# Asymptotics of $k$ dimensional spherical integrals and Applications

Alice Guionnet  Université de Lyon, ENSL, CNRS, France  Alice.Guionnet@ens-
lyon.fr and  Jonathan Husson  Université de Lyon, ENSL, CNRS, France
<EMAIL_ADDRESS>
This work was supported in part by ERC Project LDRAM : ERC-2019-ADG Project
884584

Abstract

In this article, we prove that $k$-dimensional spherical integrals are
asymptotically equivalent to the product of $1$-dimensional spherical
integrals. This allows us to generalize several large deviations principles in
random matrix theory known before only in a one-dimensional case. As examples,
we study the universality of the large deviations for $k$ extreme eigenvalues
of Wigner matrices (resp. Wishart matrices, resp. matrices with general
variance profiles) with sharp sub-Gaussian entries, as well as large
deviations principles for extreme eigenvalues of Gaussian Wigner and Wishart
matrices with a finite dimensional perturbation.

##  1\. Introduction

Spherical integrals are integrals over the unitary or orthogonal group which
can be seen as natural Fourier (or Laplace transforms) over matrices. As such,
they play a central role in random matrix theory. They can for instance be
used to express the density of the distribution of random matrices [7, 22]. In
the unitary case (and more generally when one integrates over a compact,
connected, semisimple Lie group), Harish-Chandra [14] and Itzykson and Zuber
[17] derived formulas for such integrals. However, these formulas do not allow
to estimate in general their asymptotics as the dimension goes to infinity
because they are given in terms of a determinant, so a signed sum of diverging
terms. It is however crucial to estimate such asymptotics in random matrix
theory to derive law of large numbers for matrix models or large deviations
principles. Such asymptotics also permit to see the $R$-transform as the limit
of spherical transforms, and thus of natural Laplace transform in the space of
matrices [10]. Such natural representation was recently generalized to the
$S$-transform [21]. In the case of a one dimensional spherical integral where
one integrates over a uniformly distributed vector on the sphere, such
asymptotics where derived in [10] (see also [9]). The case where the spherical
integral is taken over the whole unitary or orthogonal group was adressed in
[12]. In the case where the exponent is small enough, and the spherical
integral is $k$ dimensional, with $k$ much smaller of the dimension, the
spherical integrals was shown to be equivalent to a product of one dimensional
spherical integrals when $k$ is finite [10], or going to infinity in a
mesoscopic regime where $k$ grows like a power of the dimension [5, 15]. In
this article, we show that when $k$ is finite, this property remains true for
all ranges of parameters. Indeed, we prove that the limit of $k$-dimensional
spherical integrals is equivalent to the sum of one dimensional integrals
which are evaluated at the successive largest eigenvalues. For instance, as
foreseen in [19], the limit of a 2-dimensional spherical integral depends on
the two largest outliers in the large parameters regime, and not only the top
one. As a consequence, $k$-dimensional spherical integrals allow us to study
the universality of the large deviations for $k$ extreme eigenvalues of Wigner
matrices with sharp sub-Gaussian entries, hence generalizing the results of
[13] to finitely many extreme eigenvalues. Similarly, we extend the
universality of large deviations for Wishart matrices [13] and matrices with
general variance profile [16] with sharp sub-Gaussian tails to finitely many
extreme eigenvalues. We also prove large deviations principles for extreme
eigenvalues of Gaussian Wigner and Wishart matrices with a finite dimensional
perturbation. This generalizes the one-dimensional case derived in [18]. The
large deviations rate functions of these large deviations principles simply
decompose as a sum of the one dimensional rate functions.

The approach of this paper differs from the arguments used in [10] in the one-
dimensional case which relied heavily on the representation of the uniform law
on the sphere in terms of Gaussian variables. Instead, it is based on
considering first spherical integrals of matrices with finitely many different
eigenvalues where the uniform law on the sphere can be easily described by
Beta-distribution and where rate functions can be more simply described as
maximum over real numbers, see section 3. We then generalize our results to
matrices with continuous spectrum by density, see section 4. Applications to
large deviations principles for extreme eigenvalues of random matrices are
given in section 5.

##  2\. Statement of the results

We consider a $N\times N$ Hermitian matrix ${\bf X}_{N}$ such that the
empirical measure of its eigenvalues

$\hat{\mu}_{{\bf X}_{N}}=\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}}$

converges towards a probability measure $\mu$ with support with rightmost
point $r_{\mu}$ and leftmost point $l_{\mu}$ which are assumed to be finite.
Let $k,\ell$ be two integer numbers. Let
$\lambda_{1}^{N}\geq\lambda_{2}^{N}\geq\cdots\geq\lambda_{k}^{N}\geq r_{\mu}$
be the $\ell$ largest outliers of ${\bf X}_{N}$ counted with multiplicity one,
$\lambda_{-1}^{N}\leq\cdots\leq\lambda_{-\ell}^{N}\leq l_{\mu}$ be the
smallest outliers of ${\bf X}_{N}$ with multiplicity one (but eventually
equal). Assume that

$\lim_{N\rightarrow\infty}\lambda_{i}^{N}=\lambda_{i}>r_{\mu}\mbox{ for
}i\in[1,k],\lim_{N\rightarrow\infty}\lambda_{-i}^{N}=\lambda_{-i}<l_{\mu}\mbox{
for }i\in[1,l]$

The main result of our paper is the following. Denote by $(e_{i})_{-\ell\leq
i\leq k\atop i\neq 0}$ a family of $k+\ell$ orthonormal eigenvectors following
the uniform law on the sphere with radius one, taken with complex coordinates
if $\beta=2$ and real coordinates if $\beta=1$. Then

######  Proposition 1.

Let $\theta_{1}\geq\theta_{2}\geq\cdots\geq\theta_{k}\geq
0\geq\theta_{-\ell}\geq\cdots\geq\theta_{-1}$. Then

$\lim_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-l\atop i\neq 0}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]\qquad\qquad$
$\qquad\qquad=\frac{\beta}{2}\left(\sum_{i=1}^{k}J(\mu,\theta_{i},\lambda_{i})+\sum_{i=1}^{\ell}J(\mu,\theta_{-i},\lambda_{-i})\right)\,.$

Here,
$J(\mu,\theta,\lambda)=K\left(\mu,\theta,\lambda,v(\mu,\theta,\lambda)\right)$
with

$K(\mu,\theta,\lambda,v)=\theta\lambda+(v-\lambda)G_{\mu}(v)-\ln\theta-\int\ln|v-x|d\mu(x)-1$

and

$v(\mu,\theta,\lambda)=\left\\{\begin{array}[]{c l}\lambda&\mbox{ if
}G_{\mu}(\lambda)\leq\theta,\cr G_{\mu}^{-1}(\theta)&\mbox{ if
}G_{\mu}(\lambda)>\theta.\cr\end{array}\right.$

$G_{\mu}$ denotes the Cauchy-Stieltjes transform given, for $z$ outside the
support of $\mu$, by $G_{\mu}(z)=\int(z-x)^{-1}d\mu(x)$.

@@ As a first application, we generalize the universality of the large
deviations of the largest eigenvalue for Wigner matrices with sharp sub-
Gaussian tails [13] to the $k$-th extreme eigenvalues. We consider a Wigner
matrix ${\bf X}_{N}$ with entries $\left(\frac{X_{ij}}{\sqrt{N}}\right)_{1\leq
i,j\leq N}$ where $(X_{ij})_{i\leq j}$ are independent centered variables such
that

$\mathbb{E}[|X_{ij}|^{2}]=1,\quad i<j\mbox{ and
}\mathbb{E}[|X_{ii}|^{2}]=2^{1_{\beta=1}}$ (1)

where $\beta=1$ if the entries are real, and $\beta=2$ if they are complex. In
the complex case we assume that the real and the imaginary part of
$X_{ij},1\leq i<j\leq N,$ are independent. We moreover assume that the
$X_{ij}$ have sharp sub-Gaussian tails in the sense that

$\mathbb{E}[\exp(\Re(aX_{ij}))]\leq\exp(\frac{|a|^{2}\mathbb{E}[|X_{ij}|^{2}]}{2})$
(2)

where $a$ is any complex number in the case where $\beta=2$ and any real
number in the case $\beta=1$. We finally define the following concentration
assumption.

######  Assumption 2.

We say that ${\bf X}_{N}$ concentrates if the spectral radius of ${\bf
X}_{N}$, $||{\bf X}_{N}||$, concentrates as well as the empirical measure
$\hat{\mu}_{{\bf X}_{N}}$ of its eigenvalues in the following sense. First,
$\|X_{N}\|$ is exponentially tight at the scale $N$:

$\lim_{K\to+\infty}\limsup_{N\to+\infty}\frac{1}{N}\log\mathbb{P}\big{(}||{\bf
X}_{N}||>K\big{)}=-\infty.$ (3)

Moreover, the empirical distribution of the eigenvalues $\hat{\mu}_{{\bf
X}_{N}}$ concentrates at the scale $N$:

$\limsup_{N\to+\infty}\frac{1}{N}\log\mathbb{P}\left(d(\hat{\mu}_{{\bf
X}_{N}},\sigma)>\varepsilon(N)\right)=-\infty,$ (4)

for some $\varepsilon(N)$ goes to zero as $N$ goes to infinity, where $d$ is a
distance compatible with the weak topology and $\sigma$ is the semi-circle
law, defined by

$\sigma(dx)=\frac{1}{2\pi}\sqrt{4-x^{2}}\mathds{1}_{|x|\leq 2}dx.$

In our previous paper [13] we took $\varepsilon(N)=N^{-\kappa}$. This
hypothesis was needed to insure the continuity of spherical integrals
according to [18]. However, part of the consequences of our new approach is
that spherical integrals enjoy better continuity properties, see the Appendix
6. Assumption 2 is satisfied by all the matrix models we shall consider below
(Wigner, Wishart and variance profile) as soon as the entries satisfy log-
Sobolev inequality with uniformly bounded constant or are bounded (see [1] and
the Appendix in [13]). Examples of entries satisfying all our hypotheses
(including (2)) are Rademacher variables or uniform variables. We prove the
following universality of the large deviations for the extreme eigenvalues of
${\bf X}_{N}$:

######  Theorem 3.

Let ${\bf X}_{N}=(\frac{X_{ij}}{\sqrt{N}})_{{i,j}}$ be a $N\times N$ Hermitian
matrix where $(X_{i,j})_{i\leq j}$ are centered independent entries satisfying
(1) and (2), as well as such that ${\bf X}_{N}$ satisfies Assumption 2. Let
$\lambda_{1}^{N}\geq\lambda_{2}^{N}\geq\cdots\geq\lambda_{N}^{N}$ be the $N$
eigenvalues of ${\bf X}_{N}$. Let $k$ be a fixed integer number. Then the law
of
$\bar{\lambda}^{N}=(\lambda_{1}^{N},\lambda_{2}^{N},\ldots,\lambda_{k}^{N},\lambda_{N-k}^{N},\lambda_{N-k+1}^{N},\ldots,\lambda_{N}^{N})$
satisfies a large deviations principle in the scale $N$ and with good rate
function $I(x_{1},\ldots,x_{k},x_{-k},\ldots,x_{-1})$ which is infinite unless
$\bar{x}=(x_{1},\ldots,x_{k},x_{-k},\ldots,x_{-1})$ satisfies

$x_{1}\geq x_{2}\geq\cdots\geq x_{k}\geq 2\geq-2\geq x_{-k}\geq
x_{-k+1}\geq\cdots\geq x_{-1}$

and is given by

$I(x_{1},\ldots,x_{k},x_{-k},\ldots,x_{-1})=\frac{\beta}{2}\left(\sum_{i=1}^{k}\int_{2}^{x_{i}}\sqrt{t^{2}-4}dt+\sum_{i=-1}^{-k}\int_{2}^{-x_{-i}}\sqrt{t^{2}-4}dt\right)\,.$

Observe that this result is well known in the Gaussian case for the case
$k=1$, see [1, Section 2.6.2] and [3]. The case of more general $k$ but
Gaussian entries is a straightforward generalization, see e.g. [4]. The case
of sharp sub-Gaussian entries and $k=1$ was proven in [13, Theorem 1.4 and
Theorem 1.5]. This result can also be generalized for Wishart matrices. We
consider ${\bf G}_{L,M}$ a $L\times M$ random matrix and set $N=L+M$. We
define the Wishart matrix ${\bf W}_{L,M}=\frac{1}{{M}}{\bf G}_{L,M}{\bf
G}^{*}_{L,M}$. If $L/M$ goes to $\alpha\leq 1$, it is well known that the
spectral measure of ${\bf W}_{M,L}$ converges towards the Pastur Marchenko
distribution

$d\pi_{\alpha}(x)=\frac{1}{2\pi\alpha
x}\sqrt{(\lambda_{+}-x)(x-\lambda_{-})}dx\,.$

where $\lambda_{\pm}=(1\pm\sqrt{\alpha})^{2}$. Then we have the following :

######  Theorem 4.

Let ${\bf G}_{L,M}=(X_{ij})_{{1\leq i\leq L\atop 1\leq j\leq M}}$ be a
$L\times M$ matrix where $(X_{i,j})_{i,j}$ are centered independent entries
satisfying (1) and (2), as well as such that ${\bf W}_{{L,M}}$ satisfies
Assumption 2. Let $k\geq 0$ and $\lambda^{N}_{1}\geq...\geq\lambda^{N}_{k}$
the $k$ largest eigenvalues of ${\bf W}_{L,M}$. Assume that there exists
$\alpha\leq 1$ and $\kappa\geq 0$ so that $L/M-\alpha=o(N^{-\kappa})$. Then
$(\lambda^{N}_{1},...,\lambda^{N}_{k})$ satisfies a large deviations principle
in the scale $N$ with good rate function $J(x_{1},...,x_{k})$ which is
infinite unless $x_{1}\geq...\geq x_{k}\geq b_{\alpha}$ and :

$J(x_{1},...,x_{k})=\frac{\beta}{{4(1+\alpha)}}\sum_{i=1}^{k}\int_{\lambda_{-}}^{x_{i}}\frac{\sqrt{(y-\lambda_{-})(y-\lambda_{+})}}{y}dy$

As in the Wigner case, as soon as the entries satisfy log-Sobolev inequality
or are compactly supported, insuring that the empirical measure converges
towards the Pastur-Marchenko distribution with probability larger than any
exponential and that the norm of Wishart matrices stays bounded, yielding a
property similar to Assumption 2 for ${\bf W}_{L,M}$.

