak\ ,$ (64)

where $P_{n}$ is the Legendre polynomial. This was already found in [54]. In
particular, $\left\langle\text{tr}U^{n}\right\rangle$ diverges in the ‘t Hooft
limit in such a way that $N^{-1}\left\langle\text{tr}U^{n}\right\rangle$ is
generally finite, as can be seen from taking $q\to 1$ in (3.1). We thus have a
level density which is no longer flat but contains an oscillatory contribution
as well,

$\displaystyle\rho(\theta;y)=(2\pi)^{-1}\left[1+\frac{1}{\log
y}\sum_{n=1}^{\infty}\frac{P_{n}(2y-1)-P_{n-1}(2y-1)}{n}\cos(n\theta)\right]\leavevmode\nobreak\
.$ (65)

For $\lvert t\rvert<1$, the generating function of the Legendre polynomials
reads

$\displaystyle
P(x,t)=\sum_{n=0}^{\infty}P_{n}(x)t^{n}=\frac{1}{\sqrt{1-2xt+t^{2}}}\leavevmode\nobreak\
,$ (66)

which can be integrated to find, for $\lvert z\rvert<1$,

$\displaystyle\int_{0}^{z}P(x,t)=\sum_{n=1}^{\infty}\frac{P_{n-1}(x)z^{n}}{n}=\log\left(z-x+\sqrt{1-2xz+z^{2}}\right)-\log\left(1-x\right)\leavevmode\nobreak\
.$ (67)

Similarly,

$\displaystyle\sum_{n=1}^{\infty}\frac{P_{n}(x)z^{n}}{n}=\log
2-\log\left(1-xz+\sqrt{1-2xz+z^{2}}\right)\leavevmode\nobreak\ .$ (68)

Using Abel’s theorem, one may find that

$\displaystyle\rho(\theta;y)=\frac{1}{2\pi}+\frac{1}{2\pi\log y}\left[\log
2-\log R(\cos\theta,2y-1)\right]\leavevmode\nobreak\ ,$ (69)

where

$\displaystyle
R(z,x)=\begin{cases}(\sqrt{1+z}+\sqrt{z-x})^{2}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ x<z\\\ 1+x\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ x\geq
z\leavevmode\nobreak\ .\end{cases}$ (70)

It is easy to see that the second case listed above, $x\geq z$, gives a zero
level density, since then $\log R(z,x)=\log 2y$. In particular, a gap opens in
the spectrum555This critical angle was already found in [54], although our
level density appears to be different from the expression obtained there.,

$\displaystyle\rho(\theta;y)=0\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\theta>\theta_{c}=\arccos(2y-1)\leavevmode\nobreak\ .$ (71)

This is plotted in figure 6 for $y=.1,.2,\dots,.9$. One can see that these
level densities are approximately of semicircular form. Indeed, writing
$r(\theta;y)=\rho(0;y)\sqrt{1-\frac{\theta^{2}}{\theta_{c}^{2}}}$ , the
difference between $\rho(\theta;y)$ and $r(\theta;y)$ for
$y=0.1,0.2,\dots,0.9$ remains smaller than 0.006 for all $\theta$ and
decreases with $y$.

Figure 6: The level density in the ‘t Hooft limit plotted for various values
of $y$. For $y>0$, a gap opens at $\theta_{c}=\arccos(2y-1)$ which increases
in size with $y$, with the level remaining a convex function.

####  3.3.2 Unfolding

To compare the connected SFF in the ‘t Hooft limit with the CUE result, we
have to unfold the spectrum. Strictly speaking, unfolding involves a change of
variables to the staircase function, (see e.g. section 5.19 of [59]),

$\sigma(\theta)=\int_{\theta_{c}}^{\theta}d\theta^{\prime}\rho(\theta^{\prime})\leavevmode\nobreak\
.$ (72)

The level density in terms of $\sigma$ is a perfectly flat function. The
unfolded SFF is then given by

$\frac{1}{N}\langle|\sum_{j=1}^{N}e^{2\pi
i\sigma_{j}}|^{2}\rangle\leavevmode\nobreak\ .$ (73)

However, this unfolding procedure is often difficult in practice, and our case
is no exception. Finding $\sigma(\theta)$ using the closed form expression for
the level density obtained above is not very complicated, but the expression
in (73) is not amenable to evaluation. For this reason, we instead perform a
constant rescaling to a variable $s$ in terms of which the level density
$\rho(s)$ averaged over its support is independent of $q^{N}$, that is, we
simply rescale so that the average spacing is the same for all $y$. For any
value of $y$, we take the support of the level density and imagine we can
replace the level density by a box-shaped density of the same support. We then
rescale the support so that it is again of size $2\pi$. To do so, we write