This result can be further extended to Wigner matrices with variance profiles.
Those matrices are built by letting ${\bf
X}^{\sigma}_{N}(i,j)=\sigma_{N}(i,j)\frac{X_{i,j}}{\sqrt{N}}$ where :

  * •

either there exists $p\in\mathbb{N}$, $\alpha_{1}(N),...,\alpha_{p}(N)>0$ such
that $\sum_{1}^{p}\alpha_{i}(N)=N$ and $\lim\alpha_{i}(N)/N=\alpha_{i}>0$, and
$(\sigma_{i,j})_{i,j}\in M_{p,p}(\mathbb{R}^{+})$, $\sigma=\sigma^{T}$, such
that :

$\sigma_{N}(i,j)=\sum_{k,l=1}^{p}\sigma_{k,l}\mathds{1}_{I^{k}_{N}\times
I^{l}_{N}}(i,j)$

where $I^{1}_{N}=[0,\alpha_{1}(N)]$ and
$I^{i+1}_{N}=[\sum_{j=1}^{i}\alpha_{j}(N)+1,\sum_{j=1}^{i+1}\alpha_{j}(N)]$.
This case will be called the piecewise constant case with parameters $\sigma$
and $\alpha$.

  * •

either $\sigma_{N}(i,j)=\sigma(i/N,j/N)$ where $\sigma$ is a continuous
symetric positive function of $[0,1]^{2}$. This case will be called the
continuous case.

We will also make the following assumption on the variance profiles :

######  Assumption 5.

  * •

In the piecewise constant case, we assume that the quadratic form
$\psi\mapsto\sum_{i,j}^{p}\sigma^{2}_{i,j}\psi_{i}\psi_{j}$ is negative on the
subspace $Vect(1,...,1)^{\bot}$.

  * •

In the continuous case, we assume that the the function
$\psi\mapsto\int\sigma^{2}(x,y)d\psi(x)d\psi(y)$ is concave on the set
$\mathcal{P}([0,1])$ of probability measures on $[0,1]$.

When this Assumption as well as Assumption 4 and (2) are verified, one of the
author of this article [16] proved that the largest eigenvalue of ${\bf
X}_{N}^{\sigma}$ satisfies a large deviations principle with a good rate
function $J^{(1)}_{\sigma}$. In this article we generalize this result to the
$k$th largest eigenvalues and prove the following theorem :

######  Theorem 6.

Let ${\bf X}^{\sigma}_{N}=(\frac{\sigma_{N}(i,j)X_{ij}}{\sqrt{N}})_{{i,j}}$ be
a $N\times N$ Hermitian matrix where $(X_{i,j})_{i\leq j}$ are centered
independent entries satisfying (1) and (2), as well as such that ${\bf
X}_{N}^{\sigma}$ satisfies Assumption 2 and such that $\sigma$ verifies
Assumption 5. Let $k\geq 0$ and $\lambda^{N}_{1}\geq...\geq\lambda^{N}_{k}$ be
the $k$ largest eigenvalues of ${\bf X}_{N}^{\sigma}$. Then
$(\lambda^{N}_{1},...,\lambda^{N}_{k})$ satisfies a large deviations principle
in the scale $N$ with good rate function $J_{\sigma}^{(k)}(x_{1},...,x_{k})$
which is infinite unless $x_{1}\geq...\geq x_{k}\geq r_{\sigma}$ and in this
case equals:

$J_{\sigma}^{(k)}(x_{1},...,x_{k})=\sum_{i=1}^{k}J_{\sigma}^{(1)}(x_{i})$

where $r_{\sigma}$ is the rightmost point of the support of the limit
$\mu_{\sigma}$ of the empirical measure of ${\bf X}_{N}$ and where
$J^{(1)}_{\sigma}$ is the rate function for the large deviations of the
largest eigenvalue.

This result was proved in the case $k=1,\ell=0$ in [10, Theorem 6]. Let us now
consider ${\bf X}_{N}$ to be a GOE/GUE matrix, that is a $N\times N$ Hermitian
matrix with centered real/complex Gaussian entries satisfying (1). Let $\ell$
and $k$ be two integer numbers and let $(e_{1},\ldots
e_{k},e_{-1},\ldots,e_{-\ell})$ be orthonormal vectors following the uniform
law on the sphere. In [18], Mylene Maïda showed that the largest eigenvalue of
a Gaussian Wigner matrix perturbed by a rank one matrix satisfy a large
deviations principle. In this article we generalize this result to the $k$th
largest eigenvalues and $\ell$ smallest eigenvalues when the Gaussian matrix
is perturbed by a finite rank matrix with $k$ non-negative eigenvalues and
$\ell$ non-positive eigenvalues.

######  Proposition 7.

Let ${\bf X}_{N}$ be a GUE ($\beta=2$) or GOE ($\beta=1$) matrix. Let $\ell,k$
be two finite integer numbers. Let
$\theta_{1}\geq\theta_{2}\geq\cdots\geq\theta_{k}\geq
0\geq\theta_{-\ell}\geq\cdots\geq\theta_{-1}$ and define

${\bf X}_{N}^{\theta}={\bf X}_{N}+\sum_{-\ell\leq i\leq k\atop i\neq
0}\theta_{i}e_{i}e_{i}^{*}\,.$

Let
$\lambda_{1}^{N,\theta}\geq\lambda_{2}^{N,\theta}\geq\cdots\geq\lambda_{N}^{N,\theta}$
be the eigenvalues of ${\bf X}_{N}^{\theta}$. Then, the distribution of
$(\lambda_{1}^{N,\theta},\ldots,\lambda_{k}^{N,\theta},\lambda_{N-\ell}^{N,\theta},\ldots,\lambda_{N}^{N,\theta})$
satisfies a large deviations principle in the scale $N$ and with good rate
function which is infinite unless

$x_{1}\geq x_{2}\geq\cdots\geq x_{k}\geq 2\geq-2\geq x_{-\ell}\geq\cdots\geq
x_{-1}$

and is given then by $\beta\sum_{-\ell\leq i\leq k\atop i\neq
0}I_{\theta_{i}}(x_{i})$. Here, with
$I(y)=\frac{1}{4}x^{2}-\int\ln|x-y|d\sigma(y)$, we have set

$I_{\theta}(x)=I(x)-J(\sigma,\theta,x)-\inf_{y}(I(y)-J(\sigma,\theta,y))\,.$

A similar result holds for Wishart matrices. We next consider a $L\times M$
matrix ${\bf G}_{L,M}$ with i.i.d standard Gaussian matrices with covariance
$1$, set $N=M+L$, and assume without loss of generality that $M\geq L$. We
consider the Wishart matrix

${\bf W}_{N}^{\gamma}=\frac{1}{M}\Sigma_{L}^{1/2}{\bf G}_{L,M}{\bf
G}_{L,M}^{*}\Sigma_{L}^{1/2}$

where $\Sigma_{L}$ is a $L\times L$ covariance matrix given by
$I_{L}+\sum_{i=1}^{k}\gamma_{i}e_{i}e_{i}^{*}$ for some fixed $\gamma_{i}>-1$.
Here the $e_{i},1\leq i\leq k$ are orthonormal vectors. It is well known that
when $L/M$ goes to $\alpha\in[0,1]$, the empirical measure of the eigenvalues
of ${\bf W}_{N}^{\gamma}$ goes to the Pastur-Marhenko distribution.

Large deviations for the extreme eigenvalues in the case $\gamma_{i}=0$ are
well known, and similar to the Gaussian case, see [1, 8]. The rate function
governing the large deviations in the scale $N$ for the smallest eigenvalue is
infinite outside $[0,\lambda_{-}]$ and is given for $y\in[0,\lambda_{-}]$ by

$\displaystyle I_{\alpha}(y)$ $\displaystyle=$
$\displaystyle\frac{\beta}{4{(1+\alpha)}}(y-(1-\alpha)\ln
y-2\alpha\int\ln|y-t|d\pi_{\alpha}(y)-C)$ $\displaystyle=$
$\displaystyle\frac{\beta}{4{(1+\alpha)}}\int^{\lambda_{-}}_{y}\frac{1}{t}\sqrt{(t-\lambda_{+})(t-\lambda_{-})}dt$

where $C$ is the infimum of $y-(1-\alpha)\ln
y-2\alpha\int\ln|y-t|d\pi_{\alpha}(y)$. The same result holds for the largest
eigenvalue. We have the following analogue to Proposition 7.

######  Proposition 8.

Let $\ell\leq k$ be two finite integer numbers. Let
$\gamma_{1}\geq\gamma_{2}\geq\cdots\geq\gamma_{\ell}\geq
0\geq\gamma_{\ell+1}\geq\cdots\geq\gamma_{k}>-1$. Let
$\lambda_{1}^{N,\gamma}\geq\lambda_{2}^{N,\gamma}\geq\cdots\lambda_{M}^{N,\gamma}$
be the eigenvalues of ${\bf W}_{N}^{\gamma}$ in decreasing order. Then, the
law of
$(\lambda_{1}^{N,\gamma},\ldots,\lambda_{\ell}^{N,\gamma},\lambda_{N-k+\ell}^{N,\gamma},\ldots,\lambda_{N}^{N,\gamma})$
satisfies a large deviations principle in the scale $N$ and with good rate
function which is infinite unless

$x_{1}\geq x_{2}\geq\cdots\geq x_{\ell}\geq\lambda_{+}\geq\lambda_{-}\geq
x_{\ell+1}\geq\cdots\geq x_{k}\geq 0$

and is given otherwise by $\sum_{i=1}^{k}I_{\gamma_{i},\alpha}(x_{i})$. Here,
we have set

$I_{\gamma,\alpha}(x)=I_{\alpha}(x)-\frac{\beta}{2}J(\pi_{\alpha},\frac{\gamma}{1-\gamma},x)-\inf_{y}(I_{\alpha}(y)-\frac{\beta}{2}J(\pi_{\alpha},\frac{\gamma}{1-\gamma},y))$

We finally notice that since our results hold for any number of eigenvalues,
they capture as well the large deviations for the point processes of the
outliers. For instance, if we let $A_{i}=[a_{i},b_{i}]$ be intervals above the
bulk, $b_{i}<a_{i+1}<b_{i+1}$, if we denote $I$ the large deviation rate
function for any of the above models, the probability that there are $n_{i}$
outliers in the set $A_{i}$ has probability of order
$\exp\\{-N\sum_{i}n_{i}\inf_{A_{i}}I\\}$.

Acknowledgements We are very grateful to Mylène Maïda and Marc Potters for
preliminary discussions about the questions addressed in this paper.

##  3\. Limiting spherical integral in the discrete case

We first consider the case where ${\bf X}_{N}$ has finitely many different
eigenvalues :

${\bf
X}_{N}=diag\left(\underbrace{\eta_{-m_{1}+1},...,\eta_{-m_{1}+1}}_{N_{-m_{1}+1}},\underbrace{\eta_{-m_{1}+2},...,\eta_{-m_{1}+2}}_{N_{-m_{1}+2}},...,\underbrace{\eta_{p+m_{2}},...,\eta_{p+m_{2}}}_{N_{p+m_{2}}}\right),$

where
$\eta_{-m_{1}+1}<...<\eta_{1}\cdots<\eta_{p}<\eta_{p+1}<\cdots<\eta_{p+m_{2}}$,
and $\eta_{i}$ has multiplicity $N_{i}$ where
$\sum_{i=-m_{1}+1}^{p+m_{2}}N_{i}=N$. We assume that $N_{i}/N$ goes to a
positive limit $\alpha_{i}$ for $i\in\\{1,p\\}$ and to zero for
$i\in\\{1-m_{1},\ldots,0\\}\cup\\{p+1,\ldots,p+m_{2}\\}$, the later
representing the outliers of ${\bf X}_{N}$. $m_{1},m_{2},p$ are independent of
$N$ (with the convention that if $\alpha_{i}=0$, $\alpha_{i}\ln\alpha_{i}=0$).
In the previous notations, the eigenvalues of ${\bf X}_{N}$ are given by
$\lambda_{1}^{N}\geq\lambda_{2}^{N}\geq\cdots\geq\lambda_{N}^{N}$ with

$\lambda_{i}^{N}=\eta_{j},i\in
I_{j}=[N_{p+m_{2}}+\cdots+N_{p+m_{2}-j}+1,N_{p+m_{2}}+\cdots+N_{p+m_{2}-j}+N_{p+m_{2}-j-1}]$

and $I_{p+m_{2}}=[1,N_{p+m_{2}}]$.

######  Remark 9.

We notice that if the sequences $N_{i}$ are fixed, the spherical integrals are
$\beta/2$-Lipschitz in the $p+m_{1}+m_{2}$-uplet $(\eta_{i})_{-m_{1}+1\leq
i\leq p+m_{2}}$ with the norm $||.||_{\infty}$ .

###  3.1. Limiting 1-d spherical integral

We start by giving a new proof of [10, Theorem 6] giving the asymptotics of
spherical integrals in the one dimensional case, in the case of matrices with
$p+m_{1}+m_{2}$ different eigenvalues with multiplicity as above. This proof
will in fact extend to the higher dimensional setting in the next subsection.

######  Proposition 10.

Let $\theta\geq 0$. Then

$\lim_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}\theta\langle e,{\bf
X}_{N}e\rangle\right)\right]=\frac{\beta}{2}\sup_{\gamma_{i}\geq
0\atop\sum_{i=1-m_{1}}^{p+m_{2}}\gamma_{i}=1}\left\\{{\theta\sum_{i=-m_{1}+1}^{p+m_{2}}\eta_{i}\gamma_{i}+\sum_{i=1}^{p}\alpha_{i}\ln\frac{\gamma_{i}}{\alpha_{i}}}\right\\}$

###### Proof.

We have the following formula :

$\langle e,{\bf
X}_{N}e\rangle=\sum_{i=-m_{1}+1}^{p+m_{2}}\eta_{i}\gamma_{i}^{N}$

where we have denoted $\gamma_{j}^{N}=\sum_{i\in I_{j}}|u_{i}|^{2}$ with
$u_{i}=\langle v_{i},e\rangle$ if $v_{i}$ is the eigenvector for the $i$-th
eigenvalue of ${\bf X}_{N}$. In other words, $\gamma_{j}^{N}$ is the
$\ell^{2}$-norm of the projection of $e$ onto the eigenspace of $\eta_{j}$.
The vector $\gamma^{N}$ follows a Dirichlet law of parameters
$\frac{\beta}{2}(N_{1-m_{1}},\ldots,N_{p+m_{2}})$, that is the distribution on
$\Sigma=\\{x\in[0,1]^{m_{1}+m_{2}+p}:\sum_{i=1-m_{1}}^{m_{2}+p}x_{i}=1\\}$
given by $\gamma_{1-m_{1}}^{N}=1-\sum_{i=2-m_{1}}^{p+m_{2}}\gamma_{i}^{N}$ and

$d\mathbb{P}^{N}_{\bf N}(\gamma)=\frac{1}{Z_{\alpha}^{N}}{\bf
1}_{\sum_{i=2-m_{1}}^{p+m_{2}}\gamma_{i}\leq
1}(1-\sum_{i=2-m_{1}}^{p+m_{2}}\gamma_{i})^{\frac{\beta}{2}N_{1-m_{1}}-1}\prod_{j=2-m_{1}}^{p+m_{2}}\gamma_{j}^{\frac{\beta}{2}N_{i}-1}{\bf
1}_{\gamma_{j}\geq 0}d\gamma_{j}$ (5)

We deduce the following large deviations principle

######  Theorem 11.

Assume that $N_{i}/N$ converges towards $\alpha_{i}$ for all $i$, with
$\alpha_{i}=0$ for $i\notin[1,p]$. Then, the law of $\gamma^{N}$ satisfies a
large deviations principle with scale $N$ and good rate function $I_{\alpha}$
given for $x\in\Sigma$ by

$I_{\alpha}(x_{1-m_{1}},...,x_{p+m_{2}})=\frac{\beta}{2}\sum_{i=1}^{p}\alpha_{i}\log\frac{x_{i}}{\alpha_{i}}\,.$

The proof is a direct consequence of Laplace’s method. We deduce Proposition
10 by Varadhan’s lemma. ∎

######  Lemma 12.

For $\theta\geq 0$, $\eta=(\eta_{1-m_{1}}<\cdots<\eta_{p+m_{2}})$ and
$\alpha_{i}\in(\mathbb{R}^{+})^{p}$ such that $\sum_{i=1}^{p}\alpha_{i}=1$,

$J(\theta,\eta)=\sup_{\gamma_{i}\geq
0\atop\sum\gamma_{i}=1}\left\\{{\theta\sum_{i=-m_{1}+1}^{p+m_{2}}\eta_{i}\gamma_{i}+\sum_{i=1}^{p}\alpha_{i}\ln\frac{\gamma_{i}}{\alpha_{i}}}\right\\}$

only depends on $\eta_{p+m_{2}},\theta$ and
$\mu=\sum_{i=1}^{p}\alpha_{i}\delta_{\eta_{i}}$. It is given by

$J(\theta,\eta)=J(\mu,\theta,\eta_{p+m_{2}})=K\left(\mu,\theta,\eta_{p+m_{2}},v(\mu,\theta,\eta_{p+m_{2}})\right)$

with

$K(\mu,\theta,\lambda,v)=\theta\lambda+(v-\lambda)G_{\mu}(v)-\ln|\theta|-\int\ln|v-x|d\mu(x)-1$

and

$v(\mu,\theta,\lambda)=\left\\{\begin{array}[]{c l}\lambda&\mbox{ if
}G_{\mu}(\lambda)\leq\theta,\cr G_{\mu}^{-1}(\theta)&\mbox{ if
}G_{\mu}(\lambda)>\theta.\cr\end{array}\right.$

###### Proof.

$J(\theta,\eta)$ is the supremum of

$I_{\theta,\eta}^{p+m_{2}}(\gamma):=\theta\sum_{i=-m_{1}+1}^{p+m_{2}}\eta_{i}\gamma_{i}+\sum_{i=1}^{p}\alpha_{i}\ln\frac{\gamma_{i}}{\alpha_{i}}\,.$
(6)

The entropic term in $I_{\theta,\eta}^{p+m_{2}}$ does not depend on
$(\gamma_{i},i<1\mbox{ or }i>p)$, and the first term increases when we take
them all equal to zero except $\gamma_{m_{2}+p}$. Hence, the maximum is taken
at $\gamma_{i}=0$ for $i<1$ or $i\in[p+1,p+m_{2}-1]$. Then, putting
$\gamma_{p+m_{2}}=1-\sum_{i=1}^{p}\gamma_{i}$ we see that we need to optimize

$I_{\theta,\eta}(\gamma)=\theta\eta_{p+m_{2}}+\left\\{{\theta\sum_{i=1}^{p}(\eta_{i}-\eta_{p+m_{2}})\gamma_{i}+\sum_{i=1}^{p}\alpha_{i}\ln\frac{\gamma_{i}}{\alpha_{i}}}\right\\}$

over $\gamma_{i}\geq 0,\sum_{i=1}^{p}\gamma_{1}\leq 1$. We see that the
critical point of $I_{\theta,\eta}(\gamma)$ is

$\gamma^{*}_{i}=\frac{\alpha_{i}}{\theta(\eta_{p+m_{2}}-\eta_{i})},1\leq i\leq
p,\,\gamma_{p+m_{2}}^{*}=1-\sum_{i=1}^{p}\gamma_{i}^{*}=1-\frac{1}{\theta}G_{\mu}(\eta_{p+m_{2}})$

provided
$\sum_{i=1}^{p}\gamma_{i}^{*}=\frac{1}{\theta}G_{\mu}(\eta_{p+m_{2}})\leq 1$.
For $\theta<G_{\mu}(\eta_{p+m_{2}})$, the supremum is achieved at

$\gamma^{**}_{i}=\frac{\alpha_{i}}{\theta(G_{\mu}^{-1}(\theta)-\eta_{i})},1\leq
i\leq p,\,\gamma_{p+m_{2}}^{**}=0$

This gives the announced formula. ∎

###  3.2. Limiting 2-d spherical integral

We next consider the bi-dimensional case where $(e,f)$ are two orthonormal
vectors following the uniform law in the sphere.

######  Proposition 13.

Let $\theta_{1}\geq\theta_{2}\geq 0$. Then, if $N_{p+m_{2}}=1$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}(\theta_{1}\langle e,{\bf X}_{N}e\rangle+\theta_{2}\langle f,{\bf
X}_{N}f\rangle)\right)\right]\qquad\qquad$
$\qquad\qquad=\frac{\beta}{2}(J(\mu,\theta_{1},\eta_{p+m_{2}})+J(\mu,\theta_{2},\eta_{p+m_{2}-1}))\,.$

If $N_{m+p_{2}}\geq 2$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}(\theta_{1}\langle e,{\bf X}_{N}e\rangle+\theta_{2}\langle f,{\bf
X}_{N}f\rangle)\right)\right]\qquad\qquad$
$\qquad\qquad=\frac{\beta}{2}(J(\mu,\theta_{1},\eta_{p+m_{2}})+J(\mu,\theta_{2},\eta_{m+p_{2}}))\,.$

###### Proof.

We first assume that $N_{p+m_{2}}=1$. We can write :

$\mathbb{E}\left[\exp\left(N\frac{\beta}{2}\left(\theta_{1}\langle e,{\bf
X}_{N}e\rangle+\theta_{2}\langle f,{\bf
X}_{N}f\rangle\right)\right)\right]\qquad\qquad$
$\qquad\qquad=\mathbb{E}\left[\exp\left(N\frac{\beta}{2}\theta_{1}\langle
e,{\bf
X}_{N}e\rangle\right)\mathbb{E}\left[\exp\left(\frac{\beta}{2}\theta_{2}\langle
f,{\bf X}_{N}^{(e)}f\rangle\right)\bigg{|}e\right]\right]$

where ${\bf X}_{N}^{(e)}=P_{e^{\bot}}{\bf X}_{N}P_{e^{\bot}}$ if
$P_{e^{\bot}}=I-ee^{*}$ is the orthogonal projection onto the ortho-complement
of $e$. We can see ${\bf X}_{N}^{(e)}$ as a $(N-1)\times(N-1)$ matrix living
in $Vect(e)^{\bot}$. Its largest eigenvalue $\chi$ belongs to
$[\eta_{p+m_{2}-1},\eta_{p+m_{2}}]$ and writing that the corresponding
eigenvector $v\in Vect(e)^{\bot}$ satisfies ${\bf X}_{N}^{(e)}v=\chi v$ and
$\langle e,v\rangle=0$, we find that $\chi$ belongs to
$[\eta_{p+m_{2}-1},\eta_{p+m_{2}}]$ must satisfy

$\sum_{i=1-m_{1}}^{p+m_{2}}\frac{\gamma_{i}(e)}{\chi-\eta_{i}}=0$ (7)

if there is a solution in this interval where $\gamma_{i}(e)=\sum_{j\in
I_{i}}|\langle v_{j},e\rangle|^{2}$ with $v_{j}$ the $j$th eigenvector of
${\bf X}_{N}$. If there is no solution (which can happen only if
$\gamma_{p+m_{2}}(e)=0$) then $\chi=\eta_{p+m_{2}}$ if the rational function
is positive on this interval and $\chi=\eta_{p+m_{2}-1}$ if it is negative.
Note that $\chi=\chi(\gamma(e))$ is a continuous function of $\gamma(e)$ and
denote this function $\chi(\gamma(e))$. Moreover, the spectral measure of
${\bf X}_{N}^{(e)}$ converges towards $\mu$, the limiting spectral measure of
${\bf X}_{N}$ by Weyl interlacing property. Therefore, when $\gamma(e)$
converges towards $\kappa$, and since the empirical measure of ${\bf
X}_{N}^{(e)}$ converges toward the same limit that the empirical measure of
${\bf X}_{N}$,

$\lim_{N\rightarrow\infty}\frac{1}{N-1}\ln\mathbb{E}[e^{\theta_{2}N\frac{\beta}{2}\langle
f,{\bf X}_{N}f\rangle}|e]=\frac{\beta}{2}J(\mu,x,\chi(\kappa)).$

Moreover, the right hand side depends continuously on $\kappa$ (since $J$ is
continuous in $\chi$ and $\chi$ in $\kappa$). We now can apply the fact that
$\gamma^{N}$ follows a large deviations principle, see Theorem 11, to conclude
that

$\lim\frac{1}{N}\log{\mathbb{E}[\exp\left(N\frac{\beta}{2}\left(\theta_{1}\langle
e,{\bf X}_{N}e\rangle+\theta_{2}\langle f,{\bf
X}_{N}f\rangle\right)\right)]}\\\
=\frac{\beta}{2}\sup_{\gamma\in(\mathbb{R}^{+})^{p+m_{1}+m_{2}},\sum\gamma_{i}=1}\left(J(\mu,\theta_{2},\chi(\gamma))+\sum_{i=1}^{p}\alpha_{i}\log\frac{\gamma_{i}}{\alpha_{i}}+\sum_{i=1-m_{1}}^{p+m_{2}}\theta_{1}\eta_{i}\gamma_{i}\right)$

Since $J$ is bounded and due to the continuity of $\gamma$, we can change the
domain of the $\sup$ to $(\mathbb{R}^{+,*})^{p+m_{1}+m_{2}}$. We next complete
the proof by computing the right hand side and showing it equals the sum of
the two limiting spherical integrals as stated. We first denote by
$\tilde{\gamma}_{i}=\gamma_{i}|\chi-\eta_{i}|^{-1}$ with $\chi=\chi(\gamma)$.
By definition we have $\tilde{\gamma}_{i}>0$,
$\chi\in(\eta_{p+m_{2}-1},\eta_{p+m_{2}})$ and (7) holds so that

$\tilde{\gamma}_{p+m_{2}}=\sum_{i=1-m_{1}}^{p+m_{2}-1}\tilde{\gamma}_{i},\qquad\sum_{i=1-m_{1}}^{p+m_{2}-1}(\chi-\eta_{i})\tilde{\gamma}_{i}+(\eta_{p+m_{2}}-\chi)\tilde{\gamma}_{p+m_{2}}=1$

This simplifies into the condition

$\tilde{\gamma}_{p+m_{2}}=\sum_{i=1-m_{1}}^{p+m_{2}-1}\tilde{\gamma}_{i},\qquad\eta_{p+m_{2}}\tilde{\gamma}_{p+m_{2}}-\sum_{i=1-m_{1}}^{p+m_{2}-1}\eta_{i}\tilde{\gamma}_{i}=1$
(8)

which is independent of $\chi$. We thus first take the supremum over
$\chi\in[\eta_{p+m_{2}-1},\eta_{p+m_{2}}]$ of

$\displaystyle I(\chi,\tilde{\gamma})$ $\displaystyle=$ $\displaystyle
J(\mu,\theta_{2},\chi)+\sum_{i=1}^{p}\alpha_{i}\log[|\eta_{i}-\chi|\frac{\tilde{\gamma}_{i}}{\alpha_{i}}]-\theta_{1}(\chi-\eta_{p+m_{2}})\eta_{p+m_{2}}\tilde{\gamma}_{p+m_{2}}$
$\displaystyle+\sum_{i=1-m_{1}}^{p+m_{2}-1}\theta_{1}(\chi-\eta_{i})\eta_{i}\tilde{\gamma}_{i}$
$\displaystyle=$ $\displaystyle
H(\chi)+\sum_{i=1}^{p}\alpha_{i}\log\frac{\tilde{\gamma}_{i}}{\alpha_{i}}+\theta_{1}\eta_{p+m_{2}}^{2}\tilde{\gamma}_{p+m_{2}}-\sum_{i=1-m_{1}}^{p+m_{2}-1}\theta_{1}\eta_{i}^{2}\tilde{\gamma}_{i}$

with

$H(\chi)=J(\mu,\theta_{2},\chi)+\sum_{i=1}^{p}\alpha_{i}\log|\eta_{i}-\chi|-\chi\theta_{1}.$
(9)

Recall the formula for $J$ from Lemma 12. When $\theta_{2}\leq G_{\mu}(\chi)$,
that is $\chi\leq G_{\mu}^{-1}(\theta_{2})$, $J$ does not depend on $\chi$ and
the function $H$ increases till $G_{\mu}^{-1}(\theta_{1})$ and decreases
afterwards. When $\theta_{2}\geq G_{\mu}(\chi)$, that is $\chi\geq
G^{-1}(\theta_{2})$, Lemma 12 gives

$\displaystyle H(\chi)$ $\displaystyle=$
$\displaystyle\theta_{2}\chi-\sum_{i=1}^{p}\alpha_{i}\ln(\chi-\eta_{i})-\ln\theta_{2}-1+\sum_{i=1}^{p}\alpha_{i}\log|\eta_{i}-\chi|-\chi\theta_{1}$
$\displaystyle=$ $\displaystyle\chi(\theta_{2}-\theta_{1})-\ln\theta_{2}-1$

which is decreasing since $\theta_{1}>\theta_{2}$. Therefore, $H$ increases
till $G_{\mu}^{-1}(\theta_{1})$ and decreases afterwards. As a consequence,

$\max_{\chi\in[\eta_{p+m_{2}-1},\eta_{p+m_{2}}]}H(\chi)=\left\\{\begin{array}[]{ll}H(\eta_{p+m_{2}-1})&\mbox{
if }G_{\mu}^{-1}(\theta_{1})\leq\eta_{p+m_{2}-1},\cr
H(G_{\mu}^{-1}(\theta_{1}))&\mbox{ if
}G_{\mu}^{-1}(\theta_{1})\in[\eta_{p+m_{2}-1},\eta_{m+p_{2}}],\cr
H(\eta_{p+m_{2}})&\mbox{ if
}G_{\mu}^{-1}(\theta_{1})>\eta_{p+m_{2}}.\cr\end{array}\right.$ (10)