$s(\theta)=\frac{\pi\theta}{\theta_{c}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ s\in[0,2\pi)\leavevmode\nobreak\ ,$
(74)

so that averaging over its (rescaled) support gives

$\overline{\rho}=\frac{1}{2\pi}\int_{0}^{2\pi}ds\rho(s)=\frac{1}{2\pi}=\rho(\theta)_{\text{\tiny
CUE}}\leavevmode\nobreak\ .$ (75)

In terms of the rescaled eigenphases, the level density $\rho(s)$ is of almost
exactly the same shape for any $y>0$, that is, the various densities in 6 are
approximately related by rescaling. With this unfolding, the SFF is given by

$\tilde{F}(n)=\left\langle\left\lvert\sum_{j=1}^{N}e^{2\pi
is_{j}}\right\rvert^{2}\right\rangle=F\left(\frac{\pi
n}{\theta_{c}}\right)\leavevmode\nobreak\ .$ (76)

The discrete SFF, $F\left(\frac{\pi n}{\theta_{c}}\right)$, can only be
evaluated at integer $\frac{\pi n}{\theta_{c}}$. However, we saw in figure 5
that $F(n)_{c}$ is very close to a linear ramp with slope $\leq 1$. Since

$F(n)_{c}\approx f(y)n\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ 0\leq f(y)\leq 1\leavevmode\nobreak\ ,$ (77)

the unfolded connected SFF is approximately

$\tilde{F}(n)_{c}\approx\frac{\pi}{\theta_{c}}F(n)_{c}\eqqcolon
G(n)\leavevmode\nobreak\ ,$ (78)

which is plotted in 7.

Figure 7: The unfolded connected SFF,
$G(n)\eqqcolon\frac{\pi}{\theta_{c}}F(n)_{c}$, for $n=1,\dots,11$. At $y=0$,
we have exactly $G(n)=n$. For $y>0$, it is clear that $G(n)\approx n$ remains
true to high precision, especially for larger $n$, so that the unfolded
connected SFF is very close to a linear ramp with unit slope. Only for $y\to
1^{-}$ does $G(n)$ go to zero and is any resemblance to WD-universality lost.

It is clear that $G(n)$ closely resembles a linear ramp of unit slope for all
$y$ except close to unity, with resemblance increasing with $n$. This entails
that $f(y)\approx\frac{\theta_{c}}{\pi}$. Only for $y$ close to 1 do we get a
significant deviation from WD-universality, with $G(n)\to 0$ for $y\to 1^{-}$.
This demonstrates that the unfolded connected SFF in the ‘t Hooft limit
reproduces WD universality to high precision for $n=1,\dots,11$ and for all
$y$ except $y\approx 1$. Although it might be that deviations from WD
universality will emerge for lower values of $y$ as we increase $n$ beyond 11,
the aforementioned facts that $G(n)$ is close to a linear ramp and that this
precision in fact increases with $n$ would appear to render such deviations
rather unlikely. Conversely, it may be that deviations from WD will continue
to be squeezed into an ever smaller interval below $y=1$ as we increase $n$,
but here, too, we cannot make definitive statements. We note again that the
unfolding implemented here involved only a constant rescaling of the
eigenphases. Perhaps unfolding as in (73) would remove what deviations from
the linear ramp remain in the plot below, but this is of course speculative.
Further, spectral form factors of various systems often display non-universal
behavior close to the origin, which then disappear further away from the
origin where the SFF approaches a linear ramp. The deviations seen in the plot
below, except for the region $y\approx 1$, may be just such an effect.

As mentioned previously, the CSMM reduces to the CUE for $q\to 0$, where we
know WD-universality to hold. In our previous work[3], we demonstrated that
WD-universality holds for any $q<1$ and $N\to\infty$ as well. The ‘t Hooft
limit, which involves $q\to 1^{-}$, should then constitute the greatest
possible deviation from the CUE result, yet WD-universality reappears for all
$y$ other than $y\approx 1$ after even a very simple unfolding. This would
appear to be rather unexpected, as the CSMM was introduced and extensively
studied as a random matrix model for intermediate statistics, as described in
more detail in the introduction. We further comment on this result and its
implications in the conclusion.

###  3.4 Non-commutativity of the limit $q\to 1$ and $N\to\infty$

One can see from the expression of the SFF that the limits $q\to 1$ and
$N\to\infty$ do not commute. Such non-commutativity has been discussed in the
literature already decades ago, see [54]. In particular, if we take $q\to 1$
into expression (2) for finite $N$, the Schur polynomials simply give the
dimension of the representation, that is

$s_{\lambda}(1,1,\dots,1)=\dim\lambda\leavevmode\nobreak\ .$ (79)

Plugging this into (2) with $\lambda=(n-r,1^{r})$ and $\mu=(n-s,1^{s})$ shows
that the SFF would be simply given by