Let us also optimize on $\tilde{\gamma}$ satisfying (8) the function

$L(\tilde{\gamma})=\sum_{i=1}^{p}\alpha_{i}\log[\frac{\tilde{\gamma}_{i}}{\alpha_{i}}]+\theta_{1}\eta_{p+m_{2}}^{2}\tilde{\gamma}_{p+m_{2}}-\sum_{i=1-m_{1}}^{p+m_{2}-1}\theta_{1}\eta_{i}^{2}\tilde{\gamma}_{i}\,.$

Replacing $\tilde{\gamma}_{p+m_{2}}$ by
$\sum_{i=1-m_{1}}^{p+m_{2}-1}\tilde{\gamma}_{i}$ we get

$L(\tilde{\gamma})=\sum_{i=1}^{p}\alpha_{i}\log[\frac{\tilde{\gamma}_{i}}{\alpha_{i}}]+\theta_{1}\sum_{i=1-m_{1}}^{p+m_{2}-1}(\eta_{p+m_{2}}^{2}-\eta_{i}^{2})\tilde{\gamma}_{i}$

with by (8), $\sum(\eta_{p+m_{2}}-\eta_{i})\tilde{\gamma}_{i}=1$. We may again
do the change of variables
$\bar{\gamma}_{i}=(\eta_{p+m_{2}}-\eta_{i})\tilde{\gamma}_{i}$ which are non
negative and with mass one by (8). We get by (8)

$\displaystyle L(\tilde{\gamma})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{p}\alpha_{i}\log[\frac{\bar{\gamma}_{i}}{\alpha_{i}(\eta_{p+m_{2}}-\eta_{i})}]+\theta_{1}\sum_{i=1-m_{1}}^{p+m_{2}-1}(\eta_{p+m_{2}}+\eta_{i})\bar{\gamma}_{i}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{p}\alpha_{i}\log[\frac{\bar{\gamma}_{i}}{\alpha_{i}(\eta_{p+m_{2}}-\eta_{i})}]+\theta_{1}\sum_{i=1-m_{1}}^{p+m_{2}-1}\eta_{i}\bar{\gamma}_{i}+\theta_{1}\eta_{p+m_{2}}$
$\displaystyle=$ $\displaystyle
I_{\theta_{1},\eta}^{p+m_{2}-1}(\bar{\gamma})+\sum_{i=1}^{p}\alpha_{i}\log[\frac{1}{(\eta_{p+m_{2}}-\eta_{i})}]+\theta_{1}\eta_{p+m_{2}}$

where $I_{\theta_{1},\eta}^{p+m_{2}-1}$ is defined as in (6) with largest
outlier $\eta_{p+m_{2}-1}$. Its maximum gives
$J(\mu,\theta_{1},\eta_{p+m_{2}-1})$. We thus get

$\max
L=\sum_{i=1}^{p}\alpha_{i}\log[\frac{1}{(\eta_{p+m_{2}}-\eta_{i})}]+\theta_{1}\eta_{p+m_{2}}+J(\mu,\theta_{1},\eta_{p+m_{2}-1})$

We finally compute $\max I(\chi,\tilde{\gamma})=\max L(\tilde{\gamma})+\max
H(\chi)$.

  * •

For $G^{-1}(\theta_{1})\leq\eta_{p+m_{2}-1}\leq\eta_{p+m_{2}}$, we check that
$J(\mu,\theta_{1},\eta_{p+m_{2}})$ equals

$J(\mu,\theta_{1},\eta_{p+m_{2}-1})+\sum\alpha_{i}\ln\frac{|\eta_{i}-\eta_{p+m_{2}-1}|}{|\eta_{i}-\eta_{p+m_{2}}|}+\theta_{1}(\eta_{p+m_{2}}-\eta_{p+m_{2}-1})$
(11)

so that by (10) we find

$\max
I=J(\mu,\theta_{1},\eta_{p+m_{2}})+J(\mu,\theta_{2},\eta_{p+m_{2}-1})\,.$

  * •

For $G_{\mu}^{-1}(\theta_{1})\in[\eta_{p+m_{2}-1},\eta_{m+p_{2}}]$,
$J(\mu,\theta_{2},G_{\mu}^{-1}(\theta_{1}))=J(\mu,\theta_{2},G_{\mu}^{-1}(\theta_{2}))=J(\mu,\theta_{2},\eta_{p+m_{2}-1})$
since $\theta_{1}>\theta_{2}$ and
$\eta_{p+m_{2}-1}<G_{\mu}^{-1}(\theta_{1})<G_{\mu}^{-1}(\theta_{2})$. Moreover
as $\theta_{1}>G_{\mu}(\eta_{p+m_{2}})$,

$\max
L=J(\mu,\theta_{1},\eta_{p+m_{2}})+J(\mu,\theta_{1},\eta_{m_{2}+p-1})+\ln\theta_{1}+1$

which again does not depend on $\eta_{m_{2}+p-1}$. Hence

$\displaystyle\max I$ $\displaystyle=$ $\displaystyle
J(\mu,\theta_{2},\eta_{p+m_{2}-1})+\sum\alpha_{i}\ln|\eta_{i}-G_{\mu}^{-1}(\theta_{1})|-\theta_{1}G^{-1}_{\mu}(\theta_{1})$
$\displaystyle+J(\mu,\theta_{1},\eta_{p+m_{2}})+J(\mu,\theta_{1},G^{-1}(\theta_{1}))+\ln\theta_{1}+1$
$\displaystyle=$ $\displaystyle
J(\mu,\theta_{2},\eta_{p+m_{2}-1})+J(\mu,\theta_{1},\eta_{p+m_{2}})$

  * •

For $G_{\mu}^{-1}(\theta_{1})>\eta_{m+p_{2}}$, we compute

$\displaystyle\max I$ $\displaystyle=$ $\displaystyle
J(\mu,\theta_{2},\eta_{p+m_{2}})+J(\mu,\theta_{1},\eta_{p+m_{2}-1})$
$\displaystyle=$ $\displaystyle
J(\mu,\theta_{2},\eta_{p+m_{2}-1})+J(\mu,\theta_{1},\eta_{p+m_{2}})$

since $\theta_{2}<\theta_{1}<G_{\mu}(\eta_{p+m_{2}})<G(\eta_{p+m_{2}-1})$ so
that the above supremum does not depend on the outliers.

In the case where $N_{p+m_{2}}\geq 2$, we have $\chi=\eta_{p+m_{2}}$ by Weyl
interlacing property and therefore it does not depend on the choice of
$\gamma$. The result follows immediately after conditioning as in the proof
above. ∎

A similar (but easier) argument shows that

######  Proposition 14.

Let $\theta_{1}\geq 0\geq\theta_{2}$. Then,

$\lim_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}(\theta_{1}\langle e,{\bf X}_{N}e\rangle+\theta_{2}\langle f,{\bf
X}_{N}f\rangle)\right)\right]\qquad\qquad$
$\qquad\qquad=\frac{\beta}{2}(J(\mu,\theta_{1},\eta_{p+m_{2}})+J(\mu,\theta_{2},\eta_{1-m_{1}}))\,.$

Here $J$ is extended to negative values of $\theta_{2}$ by putting

$J(\mu,\theta_{2},\eta_{1-m_{1}})=K\left(\mu,\theta_{2},\eta_{1-m_{1}},v(\mu,\theta_{2},\eta_{1-m_{1}})\right)$

with $K$ as in Lemma 12 and for $\theta\leq 0,\eta\leq l_{\mu}$,

$v(\mu,\theta,\lambda)=\left\\{\begin{array}[]{c l}\lambda&\mbox{ if
}G_{\mu}(\lambda)\geq\theta,\cr G_{\mu}^{-1}(\theta)&\mbox{ if
}G_{\mu}(\lambda)<\theta.\cr\end{array}\right.$

The formula for $J$ for negative $\theta$ can simply be found by replacing
${\bf X}_{N}$ by $-{\bf X}_{N}$.

###  3.3. Limiting k-d spherical integrals

We now consider more general $k$-dimensional spherical integrals with $k\geq
2$. In the sequel we let
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{k}\geq 0$ be the limit of
the $k$ largest outliers of ${\bf X}_{N}$ counted with multiplicity one, and
$\lambda_{-1}\leq\cdots\leq\lambda_{-\ell}\leq 0$ be the limit of the $\ell$
smallest outliers of ${\bf X}_{N}$ counted with multiplicity one. With the
previous notations, $\lambda_{i}=\eta_{p+m_{2}}$ for $i\in[1,N_{p+m_{2}}],$
$\lambda_{i}=\eta_{p+m_{2}-1}$ for
$i\in[N_{p+m_{2}}+1,N_{p+m_{2}}+N_{p+m_{2}-1}]$.

######  Proposition 15.

Fix two integer numbers $k$ and $\ell$. Let $(e_{1},\ldots e_{k},e_{-1},\ldots
e_{-\ell})$ be $k+\ell$ orthonormal vectors following the uniform law in the
sphere and assume that the sequence ${\bf X}_{N}$ has the form described at
the beginning of this section. Let
$\theta_{1}\geq\theta_{2}\geq\cdots\geq\theta_{k}\geq
0\geq\theta_{-\ell}\geq\cdots\geq\theta_{-1}$. Then

$\lim_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-\ell,i\neq 0}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]\qquad\qquad$
$\qquad\qquad=\frac{\beta}{2}\left(\sum_{i=1}^{k}J(\mu,\theta_{i},\lambda_{i})+\sum_{i=1}^{\ell}J(\mu,\theta_{-i},\lambda_{-i})\right)\,.$

###### Proof.

For the sake of simplicity we will assume the outliers
$\lambda_{1},...,\lambda_{k}$ and $\lambda_{-1},...,\lambda_{-\ell}$ are
distinct. The general case can be deduced by equicontinuity of the spherical
integral. We will prove this proposition by induction over $k+\ell$. We know
it is true for $k+\ell\leq 2$ by the previous section. By symmetry we can
assume the proposition true for $(\ell,k-1)$ and it is enough to show it still
holds for $(\ell,k)$. Thus we will set $\eta_{p+m_{2}-i+1}=\lambda_{i}$ for
$i\in[1,k]$, and $\eta_{-m_{1}+i}=\lambda_{-i}$ and we will assume
$N_{p+m_{2}-i}=1$ for $i\in[1,k]$ and $N_{-m_{1}+i}=1$ for $i\in[1,\ell]$. We
proceed as in the $2$-dimensional case by conditioning on the vector $e_{1}$
and so we have :

$\displaystyle\mathbb{E}\left[\exp\left(\frac{\beta N}{2}\sum_{i=-\ell,i\neq
0}^{k}\theta_{i}\langle e_{i},{\bf X}_{N}e_{i}\rangle\right)\right]=$
$\displaystyle\mathbb{E}\left[\exp\left(\frac{\beta N}{2}\theta_{1}\langle
e_{1},{\bf X}_{N}e_{1}\rangle\right)\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-\ell,i\neq 0\atop i\neq 1}^{k}\theta_{i}\langle e_{i},{\bf
X}^{(e_{1})}_{N}e_{i}\rangle\right)\Big{|}e_{1}\right]\right]$

As previously, the eigenvalues of ${\bf X}_{N}^{(e_{1})}$ (seen as a
$N-1\times N-1$ matrix) are interlaced with those of ${\bf X}_{N}$. Thus if we
denote $\chi_{j}$ the $j$-th largest eigenvalue of ${\bf X}_{N}^{(e_{1})}$,
$\chi_{j}$ is the unique solution in the interval
$[\eta_{p+m_{2}-j},\eta_{p+m_{2}-j+1}]$ of the equation :

$\sum_{i=1-m_{1}}^{p+m_{2}}\frac{\gamma_{i}(e)}{\chi_{j}-\eta_{i}}=0$ (12)

for $j\in[1,k-1]$. The same equation holds for the $\ell$ smallest eigenvalues
below the bulk : if we denote $\chi_{-j}$ the $j$-th smallest eigenvalue of
${\bf X}_{N}^{(e)}$, it is solution of the same equation in
$[\eta_{-m_{1}+j},\eta_{-m_{1}+j+1}]$. Observe that unless $\gamma_{i}(e)$
vanishes, $\chi_{i}$ can not be equal to $\eta_{i}$. So, if we denote for
$i=-m_{1}+l+1,...,p+m_{2}-k$, $\delta_{i}$ the solution of the same
interlacing equation in $[\eta_{i},\eta_{i}+1]$, up to diagonalization, ${\bf
X}_{N}^{(e_{1})}$ has the following form :

${\bf
X}_{N}^{(e_{1})}=diag(\chi_{-1},...,\chi_{-l},\underbrace{\eta_{-m_{1}+l+1}}_{N_{-m_{1}+l+1}-1},\delta_{-m_{1}+l+1},\underbrace{\eta_{-m_{1}+l+2}}_{N_{-m_{1}+l+2}-1},...,\underbrace{\eta_{p+m_{2}-k}}_{N_{p+m_{2}-k}-1},\delta_{p+m_{2}-k},\chi_{k-1},...,\chi_{1})$

where the $\delta_{j}$ and the $\chi_{i}$ being continuous functions of
$\gamma(e)$. If we denote $\chi_{i}(\kappa)$ the value of $\chi_{i}$ when
$\gamma(e)=\kappa$, we deduce by induction and using the continuity in Remark
9 that when $\gamma({e_{1}})$ converges toward $\kappa$ :

$\lim_{N\to\infty}\frac{1}{N-1}\ln\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-\ell,i\neq 0,1}^{k}\theta_{i}\langle e_{i},{\bf
X}^{(e_{1})}_{N}e_{i}\rangle\right)\Big{|}e_{1}\right]=\frac{\beta}{2}\sum_{i=-\ell,i\neq
0,1}^{k}J(\mu,\theta,\chi_{i}(\kappa))$

Then, using again that $\gamma^{N}$ satisfies a large deviations principle we
can write that :

$\displaystyle\lim_{N\to\infty}\frac{1}{N}\ln\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-\ell,i\neq 0}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]$ (13)
$\displaystyle=\frac{\beta}{2}\sup_{\gamma\in(\mathbb{R}^{+})^{p+m_{1}+m_{2}},\sum\gamma_{i}=1}\bigg{\\{}\sum_{i=-\ell}^{-1}J(\mu,\theta_{i},\chi_{i}(\gamma))+\sum_{i=2}^{k}J(\mu,\theta_{i},\chi_{i-1}(\gamma))$
$\displaystyle\qquad\qquad+\sum_{i=1}^{p}\alpha_{i}\log\frac{\gamma_{i}}{\alpha_{i}}+\sum_{i=1-m_{1}}^{p+m_{2}}\theta_{1}\eta_{i}\gamma_{i}\bigg{\\}}$

By continuity, taking this supremum only on the set of $\gamma$ summing up to
$1$ and such that $\gamma_{i}>0$ does not change its value. Notice that for
such $\gamma$ we have for all $i$ and $j$ $\chi_{i}(\gamma)\neq\eta_{j}$. We
set $I_{-j}=]\eta_{-m_{1}+j},\eta_{-m_{1}+j+1}[$ for $j=1,...,\ell$ and
$I_{j}=]\eta_{m_{2}+p-j},\eta_{m_{2}+p-j+1}[$ for $j=1,...,k-1$ and we define
:

$D=\Big{\\{}(\gamma,\chi)\in\prod_{j=-\ell,j\neq
0}^{k-1}I_{j}\times(\mathbb{R}^{+,*})^{m_{1}+m_{2}+p}:\sum_{i}\gamma_{i}=1,\sum_{i}\frac{\gamma_{i}}{\chi_{j}-\eta_{i}}=0\,\forall
j\in[-\ell,k-1]\setminus\\{0\\}\Big{\\}}$

Therefore we have :

$\displaystyle\lim_{N\to\infty}\frac{1}{N}\ln\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-\ell,i\neq 0}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]$ (14)
$\displaystyle=\frac{\beta}{2}\sup_{(\chi,\gamma)\in
D}\bigg{\\{}\sum_{i=-\ell}^{-1}J(\mu,\theta_{i},\chi_{i})+\sum_{i=2}^{k}J(\mu,\theta_{i},\chi_{i-1})+\sum_{i=1}^{p}\alpha_{i}\log\frac{\gamma_{i}}{\alpha_{i}}+\sum_{i=1-m_{1}}^{p+m_{2}}\theta_{1}\eta_{i}\gamma_{i}\bigg{\\}}$

For $i=-m_{1}+\ell,...,m_{2}+p-k$, we set :

$\bar{\gamma}_{i}=\frac{\prod_{j=1}^{\ell}(\eta_{-m_{1}+j}-\eta_{i})\prod_{j=1}^{k-1}(\eta_{m_{2}+p-j+1}-\eta_{i})}{\prod_{j=1}^{\ell}(\chi_{-j}-\eta_{i})\prod_{j=1}^{k-1}(\chi_{j}-\eta_{i})}\gamma_{i}$

We have that if $\gamma_{i}>0$ for all $i$ then $\bar{\gamma}_{i}>0$ for all
$i$ and $\bar{\gamma}_{i}$ vanishes at the outliers. We want to prove that
this definition provides a one to one correspondance between the set $D$ of
parameters $(\chi,\gamma)$ and the set $\bar{D}$ parameters
$(\chi,\bar{\gamma})$ defined as follows :

$\bar{D}=\Big{\\{}(\chi,\bar{\gamma})\in\prod_{j=-\ell,j\neq
0}^{k-1}I_{j}\times(\mathbb{R}^{+,*})^{m_{1}+m_{2}+p-k-\ell+1},\sum_{i}\bar{\gamma}_{i}=1\Big{\\}}$

Note that $\bar{\gamma}$ lives a priori in a set of $k+\ell-1$ dimension
smaller but $\gamma$ was satisfying as well $k+\ell-1$ additional equations.
First, let us prove that if $(\chi,\gamma)\in D$ the
$\bar{\gamma}_{i}^{\prime}s$ sum up to $1$. We let for a real number $X$, $F$
to be the rational function

$F(X)=\frac{\prod_{j=1}^{\ell}(\eta_{-m_{1}+j}-X)\prod_{j=1}^{k-1}(\eta_{m_{2}+p-j+1}-X)}{\prod_{j=1}^{\ell}(\chi_{-j}-X)\prod_{j=1}^{k-1}(\chi_{j}-X)}$

so that $\bar{\gamma}_{i}=F(\eta_{i})\gamma_{i}$ for
$i\in[-m_{1}+\ell,m_{2}+p-k]$ and $F(\eta_{i})=0$ for the other values of $i$.
Let us decompose $F$ in partial fractions : as it goes to one at infinity, we
find

$F(X)=1+\sum_{j=-\ell,j\neq 0}^{k-1}\frac{a_{j}}{\chi_{j}-X}$

for some real numbers $a_{j}$. Then, since $F(\eta_{i})=0$ for $i\neq-
m_{1}+\ell,...,m_{2}+p-k$ we have :

$\displaystyle\sum_{i=-m_{1}+\ell}^{m_{2}+p-k}\bar{\gamma}_{i}$
$\displaystyle=$
$\displaystyle\sum_{i=-m_{1}+1}^{m_{2}+p}F(\eta_{i})\gamma_{i}$
$\displaystyle=$
$\displaystyle\sum_{i=-m_{1}+1}^{m_{2}+p}\gamma_{i}+\sum_{j=-\ell,j\neq
0}^{k-1}a_{j}\sum_{i=-m_{1}+1}^{m_{2}+p}\frac{\gamma_{i}}{\chi_{j}-\eta_{i}}$
$\displaystyle=$ $\displaystyle 1$

where we used the interlacing relations. Therefore, since when $\chi$ is fixed
the fonction $\gamma\mapsto\bar{\gamma}$ is an affine map between the affine
subspace $E$ of $\mathbb{R}^{p+m_{1}+m_{2}}$ defined by the $k+\ell-1$
interlacing relations and the condition of sum one and the affine subspace $F$
of $\mathbb{R}^{p+m_{1}+m_{2}-k-\ell+1}$ defined by the condition of sum one.
Since these spaces have the same dimension to conclude we only need to prove
that this map is injective and that for all $\gamma\in E$,
$\bar{\gamma}_{i}>0$ for all $i$ implies $\gamma_{i}>0$ for all $i$. To prove
injectivity first notice that $\gamma_{i}=F(\eta_{i})^{-1}\bar{\gamma}_{i}$
for $i\in[-m_{1}+\ell,m_{2}+p-k]$. We next show how to reconstruct
$\gamma_{i}$ for $i\in[-m_{1}+\ell,m_{2}+p-k]^{c}$, and more precisely
$i\in[m_{2}+p-k+1,m_{2}+p]$. To this end, for $j=1,...,k-1$, we let
$G_{j}(X)=\frac{F(X)}{(\eta_{p+m_{2}-j+1}-X)}$ and for $j=1,...,\ell$, we let
$G_{-j}(X)=\frac{F(X)}{(\eta_{-m_{1}+j}-X)}$. Let us suppose $j>0$ (the $j<0$
case is similar). Then again decomposing $G_{j}$ in partial fractions, we have

$G_{j}(X)=\sum_{j=-\ell,j^{\prime}\neq
0,j}^{k-1}\frac{b_{j^{\prime}}}{\chi_{j^{\prime}}-X}$

for some real numbers $b_{j}$. Again by the interlacing relations

$\sum_{i=-m_{1}+1}^{p+m_{2}}G_{j}(\eta_{i})\gamma_{i}=0\,.$

But we can also write :

$\sum_{i=-m_{1}+1}^{p+m_{2}}G_{j}(\eta_{i})\gamma_{i}=\sum_{i=-m_{1}+\ell}^{p+m_{2}-k}\frac{\bar{\gamma_{i}}}{\chi_{j}-\eta_{i}}+G_{j}(\eta_{p+m_{2}-j+1})\gamma_{p+m_{2}-j+1}$

so that we deduce

$\gamma_{p+m_{2}-j+1}=-(\sum_{i=-m_{1}=\ell}^{p+m_{2}-k+1}\frac{\bar{\gamma_{i}}}{\chi_{j}-\eta_{i}})/G_{j}(\eta_{p+m_{2}-j+1})\,.$

As a consequence, the map $\gamma\mapsto\bar{\gamma}$ is injective.
Furthermore if $\bar{\gamma}_{i}>0$ for all $i$, then $\gamma_{j}>0$ since
$G_{j}(\eta_{j})<0$. The same remains true for $j$ negative. Therefore we have
that the change of variables from $(\chi,\gamma)\in D$ to
$(\chi,\bar{\gamma})\in\bar{D}$ is one to one. But before changing variables,
let us compare $\sum\eta_{i}\gamma_{i}$ and $\sum\eta_{i}\bar{\gamma_{i}}$. We
use the following decomposition :

$XF(X)=X+S+\sum_{j=-\ell,j\neq 0}^{k-1}\frac{c_{j}}{\chi_{j}-X}$

for some real numbers $c_{j}$ and where

$S=\sum_{j=-\ell,j\neq
0}^{k-1}\chi_{j}-\sum_{j=1}^{\ell}\eta_{-m_{1}+j}-\sum_{j=1}^{k-1}\eta_{m_{2}+p-j+1}\,.$

We deduce that

$\sum_{-m_{1}+l}^{m_{2}+p-k+1}\eta_{i}\bar{\gamma}_{i}=\sum_{-m_{1}+1}^{m_{2}+p}\eta_{i}F(\eta_{i})\gamma_{i}=\sum_{-m_{1}+1}^{m_{2}+p}\eta_{i}\gamma_{i}+S$

where we used again the interlacing relationships and the fact that the
$\gamma_{i}$’s sum up to $1$. Coming back to (13), we have to take the
supremum of the following function $I$ for $(\chi,\bar{\gamma})\in\bar{D}$:

$\displaystyle I(\bar{\gamma},\chi)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{k-1}\left[J(\mu,\theta_{j+1},\chi_{j})+\sum_{i=1}^{p}\alpha_{i}\log|\chi_{j}-\eta_{i}|-\sum_{i=1}^{p}\alpha_{i}\log|\eta_{m_{2}+p-j+1}-\eta_{i}|\right]$
$\displaystyle+$
$\displaystyle\sum_{j=-\ell}^{-1}\left[J(\mu,\theta_{j},\chi_{j})+\sum_{i=1}^{p}\alpha_{i}\log|\chi_{j}-\eta_{i}|-\sum_{i=1}^{p}\alpha_{i}\log|\eta_{-m_{1}-j}-\eta_{i}|\right]$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{p}\alpha_{i}\ln\frac{\bar{\gamma_{i}}}{\alpha_{i}}+\theta_{1}\left(\sum_{i=-m_{1}+1}^{p+m_{2}}\eta_{i}\bar{\gamma}_{i}-\sum_{j=-\ell,j\neq
0}^{k-1}\chi_{j}+\sum_{j=1}^{\ell}\eta_{-m_{1}+j}+\sum_{j=1}^{k-1}\eta_{m_{2}+p-j+1}\right)$

Therefore we have :

$\displaystyle I(\bar{\gamma},\chi)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{k-1}H(\chi_{i},\theta_{i+1})+\sum_{i=-\ell}^{-1}H(\chi_{i},\theta_{i})+\sum_{i=1}^{p}\alpha_{i}\log\frac{\bar{\gamma}_{i}}{\alpha_{i}}+\theta_{1}\sum_{i=m_{1}-\ell}^{p+m_{2}-k+1}\eta_{i}\bar{\gamma}_{i}$
$\displaystyle-$
$\displaystyle\sum_{j=1}^{k-1}\sum_{i=1}^{p}\alpha_{i}\log|\eta_{m_{2}+p-j+1}-\eta_{i}|-\sum_{j=-\ell}^{-1}\sum_{i=1}^{p}\alpha_{i}\log|\eta_{-m_{1}-j}-\eta_{i}|$
$\displaystyle\color[rgb]{0,0,1}+\color[rgb]{0,0,0}$
$\displaystyle\theta_{1}\sum_{j=1}^{\ell}\eta_{-m_{1}+j}\color[rgb]{0,0,1}+\color[rgb]{0,0,0}\theta_{1}\sum_{j=1}^{k-1}\eta_{m_{2}+p-j+1}$

where we set :

$H(\chi,\theta)=J(\mu,\theta,\chi)+\sum_{i=1}^{p}\alpha_{i}\ln|\chi-\eta_{i}|-\chi\theta_{1}\,.$

The supremum over $\bar{\gamma}$ and $\chi$ are now decoupled and the
$\chi_{i}$ belongs to $]\eta_{-m_{1}+i},\eta_{-m_{1}+i+1}[$ if
$i\in[-\ell,-1]$ and $]\eta_{p+m_{2}-i+1},\eta_{p+m_{2}-i+2}[$ if $i\in[1,k]$.
As in the two-dimensional case we can compute for $i=2,...,k$,

$\sup_{\chi\in]\eta_{p+m_{2}-i+1},\eta_{p+m_{2}-i+2}[}H(\chi,\theta_{i})=\left\\{\begin{array}[]{ll}H(\eta_{p+m_{2}-i+1},\theta_{i})&\mbox{
if }G_{\mu}^{-1}(\theta_{1})\leq\eta_{p+m_{2}-i+1},\cr
H(G_{\mu}^{-1}(\theta_{1}),\theta_{i})&\mbox{ if
}G_{\mu}^{-1}(\theta_{1})\in[\eta_{p+m_{2}-i+1},\eta_{m+p_{2}-i+2}],\cr
H(\eta_{p+m_{2}-i+2},\theta_{i})&\mbox{ if
}G_{\mu}^{-1}(\theta_{1})>\eta_{p+m_{2}-i+2}.\cr\end{array}\right.$

Moreover, for $i=1,...,\ell$, $H(\chi,\theta_{-i})$ is a decreasing function
of $\chi$ since $\theta_{-i}$ is negative and so

$\sup_{\chi\in]\eta_{-m_{1}+i},\eta_{-m_{1}+i+1}[}H(\chi,\theta_{-i})=H(\eta_{-m_{1}+i},\theta_{-i})\,.$

It remains to optimize the sum of the third and fourth term in
$I(\bar{\gamma},\chi)$. But this sum is equal to
$I_{\theta_{1},\eta_{p+m_{2}-k+1}}^{p+m_{2}-k+1}(\bar{\gamma})$, see (6).
Thus, taking the supremum for $\bar{\gamma}_{i}>0$ and
$\sum\bar{\gamma}_{i}=1$ gives $J(\mu,\theta_{1},\eta_{p+m_{2}-k+1})$.