$\left(\sum_{r=0}^{n-1}(-1)\frac{(N+n-r-1)!}{(N-r-1)!(n-r-1)!r!n}\right)^{2}=N^{2}\leavevmode\nobreak\
.$ (80)

for all $n$. In terms of knot theory, we see that taking $q\to 1$ for $N$
finite breaks the $(2n,2)$-torus link that is the SFF up into its separate
$(n,1)$-torus knot components, as we have

$\lim_{q\to 1}\langle\lvert\text{tr}U^{n}\rvert^{2}\rangle=\lim_{q\to
1}\langle\text{tr}U^{n}\rangle\langle\text{tr}U^{-n}\rangle=\lim_{q\to
1}\left(\langle\text{tr}U^{n}\rangle\right)^{2}\leavevmode\nobreak\ .$ (81)

The connected SFF then equals zero, as was the case in section 3.3. We
consider the case where $t=Ng_{s}\ll 1$ is very small. This allows us to use
the following expansion [[57] I.3, example 10],

$s_{\lambda}(1+x_{1},1+x_{2},\dots,1+x_{N})=\sum_{\mu}d_{\lambda\mu}s_{\mu}(x_{1},\dots,x_{N})\leavevmode\nobreak\
,$ (82)

where we sum over all $\mu\subseteq\lambda$ and where

$d_{\lambda\mu}=\det\begin{pmatrix}\lambda_{i}+n-1\\\
\mu_{j}+n-j\end{pmatrix}_{1\leq i,j\leq N}\leavevmode\nobreak\ .$ (83)

Some simple examples are given by (see e.g. [60])

$d_{\lambda\emptyset}=\dim\lambda\leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
d_{\lambda\Box}=\dim\lambda\frac{c_{1}(\lambda)}{N}\leavevmode\nobreak\ ,$
(84)

where the first Casimir invariant is given by
$c_{1}(\lambda)=\lvert\lambda\rvert=\sum_{i}\lambda_{i}$. For $q=e^{-g_{s}}$
close to 1 and $Ng_{s}\ll 1$, we have

$s_{(a,1^{b})}(q^{j-1})\simeq
s_{(a,1^{b})}(1,1-g_{s},1-2g_{s},\dots)=\sum_{\mu}d_{(a,1^{b})\mu}s_{\mu}(0,-g_{s},-2g_{s},\dots)\leavevmode\nobreak\
.$ (85)

Expanding up to linear order in $g_{s}$, we only get contributions for
$\mu=\emptyset$ and $\mu=\Box$, which gives

$\displaystyle s_{(a,1^{b})}(0,-g_{s},-2g_{s},\dots)$
$\displaystyle=\dim(a,1^{b})\left(1+\frac{a+b}{N}(-g_{s}-2g_{s}-\dots)\right)$
$\displaystyle=\dim(a,1^{b})\left(1-\frac{(a+b)(N-1)}{2}g_{s}\right)\leavevmode\nobreak\
.$ (86)

Further,

$\displaystyle s_{(a,1^{b})}(q^{-c},1,q,\dots,q^{d-1},q^{d+1},\dots,q^{N-1}))$
$\displaystyle\simeq
s_{(a,1^{b})}(cg_{s},-g_{s},\dots,-(d-1)g_{s},-(d+1)g_{s},\dots)$
$\displaystyle=\dim(a,1^{b})\left[1+(a+b)\left(\frac{1-N}{2}+\frac{c+d}{N}\right)g_{s}\right]\leavevmode\nobreak\
.$ (87)

Plugging this into (2) for $\lambda=(n-r,1^{r})$ and $\mu=(n-s,1^{s})$ gives

$\displaystyle\langle W_{\lambda\mu}\rangle-\langle W_{\lambda}\rangle\langle
W_{\mu}\rangle=\dim(n-r,1^{r})\dim(n-s,1^{s})\frac{n^{2}g_{s}}{N}+\mathcal{O}(g_{s}^{2})$
(88)

We thus see that, to first order in $Ng_{s}$, the Wilson loop factorizes, so
that

$\displaystyle F(n)_{c}=tn^{2}+\mathcal{O}(t^{2})\leavevmode\nobreak\ .$ (89)

If we now take $N\to\infty$ in such a way that $t$ remains small, we clearly
get a very different result from the linear ramp $F(n)_{c}=n$ that is found
when taking $q\to 1$ after $N\to\infty$. Indeed, one may check that the
connected SFF in the ‘t Hooft limit for small $n$ and $y\lesssim 1$ is very
close to $tn^{2}$.