To conclude, we need to look at the position of $G^{-1}_{\mu}(\theta_{1})$
relatively to the $k$ largest outliers. Let us denote $H_{i}=\max
H(.,\theta_{i})$ for $i<0$ and $H_{i}=\max H(.,\theta_{i+1})$ for $i>0$. We
have

$\displaystyle\sup I$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{k-1}\left(H_{j}-\sum\alpha_{i}\log|\eta_{i}-\eta_{p+m_{2}-j+1}|+\theta_{1}\eta_{p+m_{2}-j+1}\right)$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{\ell}\left(H_{-j}-\sum\alpha_{i}\log|\eta_{i}-\eta_{-m_{1}+j}|+\theta_{1}\eta_{-m_{1}+j}\right)$
$\displaystyle+$ $\displaystyle J(\mu,\theta_{1},\eta_{p+m_{2}-k+1})$

Here we will treat the case where
$G^{-1}_{\mu}(\theta_{1})\in[\eta_{p+m_{2}-k+1},\eta_{p+m_{2}}]$ which is the
most complex one. First of all, since for $j=1,...,\ell$
$H_{-j}=H(\eta_{-m_{1}+j},\theta_{-j})$, we have that in the second sum, the
term of index $j$ is indeed equal to $J(\mu,\theta_{-j},\eta_{-m_{1}+j})$. If
$j^{\prime}$ is the index such that,
$G^{-1}_{\mu}(\theta_{1})\in[\eta_{p+m_{2}-j^{\prime}},\eta_{p+m_{2}-j^{\prime}+1}]$
then for $j<j^{\prime}$, $H_{j}=H(\eta_{p+m_{2}-j},\theta_{j+1})$ and the term
of index $j$ of the first sum is :

$J(\mu,\theta_{j+1},\eta_{p+m_{2}-j})+\sum\alpha_{i}\log\frac{|\eta_{p+m_{2}-j}-\eta_{i}|}{|\eta_{p+m_{2}-j+1}-\eta_{i}|}+\theta_{1}(\eta_{p+m_{2}-j}-\eta_{p+m_{2}-j+1})$

The term of index $j^{\prime}$ is equal to :

$J(\mu,\theta_{j+1},G^{-1}_{\mu}(\theta_{1}))+\sum_{i}\alpha_{i}\log\frac{|G^{-1}_{\mu}(\theta_{1})-\eta_{i}|}{|\eta_{p+m_{1}-j^{\prime}+1}-\eta_{i}|}+\theta_{1}(\eta_{p+m_{2}-j^{\prime}}-G^{-1}_{\mu}(\theta_{1}))$

And the terms $j>j^{\prime}$ are equal to
$J(\mu,\theta_{j+1},\eta_{p+m_{2}-j+1})$. Since $\theta_{j+1}\leq\theta_{1}$,
$G_{\mu}^{-1}(\theta_{j+1})\geq G_{\mu}^{-1}(\theta_{1})$, so we have that for
$j>j^{\prime}$

$J(\mu,\theta_{j+1},\eta_{p+m_{2}-j+1})=J(\mu,\theta_{j+1},\eta_{p+m_{2}-j})\mbox{
and
}J(\mu,\theta_{j^{\prime}+1},G^{-1}_{\mu}(\theta_{1}))=J(\mu,\theta_{j^{\prime}+1},\eta_{p+m_{2}-j^{\prime}})$

Therefore the whole sum can be simplified as follows :

$\displaystyle\max I$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{k-1}J(\mu,\theta_{j+1},\eta_{p+m_{2}-j})+\sum_{j=1}^{\ell}J(\mu,\theta_{-j},\eta_{-m_{1}+j})+\sum_{i}\alpha_{i}\ln\frac{|G^{-1}_{\mu}(\theta_{1})-\eta_{i}|}{|\eta_{p+m_{2}}-\eta_{i}|}$
$\displaystyle+$
$\displaystyle\theta_{1}(\eta_{p+m_{2}}-G^{-1}_{\mu}(\theta_{1}))+J(\mu,\theta_{1},\eta_{p+m_{2}-k+1})$

Then we notice that
$J(\mu,\theta_{1},\eta_{p+m_{2}-k+1})=J(\mu,\theta_{1},G^{-1}_{\mu}(\theta_{1}))$
and conclude since :

$J(\mu,\theta_{1},\eta_{p+m_{2}})=J(\mu,\theta_{1},G^{-1}_{\mu}(\theta_{1}))+\sum_{i}\alpha_{i}\ln\frac{|G^{-1}_{\mu}(\theta_{1})-\eta_{i}|}{|\eta_{p+m_{2}}-\eta_{i}|}+\theta_{1}(\eta_{p+m_{2}}-G^{-1}_{\mu}(\theta_{1}))\,.$

∎

##  4\. Diffuse spectrum

We next consider the general case where ${\bf X}_{N}$ is a Hermitian matrix
such that

$\hat{\mu}_{{\bf X}_{N}}=\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}}$

converges towards a probability measure $\mu$ with support with rightmost
point $r_{\mu}$ and leftmost point $l_{\mu}$ which are assumed to be finite.
Let $\lambda_{1}^{N}\geq\lambda_{2}^{N}\geq\cdots\geq\lambda_{k}^{N}\geq
r_{\mu}$ be the $k$ largest outliers of ${\bf X}_{N}$ counted with
multiplicity one, $\lambda_{N}^{N}\leq\cdots\leq\lambda_{N-\ell+1}^{N}\leq 0$
be the smallest outliers of ${\bf X}_{N}$ with multiplicity one (but
eventually equal). Assume that

$\lim_{N\rightarrow\infty}\lambda_{i}^{N}=\lambda_{i}>r_{\mu}\mbox{ for
}i\in[1,k],\lim_{N\rightarrow\infty}\lambda_{N-i+1}^{N}=\lambda_{-i}<l_{\mu}\mbox{
for }i\in[1,\ell]\,.$

If the above assumption is true for the $k$ largest outliers and we want to
study the spherical integral with non-negative $\theta_{i}$’s while relaxing
the assumption on the smallest ones, we still need to assume that the latter
are bounded.

We are going to prove that

######  Proposition 16.

Fix two integer numbers $k,\ell$. Let
$\theta_{1}\geq\theta_{2}\geq\cdots\geq\theta_{k}\geq
0\geq\theta_{-\ell}\geq\cdots\geq\theta_{-1}$. Then

$\lim_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-\ell,i\neq 0}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]\qquad\qquad$
$\qquad\qquad=\frac{\beta}{2}\left(\sum_{i=1}^{k}J(\mu,\theta_{i},\lambda_{i})+\sum_{i=1}^{\ell}J(\mu,\theta_{-i},\lambda_{-i})\right)\,.$

###### Proof.

We first remark that we can assume ${\bf X}_{N}$ diagonal without loss of
generality. In a first step, we assume that the the partition function of
$\mu$ is continuous and that ${\bf X}_{N}$ has bounded norm. We fix
$\varepsilon>0$ . We know by assumption that for $N$ large enough, the
spectrum of ${\bf X}_{N}$ is included into
$[\lambda_{-\ell}-\varepsilon,\lambda_{k}+\varepsilon[$. For $j\geq-1$ we let
$n_{j}^{\varepsilon,N}$ be the number of eigenvalues of ${\bf X}_{N}$ in
$[\lambda_{-\ell}+j\varepsilon,\lambda_{-\ell}+(j+1)\varepsilon[$, until the
first $j$ so that
$\lambda_{-\ell}+(j+1)\varepsilon\geq\lambda_{k}+\varepsilon$. We let ${\bf
X}_{N}^{\varepsilon}$ be the diagonal matrix with eigenvalues
$\lambda_{-\ell}+j\varepsilon$ with multiplicity $n_{j}^{\varepsilon,N}$.
Since we assumed the extreme eigenvalues distinct, for $N$ large enough the
extreme eigenvalues of ${\bf X}_{N}^{\varepsilon}$ are $(\lambda_{i},i\leq
k,\lambda_{-j},j\leq\ell)$. $n_{j}^{\varepsilon,N}/N$ converges towards
$\mu([\lambda_{-\ell}+j\varepsilon,\lambda_{-\ell}+(j+1)\varepsilon])$ since
we assumed the partition function of $\mu$ to be continuous. Hence, ${\bf
X}_{N}^{\varepsilon}$ satisfies the hypotheses of Proposition 15. Moreover, by
definition, for $N$ large enough we know that

$\|{\bf X}_{N}-{\bf X}_{N}^{\varepsilon}\|\leq\varepsilon$

Therefore,

$\left|\frac{1}{N}\log\frac{\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-\ell,i\neq 0}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]}{\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-\ell,i\neq 0}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}^{\varepsilon}e_{i}\rangle\right)\right]}\right|\leq\frac{\beta}{2}\sum|\theta_{i}|\varepsilon\,.$

On the other hand, Proposition 15 implies

$\lim_{N\rightarrow\infty}\frac{1}{N}\log\mathbb{E}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-\ell,i\neq 0}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}^{\varepsilon}e_{i}\rangle\right)\right]=\frac{\beta}{2}\left(\sum_{i=-\ell,i\neq
0}^{k}J(\mu^{\varepsilon},\theta_{i},\lambda_{i})\right)\,$

with
$\mu^{\varepsilon}=\sum_{j}\mu([\lambda_{-\ell}+j\varepsilon,\lambda_{-\ell}+(j+1)\varepsilon])\delta_{\lambda_{-\ell}+j\varepsilon}$.
By continuity of $\mu\rightarrow J(\mu,\theta_{i},\lambda_{i})$, see [18] or
the Appendix, and the weak convergence of $\mu^{\varepsilon}$ towards $\mu$,
the conclusion follows.

Finally, to remove the condition that $\mu$ has a continuous partition
function we note that we can always add a small matrix to ${\bf X}_{N}$ and
its contribution will go to zero as its norm goes to zero after $N$ goes to
infinity. We again assume ${\bf X}_{N}$ diagonal and replace it by the
diagonal matrix with the same outliers and in the bulk the entries are added
independent uniform variables with uniform distribution on $[0,\varepsilon]$.
Again ${\bf X}_{N}^{\varepsilon}-{\bf X}_{N}$ has norm bounded by
$\varepsilon$. Moreover, the spectral measure of ${\bf X}_{N}^{\varepsilon}$
converges towards $\mu*1_{[0,\varepsilon]}du/\varepsilon$ whose partition
function is continuous. Hence, we can apply our result to this new matrix and
then let $\varepsilon$ go to zero to conclude. ∎

##  5\. Applications to large deviations for the extreme eigenvalues of random
matrices

###  5.1. Universality of the large deviations for the $k$ extreme eigenvalues
of Wigner matrices with sharp sub-Gaussian entries

In this section, we prove Theorem 3. The proof follows the ideas of [13] quite
closely: we simply sketch the main arguments and changes. First note that it
is enough to prove a weak large deviations principle thanks to our assumption
which insures that exponential tightness holds. Moreover. let
$\bar{\lambda}^{N}=(\lambda_{1},\ldots,\lambda_{k},\lambda_{N-k},\ldots,\lambda_{N})$
be the $k$ extreme eigenvalues of ${\bf X}_{N}$. To get a weak large
deviations upper bound, we tilt the measure by spherical integrals as above :
if $(e_{i})_{-k\leq i\leq k}$ follows the uniform law on the set of $2k$
orthonormal vectors on the sphere ($\theta_{0}=0$ and $e_{0}=0$ is added to
shorten the notations), $\theta_{i}$ are real numbers of the same sign than
$i\in[-k,k]$ to be chosen later, we write

$\displaystyle\mathbb{P}\left(\|\bar{\lambda}^{N}-\bar{x}\|_{2}\leq\varepsilon\right)$
$\displaystyle\leq$ $\displaystyle\mathbb{E}_{{\bf
X}_{N}}\left[1_{\|\bar{\lambda}^{N}-\bar{x}\|_{2}\leq\varepsilon}\frac{\mathbb{E}_{e}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-k}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]}{\mathbb{E}_{e}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-k}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]}\right]$ $\displaystyle\leq$ $\displaystyle
e^{-N\frac{\beta}{2}F(\bar{x},\bar{\theta})+o(\varepsilon)N}\mathbb{E}_{{\bf
X}_{N}}\mathbb{E}_{e}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-k}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]$

where

$F(\bar{x},\bar{\theta})=\sum_{i=-k}^{k}J(\sigma,\theta_{i},x_{i})\,.$

We used in the second line that by Theorem 22, the spherical integrals are
uniformly continuous and are asymptotically given by
$F(\bar{x},\bar{\theta})$, and our assumption that the spectral measure of
${\bf X}_{N}$ converges towards the semi-circle law $\sigma$ faster than any
exponential. Here $o(\varepsilon)$ goes to zero when $\varepsilon$ does. We
also used the bound

$\frac{\mathbb{E}_{{\bf
X}_{N}}\left[1_{\|\bar{\lambda}^{N}-\bar{x}\|_{2}\leq\varepsilon}\mathbb{E}_{e}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-k}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]\right]}{\mathbb{E}_{{\bf
X}_{N}}\left[\mathbb{E}_{e}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-k}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]\right]}\leq 1$ (15)

We next compute the expectation of the spherical integral by using that our
entries are sharp-subgaussian :

$\displaystyle\mathbb{E}_{{\bf
X}_{N}}\left[\mathbb{E}_{e}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-k}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]\right]$
$\displaystyle\leq\mathbb{E}_{e}[\exp\\{\frac{\beta}{4}N\sum_{i=-k}^{k}\sum_{k\leq
j}2^{k\neq
j}|\sum\theta_{i}e_{i}(k)e_{i}(j)|^{2}\\}]=\exp\\{\frac{\beta}{4}\sum_{j=-k}^{k}\theta_{j}^{2}\\}$

We hence get the upper bound

$\limsup_{\varepsilon{\rightarrow}0}\limsup_{N{\rightarrow}\infty}\frac{1}{N}\ln\mathbb{P}\left(\|\bar{\lambda}^{N}-\bar{x}\|_{2}\leq\varepsilon\right)\leq-\frac{\beta}{2}\sup_{\theta_{i}}\\{\sum_{j=-k}^{k}\frac{\theta_{j}^{2}}{2}-F(\bar{x},\bar{\theta})\\}$

where we take the supremum over non-negative $\theta_{i}$ for $i\in[1,k]$ and
non-positive $\theta_{i}$’s for $i\in[-k,-1]$. Finally we observe that the
supremum decouples and recall from [13, Section 4.1] that the supremum over
each $\theta_{i}$ of $\theta_{i}^{2}/2-J(\sigma,\theta_{i},x_{i})$ gives
$\int_{2}^{|x_{i}|}\sqrt{t^{2}-4}dt$. To get the lower bound, we need to show
that there exists
$\bar{\theta}=(\theta_{-k},\ldots,\theta_{-1},\theta_{1},\ldots,\theta_{k})$
such that (15) is almost an equality in the sense that for every
$\varepsilon>0$

$\liminf_{\mathbb{N}{\rightarrow}\infty}\frac{1}{N}\ln\frac{\mathbb{E}_{{\bf
X}_{N}}\left[1_{\|\bar{\lambda}^{N}-\bar{x}\|_{2}\leq\varepsilon}\mathbb{E}_{e}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-k}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]\right]}{\mathbb{E}_{{\bf
X}_{N}}\left[\mathbb{E}_{e}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-k}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]\right]}\geq 0$ (16)

and

$\liminf_{\mathbb{N}{\rightarrow}\infty}\frac{1}{N}\ln\mathbb{E}_{{\bf
X}_{N}}\left[\mathbb{E}_{e}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-k}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]\right]\geq\frac{\beta}{2}\sum_{j=-k}^{k}\theta_{j}^{2}\,.$
(17)

In both cases we use the fact that under the uniform measure, the vectors
$e_{i}$ are delocalized with overwhelming probability, namely if
$V_{N}^{\kappa}=\cap_{1\leq i\leq k}\\{\|e_{i}\|_{\infty}\leq
N^{-1/4-\kappa}\\}$ then $\mathbb{P}(V_{N}^{\kappa})$ goes to one for any
$\kappa\in(0,1/4)$. Therefore, to prove (17) we notice that

$\displaystyle\mathbb{E}_{{\bf
X}_{N}}\left[\mathbb{E}_{e}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=-k}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]\right]$ $\displaystyle\geq$
$\displaystyle\mathbb{E}_{e}\left[1_{e\in V_{N}^{\kappa}}\prod_{i\leq
j}\mathbb{E}[\exp\\{\frac{\beta}{2}N2^{1_{i\neq
j}}\sum_{r}\theta_{r}\Re(e_{r}(i)\bar{e}_{r}(j)X_{ij})]\right]$
$\displaystyle\geq$
$\displaystyle\exp\\{N\frac{\beta}{2}\sum_{r=-k}^{k}\theta_{j}^{2}+O(N^{1-2\kappa})\\}\mathbb{P}(V_{N}^{\kappa})$

where we used that $\sum_{r}\theta_{r}e_{r}(i)\bar{e}_{r}(j)$ is of order at
most $N^{-1/2-2\kappa}$ on $V_{N}^{\varepsilon}$ so that we can expand the
Laplace transform of the entries around the origin. This proves (17). To prove
(16) we notice that it is enough to show that for $N$ large enough

$\inf_{\bar{e}\in V_{N}^{\varepsilon}}\frac{\mathbb{E}_{{\bf
X}_{N}}\left[1_{\|\bar{\lambda}^{N}-\bar{x}\|_{2}\leq\varepsilon}\exp\left(\frac{\beta
N}{2}\sum_{i=-k}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\right)\right]}{\mathbb{E}_{{\bf
X}_{N}}\left[\exp\left(\frac{\beta N}{2}\sum_{i=-k}^{k}\theta_{i}\langle
e_{i},{\bf X}_{N}e_{i}\rangle\right)\right]}\geq\frac{1}{2}\,.$

But under the law tilted by $\exp\left(\frac{\beta
N}{2}\sum_{i=-k}^{k}\theta_{i}\langle e_{i},{\bf X}_{N}e_{i}\rangle\right)$,
${\bf X}_{N}$ still has independent entries. We can compute its mean and
covariance under the tilted law and using again that
$\sum_{r}\theta_{r}e_{r}(i)\bar{e}_{r}(j)$ is of order at most
$N^{-1/2-2\kappa}$, we see that its mean is $\sum\theta_{i}e_{i}e_{i}^{*}$ and
its covariance is close to $1/N$. We deduce as in [13] and the BBP transition
[2, 20] that under this tilted law the outliers of ${\bf X}_{N}$ are given by
$\theta_{i}+\theta_{i}^{-1}$: it is therefore sufficient to choose
$\theta_{i}=\frac{1}{2}(x_{i}\pm\sqrt{x_{i}^{2}-4})$. We refer the reader to
[13] for more details.

###  5.2. Universality of the large deviations for the $k$ largest eigenvalues
of Wishart matrices with sharp sub-Gaussian entries

We here prove Theorem 6 and, as in the previous subsection, we will only
sketch the changes from the proof in [13]. As in [13] we will study the
largest eigenvalues of the linearized matrix ${\bf Y}_{N}$ of the matrix
${N^{-1/2}}{\bf G}_{L,M}{\bf G}_{L,M}^{*}$ :

${\bf Y}_{N}=\begin{pmatrix}0_{L\times L}&\frac{1}{\sqrt{N}}{\bf G}_{L,M}\\\
\frac{1}{\sqrt{N}}{\bf G}_{L,M}^{*}&0_{M\times M}\end{pmatrix}$

Up to a factor $(N/L)^{1/2}=((1+\alpha)+o(N^{-\kappa}))^{1/2}$ ${\bf Y}_{N}$
is the linearization of ${\bf W}_{L,M}$. The main difference with the proof
for Wigner matrices will be that computing the asymptotics of the annealed
spherical integral requires more skill as it depends on the large deviations
for the scalar products of projections of vectors uniformly distributed on the
sphere : we can not merely assume that they are delocalized since this could a
priori change the large deviations weight. To be more precise, let
$\Lambda_{N}$ be the annealed spherical integral given for
$\bar{\theta}=(\theta_{1},\ldots,\theta_{k})\in(\mathbb{R}^{+})^{k}$ by

$\Lambda_{N}(\bar{\theta})=\frac{1}{N}\log\mathbb{E}_{{\bf
X}_{N}}\left[\mathbb{E}_{e}\left[\exp\left(\frac{\beta
N}{2}\sum_{i=1}^{k}\theta_{i}\langle e_{i},{\bf
Y}_{N}e_{i}\rangle\right)\right]\right]\,.$

We shall prove that

$\lim_{N\rightarrow\infty}\frac{1}{N}\log\Lambda_{N}(\bar{\theta})=\Lambda(\bar{\theta})=\sum_{i=1}^{k}\Lambda(\theta_{i})$
(18)

with, if $\alpha^{\prime}=(1+\alpha)^{-1}$, and with $\alpha$ the limit of
$M/N$,

$\Lambda(\theta)=\sup_{a\in]0,1[}\Big{(}\theta^{2}a(1-a)+\alpha^{\prime}\ln\frac{a}{\alpha^{\prime}}+(1-\alpha^{\prime})\ln\frac{1-a}{1-\alpha^{\prime}}\Big{)}\,.$

The above supremum is achieved at $x_{\theta,\alpha}$, as defined in Lemma 3.4
of [13]. We first prove the upper bound in (18).

$\displaystyle e^{N\Lambda_{N}(\bar{\theta})}$ $\displaystyle=$
$\displaystyle\mathbb{E}_{e}\mathbb{E}_{{\bf
Y}}\Big{[}\Big{[}\exp\Big{(}\frac{\beta N}{2}\sum_{i=1}^{k}\theta_{i}\langle
e_{i},{\bf Y}_{N}e_{i}\rangle\Big{)}\Big{]}\Big{]}$ $\displaystyle=$
$\displaystyle\mathbb{E}_{e}\mathbb{E}_{{\bf
Y}}\Big{[}\Big{[}\exp\Big{(}\frac{\beta}{2}\sum_{i=1}^{k}\sum_{l=1,...,L\atop
m=L+1,...,N}\theta_{i}\sqrt{N}\Re(e_{i}(l)\bar{e}_{i}(m)X_{l,m})\Big{)}\Big{]}\Big{]}$
$\displaystyle\leq$ $\displaystyle\mathbb{E}_{e}\Big{[}\exp\Big{(}\frac{\beta
N}{4}\sum_{l=1,...,L\atop
m=L+1,...,N}\sum_{i,j=1}^{k}\theta_{i}\theta_{j}\Re(e_{i}(l)\bar{e}_{i}(m)\bar{e}_{j}(l)e_{j}(m))\Big{)}\Big{]}$

where we used that the entries are sharp sub-Gaussian. Now, let us call
$e^{(1)}$ the vector of $\mathbb{C}^{L}$ whose coordinnates are the $L$ first
coordinates of $e$ and $e^{(2)}$ the vector of $\mathbb{C}^{M}$ whose
coordinates are the $M$ last of $e$. If we let $\psi^{(p)}_{l,m}=\langle
e_{l}^{(p)},e_{m}^{(p)}\rangle$, the upper bound gives :

$\displaystyle\Lambda_{N}(\bar{\theta})$ $\displaystyle\leq$
$\displaystyle\frac{1}{N}\log\mathbb{E}_{e}\Big{[}\exp\Big{(}\frac{\beta
N}{2}\sum_{i,j=1}^{k}\theta_{i}\theta_{j}\psi^{(1)}_{i,j}\psi^{(2)}_{j,i}\Big{)}\Big{]}$

but since the $e_{i}$ are unitary and orthogonal , if we let
$\Psi^{(p)}=(\psi^{(p)}_{i,j})_{1\leq i,j\leq k}$ we have
$\Psi^{(1)}+\Psi^{(2)}=I_{2k}$ and so
$\psi^{(1)}_{i,j}\psi^{(2)}_{j,i}=\psi^{(1)}_{i,j}(\mathds{1}_{i=j}-\bar{\psi}^{(1)}_{i,j})$.
Furthermore the $\Psi^{(1)}$ is an element of a Jacobi ensemble as the
following lemma states :

######  Lemma 17.

The distribution of the matrix $\Psi^{(1)}$ when $N>k$ is given by the
following density for the Lebesgue measure on the set of symmetric/Hermitian
matrices :

$\frac{1}{Z}\det(\Psi^{(1)})^{\beta\frac{L-k+1}{2}-1}\det(I_{k}-\Psi^{(1)})^{\beta\frac{M-k+1}{2}-1}\mathds{1}_{0\leq\Psi^{(1)}\leq
I_{k}}d\Psi^{(1)}$

###### Proof.

Let $U$ be a orthohogonal/unitary $N\times N$ Haar matrix, $U_{1}$ its
$L\times k$ top left block. Then $\Psi^{(1)}$ has the same law as
$U_{1}^{*}U_{1}$. If we denote $\Pi$ the matrix
$diag(\underbrace{1,...,1}_{\text{L times}},0,...0)$ and $\Pi^{\prime}$ the
matrix $diag(\underbrace{1,...,1}_{\text{k times}},0,...0)$, then
$U_{1}^{*}U_{1}=\Pi^{\prime}U^{*}\Pi U\Pi^{\prime}$. Then we can apply [6,
Theorem 2.2] (up to adapt this theorem to the real case). ∎

Therefore, using Laplace’s method, we see that the distribution of
$\Psi^{(1)}$ satisfies a large deviations principle with rate function $I$ :

$I(M)=\begin{cases}-\frac{\beta}{2}\Big{[}\frac{1}{1+\alpha}\ln\det(M)+\frac{\alpha}{1+\alpha}\ln\det(I_{k}-M)\Big{]}-Z\text{
if }0\leq M\leq I_{k},\\\ +\infty\text{ otherwise.}\end{cases}$

where $Z$ is such that $\min I=0$. As a consequence, Varadhan’s lemma implies
that

$\limsup\Lambda_{N}(\bar{\theta})\leq\Lambda(\bar{\theta})$

where :

$\Lambda(\bar{\theta})=\sup_{M}[f(M)-I(M)]$

with
$f(M)=\frac{\beta}{4}\sum_{i,j=1}^{k}\theta_{i}\theta_{j}M_{i,j}(I_{k}-M)_{j,i}$.
We notice by taking $M=\alpha^{\prime}I_{k}$ that

$Z\leq-\frac{k\beta}{2}\Big{(}\alpha\log\alpha^{\prime}+(1-\alpha^{\prime})\log(1-\alpha^{\prime})\Big{)}\,.$

On the other hand, because $\det(M)\leq\prod M_{ii}$ for any positive self-
adjoint matrix $M$,

$I(M)\geq-\frac{\beta}{2}\Big{[}\sum_{i=1}^{k}\\{\alpha^{\prime}\ln(M_{i,i})+(1-\alpha^{\prime})\ln(1-M_{i,i})\\}\Big{]}+Z$

whereas
$f(M)\geq\frac{\beta}{4}\sum_{i=1}^{k}\theta_{i}\theta_{j}M_{i,i}(I_{k}-M)_{i,i}$
since the off-diagonal terms are non-positive (because $M$ is symmetric and
the $\theta_{i}$’s non-negative). We deduce (with $M_{i,i}=a_{i}$) that

$\Lambda(\bar{\theta})\leq\frac{\beta}{2}\sup_{(a_{i})_{i=1}^{k}\in]0,1[^{k}}\sum_{i=1}^{k}\Big{(}\theta_{i}^{2}a_{i}(1-a_{i})+\alpha^{\prime}\ln\frac{a_{i}}{\alpha^{\prime}}+(1-\alpha^{\prime})\ln\frac{1-a_{i}}{1-\alpha^{\prime}}\Big{)}=\sum_{i=1}^{k}\Lambda(\theta_{i})$

To obtain the lower bound on $\liminf_{N}\Lambda_{N}(\bar{\theta})$ as in
[13], it is enough to find a sequence of events $V_{N}^{\kappa}$ independent
of $\Psi^{(1)}$ such that on these events $|e_{i}(l)|\leq CN^{-1/4-\kappa}$
for some $\kappa>0$ and all $i$ and $l$ since then we will be in the regime
where the sharp sub-Gaussian bound is also a lower bound. Note here that
$\Psi^{(2)}$ is determined by $\Psi^{(1)}$, so we only condition on
$\Psi^{(1)}$. To do that let us denote $U$ the $k\times L$ matrix with column
vectors $(e^{(1)}_{i},1\leq i\leq k)$. Then

$U=(\Psi^{(1)})^{1/2}V$

and conditionally to $\Psi^{(1)}$, $V=(v_{1},\ldots,v_{k})$ follows the
uniform law on the set of $k$ orthonormal vectors on the sphere
$\mathbb{S}_{L}$. We can then let
$V_{N}^{\kappa}=\\{\max_{i}\max_{l}|v_{i}(l)|\leq N^{-1/4-\kappa}\\}$. On this
set, $\max_{i}\max_{l}|e_{i}(l)|\leq CN^{-1/4-\kappa}$ so that

$\displaystyle\Lambda_{N}(\bar{\theta})$ $\displaystyle\geq$
$\displaystyle\mathbb{E}_{e}\Big{[}1_{e\in
V_{N}^{\kappa}}\exp\Big{(}\frac{\beta N}{4}(1+o(1))\sum_{l=1,...,L\atop
m=L+1,...,N}\sum_{i,j=1}^{k}\theta_{i}\theta_{j}\Re(e_{i}(l)\bar{e}_{i}(m)\bar{e}_{j}(l)e_{j}(m))\Big{)}\Big{]}$
$\displaystyle=$ $\displaystyle\mathbb{E}_{e}\Big{[}1_{e\in
V_{N}^{\kappa}}\exp\Big{(}\frac{\beta
N}{2}\sum_{i,j=1}^{k}\theta_{i}\theta_{j}\psi^{(1)}_{i,j}\psi^{(2)}_{j,i}\Big{)}\Big{]}$

where we expended the Laplace transform of the entries close to the origin. We
finally notice that $V_{N}^{\kappa}$ is independent of $\Psi^{(1)}$ and with
probability going to one. We can therefore apply the large deviations
principle to deduce that