##  4 Conclusion

In this work, we calculate the SFF of the CSMM for general $q=e^{-g_{s}}$ and
matrix size $N$. We find that, as $y=q^{N}\to 0$, we recover our results in
[3] for the case of the CSMM. That is, the connected SFF,
$F(n)_{c}=\left\langle\lvert\text{tr}U^{n}\rvert^{2}\right\rangle-\left\langle\text{tr}U^{n}\right\rangle^{2}$,
is exactly given by a linear ramp of unit slope. For $y$ different from zero,
we see that the (approximate) slope of the SFF becomes smaller than one as we
increase $q$. For $n>N$, the SFF eventually saturates at a plateau, which
takes longer for larger $q$ due to the fact that the slope decreases with $q$.
The emergence of the linear ramp and its saturation at a plateau is shown to
arise from the properties of Schur bilinears appearing in the character
expansion of the SFF.

In the ‘t Hooft limit, the connected SFF reduces to a sequence of polynomials
of degree $2n-1$ in $y$, which we calculated up to $n=11$. As far as the
authors are aware, this sequence has not appeared in the literature thus far.
Before unfolding, the connected SFF is again approximately given by a linear
ramp, but now with slope $\leq 1$. As described in section 3.3, the SFF is
symmetric around $y=1/2$, where we have $F(n)_{c}=\frac{n}{2}$. Further,
$N^{-1}\left\langle\text{tr}U^{n}\right\rangle$ is generally finite, so that
the level density is no longer flat. In particular, a gap opens up for $y>0$,
so that the support of the level density is given by
$(-\theta_{c},\theta_{c})$ for $\theta_{c}=\arccos(2y-1)$. To unfold the
spectrum, we rescale the eigenphases so that the support is again an interval
of length $2\pi$. Upon unfolding, the SFF is again very close to a linear ramp
for all $y$ except $y$ close to 1, with precision increasing with $n$. That
is, we recover Wigner-Dyson universality even in the ‘t Hooft limit as long as
$y$ is not too close to unity. It would be interesting to see whether
deviations from WD for $y\to 1^{-}$ persist in $F(n)_{c}$ for $n\geq 12$,
whether they continue to be squeezed in a smaller interval below $y=1$, or
whether they perhaps disappear altogether. The fact that we have to base our
conclusions on 11 instances of the connected SFF prevents us from making
definitive statements on this point, but perhaps future investigation will
shed further light on it.

As described in the introduction, the CSMM was originally introduced [1] to
describe the intermediate statistics of disordered electrons at the mobility
edge, and there is a significant amount of literature on this application of
the CSMM and related ensembles, see e.g. [32], [31], [37], [36], [61], [62],
[63]. Indeed, the ‘t Hooft limit involves $q\to 1$, which is the opposite
extreme of the CUE limit, $q\to 0$, so it should come as a surprise that WD-
universality is recovered even here, and indeed for all $y$ except $y\approx
1$. We emphasize that we do not mean to cast doubt on the results derived in
the aforementioned papers. Although their results may appear hard to reconcile
with ours at first sight, their analysis involves a different set of limits
and approximations, the approximation in our case being the unfolding by a
constant rescaling and in their case e.g. considering only a small eigenvalue
window around the origin. Their analysis centers on the hermitian version of
the CSMM where the weight function is $\sim e^{-\alpha\log^{2}(x)}$ for
$x\in\mathds{R}$, and it employs the theory of orthogonal polynomials rather
than the character expansion used here. The latter two points are not of
fundamental importance as they should be merely different descriptions of the
same underlying structure, yet they do complicate the comparison between these
sets of results. Further, it was shown very recently [64] that defining the
SFF as $K(t)$ with $t\in\mathds{C}$ and taking $\text{Im}(t)\to 0$ generally
leads to different SFF than when $t$ is taken to be always real-valued, so it
is clear that we have not exhausted the different limits in which one can
calculate the SFF. It would be very interesting to understand under what
conditions the CSMM or related models exhibit deviations from WD-universality,
as apparently it is not enough to take the ‘disorder parameter’ $q$ to its
maximum value.

The authors believe that some of the results derived here are also of
mathematical interest. As mentioned in the introduction, the SFF is
proportional to the HOMFLY invariant of a $(2n,2)$-torus link with components
carrying fundamental and antifundamental representations. The calculation of
the SFF thus provides explicit expressions for new link invariants, both for
general $q,\leavevmode\nobreak\ N$ as well as in the ‘t Hooft limit. Due to
the appearance of $U(N)$ Chern-Simons theory at large $N$ in the form of
various topological string theories described in the the introduction, and the
relation of these large $N$ dualities to enumerative geometry and intersection
theory, the results derived here could be of mathematical interest beyond knot
theory.

##  5 Acknowledgements

We are greatly indebted to Wouter Buijsman for helping us out with the
numerical calculations in this work. Further, we are grateful to Aleksandr
Garkun, Vladimir Kravtsov, and Denis Kurlov for useful discussions and for
their help at different stages of this project. This work is part of the
DeltaITP consortium, a program of the Netherlands Organization for Scientific
Research (NWO) funded by the Dutch Ministry of Education, Culture and Science
(OCW).

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