$\limsup\Lambda_{N}(\bar{\theta})\geq\sup_{M}[f(M)-I(M)]\,.$

We finally conclude by taking $M$ diagonal that the above right hand side is
bounded below by $\sum\Lambda(\theta_{i})$, which completes the proof of (18).
To deduce the large deviations principle for the $k$ largest eigenvalues of
Wishart matrices, we first obtain a large deviations upper bound by tilting
the measure by the $k$-dimensional spherical integral. Because it factorizes
as well as $\Lambda(\bar{\theta})$ the upper bound has a rate function given
by the sum of the rate functions for each outliers. To obtain the large
deviations upper bound, we tilt again the measure by $\exp(\frac{\beta
N}{2}\sum_{i=1}^{k}\theta_{i}\langle e_{i},{\bf Y}_{N}e_{i}\rangle)$ with
$e_{i}\in V_{N}^{\epsilon}$. Under this tilted measure, we have the following
expectations :

$\mathbb{E}^{(e,\theta)}[{\bf
Y}_{N}]=\sum_{i=1}^{k}\theta_{i}\Big{(}e_{i}^{(1)}(e_{i}^{(2)})^{*}+e_{i}^{(2)}(e_{i}^{(1)})^{*}\Big{)}$

( where we identify $\mathbb{C}^{L}$ and $\mathbb{C}^{M}$ respectively with
$\mathbb{C}^{L}\times\\{0\\}^{M}$ and $\\{0\\}^{L}\times\mathbb{C}^{M}$). We
can then write

${\bf Y}_{N}=\tilde{{\bf
Y}}_{N}+\sum_{i=1}^{k}\theta_{i}\Big{(}e_{i}^{(1)}(e_{i}^{(2)})^{*}+e_{i}^{(2)}(e_{i}^{(1)})^{*}\Big{)}+o(1)$

where $\tilde{{\bf Y}}_{N}$ has the same form as ${\bf Y}_{N}$ under the
original measure. Then to identify the eigenvalues of ${\bf Y}_{N}$ outside
the bulk of the limit measure we need to solve the following equation

$\det\Big{(}I_{N}+(\tilde{Y}_{N}-z)^{-1}\sum_{i=1}^{k}\theta_{i}\big{(}e_{i}^{(1)}(e_{i}^{(2)})^{*}+e_{i}^{(2)}(e_{i}^{(1)})^{*}\big{)}\Big{)}=0$

Note that the above arguments also show that in
$\mathbb{P}^{\bar{\theta}}$-probability $\Psi^{(1)}$ converges towards the
diagonal matrix with entries $(x_{\theta_{i},\alpha})_{1\leq i\leq k}$. We
also have local laws for $(z-\tilde{Y}_{N})^{-1}$ under
$\mathbb{P}^{\bar{\theta}}$. Therefore, if we denote
$\tilde{\lambda_{+}}=\sqrt{(1+\alpha)^{-1}\lambda_{+}}$ (which is the
rightmost point of the support of the limit measure of ${\bf Y}_{N}$), the
left hand side converges uniformly on any band
$\\{z\in\mathbb{C}:\tilde{\lambda}_{+}+\epsilon\leq\Im z\leq A,|\Re z|\leq
1\\}$ toward :

$g:z\mapsto\prod_{i=1}^{k}\Big{(}1-\theta^{2}z^{2}x_{\theta_{i},\alpha}(1-x_{\theta_{i},\alpha})(1+\alpha)^{2}G_{MP(\alpha)}((1+\alpha)z^{2})G_{MP(1/\alpha)}((1+\alpha)z^{2})\Big{)}$

where $MP(\alpha)$ is the Marchenko-Pastur distribution of parameter $\alpha$.
Using the fact the these functions are holomorphic, we have the $k$ largest
eigenvalue converges toward $z_{\theta_{1},\alpha}\geq z_{\theta_{k},\alpha}$
where $z_{\theta,\alpha}$ is defined as the unique solution of

$1-\theta^{2}x_{\theta,\alpha}(1-x_{\theta,\alpha})(1+\alpha)^{2}z^{2}G_{MP(\alpha)}((1+\alpha)z^{2})G_{MP(1/\alpha)}((1+\alpha)z^{2})=0$

on $]\tilde{\lambda}_{+},+\infty[$ (see [13] for details).

###  5.3.  Universality of the large deviations for the $k$ largest
eigenvalues of Hermitian matrices with variance profiles and sharp sub-
Gaussian entries

We consider in this section the setting of Theorem 6, which generalizes the
previous subsection. We will proceed as in [16] and we will first deal with
the piecewise constant case with the supplementary technical assumption that
the variance profile is non-negative.

The main point is to prove the following estimate for the annealed spherical
integral.

######  Lemma 18.

Let

$\Lambda_{N}^{\sigma}(\bar{\theta})=\frac{1}{N}\log\mathbb{E}_{{\bf
X},e}[\exp(N\sum_{i=1}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}^{\sigma}e_{i}\rangle)]$

Then, let $\sigma$ be piecewise constant and under the assumptions of Theorem
6, for all $\theta_{i}\in\mathbb{R}^{+}$

$\lim_{N{\rightarrow}\infty}\Lambda^{\sigma}_{N}(\bar{\theta})=\sum_{i=1}^{k}\Lambda^{\sigma}(\theta_{i})$

with, if $R_{ij}:=\sigma_{ij}^{2}$,

$\Lambda^{\sigma}(\theta)=\frac{\beta}{2}\sup_{\psi\in
S}\Big{[}\frac{\theta^{2}}{2}\langle\psi_{j},R\psi_{j}\rangle+\sum_{i=1}^{p}\alpha_{i}\log\frac{\psi(j)}{\alpha_{i}}\Big{]}$

Indeed, let us define for $e\in\mathbb{S}^{\beta N-1}$ and $j\in[1,p]$,
$e^{(p)}$ the vector of $\mathbb{C}^{\alpha_{i}(N)}$ whose coordinates are the
coordinates of $e$ whose indices lie in $I_{N}^{i}$. We then define for
$j=1,...,p$ the random matrix $\Psi^{(j)}=(\langle
e^{(j)}_{i},e^{(j)}_{j}\rangle)$. Following the same computations as before
and using the sharp sub-Gaussian character of the entries, we have :

$\Lambda^{\sigma}_{N}(\bar{\theta})\leq\mathbb{E}_{e}\Big{[}\exp\Big{(}\frac{\beta
N}{4}\sum_{l,m=1}^{k}\sum_{i,j=1}^{p}\theta_{l}\theta_{m}\Psi^{(i)}_{l,m}\bar{\Psi}^{(j)}_{l,m}\sigma_{i,j}^{2}\Big{)}\Big{]}$

Notice that the $\Psi^{(j)}$ are Gram matrices (hence self-adjoint and
positive) and that their sum is $I_{k}$. There again we will use a slightly
improved version of the Lemma 17 to determine the distribution of the
$\Psi^{(j)}$ :

######  Lemma 19.

The joint distribution of the matrices $\Psi^{(1)},...,\Psi^{(p-1)}$ when
$\alpha_{1}(N),...,\alpha_{p}(N)>k$ is given by the following density for the
Lebesgue measure on the set of symmetric/Hermitian matrices :

$\frac{1}{Z}\prod_{i=1}^{p-1}\Big{(}\mathds{1}_{0\leq\Psi^{(i)}}\det(\Psi^{(i)})^{\beta\frac{\alpha_{i}(N)-k+1}{2}-1}\Big{)}\det(I_{k}-\sum_{i=1}^{p-1}\Psi^{(i)})^{\beta\frac{\alpha_{p}(N)-k+1}{2}-1}\mathds{1}_{\sum_{i=1}^{p-1}\Psi^{(i)}\leq
I_{k}}\prod_{i=1}^{p-1}d\Psi^{(i)}$

###### Proof.

Here we need an improved version of [6, Theorem 2.2] which states as follows.
Let $U$ be a $N\times N$ Haar-distributed orthogonal or unitary matrix,
$n_{0}=0<n_{1}<...<n_{p}=N$ a $p$-uplet of integers and for $i\in[1,p]$,
$\tilde{\pi}_{i}$ the orthogonal projection on the vector span of the columns
of $U$ with indices between $n_{i-1}+1$ and $n_{i}$. Let $\pi$ be a constant
projection of rank $k$. Then, if we identify $\pi S_{N}(\mathbb{R})\pi$
(respectively $\pi H_{N}(\mathbb{C})\pi$) to $S_{k}(\mathbb{R})$ (respectively
$H_{k}(\mathbb{C})$), the joint distribution of
$(M_{1},...,M_{p-1})=\pi\tilde{\pi}_{1}\pi,...,\pi\tilde{\pi}_{p-1}\pi$ has
the following density on $S_{N}(\mathbb{R})^{p-1}$ ( resp.
$H_{N}(\mathbb{C})^{p-1}$) :

$\frac{1}{Z}\prod_{i=1}^{p-1}\Big{(}\mathds{1}_{0\leq
M_{i}}\det(M_{i})^{\beta\frac{m_{i}-k+1}{2}-1}\Big{)}\det(I_{k}-\sum_{i=1}^{p-1}M_{i})^{\beta\frac{m_{p}-k+1}{2}-1}\mathds{1}_{\sum_{i=1}^{p-1}M_{i}\leq
I_{k}}\prod_{i=1}^{p-1}dM_{i}$

where $m_{i}=n_{i}-n_{i-1}$.

The proof of this result is the same as the proof of [6, Theorem 2.2]. The
difference is that one need to prove that $(\pi\tilde{\pi}_{i}\pi)_{1\leq
p-1}$ has the same law as $(\Sigma^{-1/2}X_{i}\Sigma^{-1/2})$ where the
$X_{i}$ are independent Gaussian Wishart of parameters $(k,m_{i})$ and
$\Sigma=X_{1}+...+X_{p}$. Once we have this result, we take a Haar-distributed
unitary matrix $U$ and we denote $U_{i}$ the $\alpha_{i}(N)\times k$ matrix
extracted from $U$ by taking its $k$ first columns and its rows of indices in
$I_{N}^{i}$. We denote $\Pi^{\prime}={\rm diag}(\underbrace{1,...,1}_{\text{ k
times}},0,..,0)$ and $\Pi_{i}$ the diagonal matrix with entries equal to $1$
for indices in $I_{N}^{i}$ and $0$ elsewhere. Then, since $(\Psi^{(i)})_{1\leq
p-1}$ has the same law as $(\Pi^{\prime}U^{*}\Pi_{i}U\Pi^{\prime})_{i\leq
p-1}$, we can use the previous theorem. ∎

We deduce from this explicit distribution of the $p-1$-uplet
$(\Psi^{(i)})_{1\leq i\leq p}$ that it follows a large deviations principle
with rate function :

$I((M_{i})_{1\leq i\leq
p-1})=\begin{cases}-\frac{\beta}{2}\Big{[}\sum_{i=1}^{p}\alpha_{i}\ln\det(M_{i})\Big{]}-C\text{
if }\forall i\in[1,p],0\leq M_{i}\leq I_{k},\text{ and
}\sum_{i=1}^{p}M_{i}=I_{k}\\\ +\infty\text{ otherwise.}\end{cases}$

Then we have using Varadhan’s lemma :

$\limsup_{N}\Lambda^{\sigma}_{N}(\bar{\theta})\leq\Lambda^{\sigma}(\bar{\theta})$

where :

$\Lambda^{\sigma}(\bar{\theta})=\sup_{(M_{i})_{1\leq i\leq
p}}[f((M_{i}))-I((M_{i}))]$

with $f((M_{i}))=\frac{\beta}{4}\sum_{i,j=1}^{k}\theta_{i}\theta_{j}\langle
M^{i,j},RM^{i,j}\rangle$ where $R$ is the $p\times p$ matrix
$(\sigma_{i,j}^{2})$ and $M^{i,j}$ is the vector
$(M_{1}(i,j),...,M_{p}(i,j))$. But, as before, if $d(M)$ represents the
diagonal matrix with entries $(M_{ii})_{1\leq i\leq k}$ we have that

$I((M_{i}))\geq I((d(M_{i})))\mbox{ and for }i\neq j\langle
M^{i,j},RM^{i,j}\rangle\leq 0$

where the last inequality is due to Assumption 5 which implies
$\sum_{i=1}^{p}M_{i}=I$ and therefore that for $i\neq j$
$\sum_{l=1}^{p}M^{i,j}=0$. Therefore we can again restrict the $\sup$ to
diagonal matrices and it then decouples into

$\Lambda^{\sigma}(\bar{\theta})=\frac{\beta}{2}\sup_{(\psi_{i})_{1\leq i\leq
k}\in
S^{k}}\Big{[}\sum_{j=1}^{k}\frac{\theta_{i}^{2}}{2}\langle\psi_{j},R\psi_{j}\rangle+\sum_{j=1}^{k}\sum_{i=1}^{p}\alpha_{i}\log\frac{\psi_{i}(j)}{\alpha_{i}}\Big{]}=\sum_{j=1}^{k}\Lambda^{\sigma}(\theta_{j})$

where $S:=\\{\psi\in(\mathbb{R}^{+})^{p}:\psi_{1}+...+\psi_{p}=1\\}$. In
particular, since the function $\psi\mapsto\langle\psi,R\psi\rangle$ is
concave on $S$ thanks to Assumption 5, the function optimized is strictly
concave and thus is maximum at a unique $\psi$. Furthermore $\psi_{j}$ only
depends on $\theta_{j}$ so that we will denote $\psi_{j}=\psi_{\theta_{j}}$.
Using again the strict concavity and the implicit function theorem, we have
that the function $\theta\mapsto\psi_{\theta}$ is analytic in $\theta$.
Furthermore, if we tilt our measure by $\mathbb{E}_{{\bf
X}}[\exp(N\sum\theta_{i}\langle e_{i},{\bf X}_{N}^{\sigma}e_{i}\rangle)]$, the
$\Psi^{(i)}$’s follow a large deviations principle and converges respectively
toward ${\rm diag}(\psi_{\theta_{1}}(i),...,\psi_{\theta_{k}}(i))$. For the
lower bound we restrict the integral as in the preceding subsection to
delocalized vectors with fixed $\Psi$ and conclude similarly.

To prove the large deviations principle, we first observe that the large
deviations upper bound is direct after a tilt by spherical integrals and
decoupling of the annealed spherical integrals. For the large deviations lower
bound, we tilt by $\exp(N\sum_{i=1}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}^{\sigma}e_{i}\rangle)$. Under this tilted measure
$\mathbb{P}^{e,\bar{\theta}}$, we have the following expectation
$\mathbb{E}^{e,\bar{\theta}}[{\bf
X}_{N}^{\sigma}]=\sum_{i=1}^{k}\theta_{i}\sum_{l,m=1}^{p}\sigma_{l,m}^{2}e_{i}^{(l)}(e_{i}^{(m)})^{*}$.
Using the BBP transition phenomenon, the local law for ${\bf X}_{N}$ as in
[16, Lemma 5.6] and the fact that the $\Psi^{(k)}$ converges in
$\mathbb{P}^{\bar{\theta}}$ \- probability, we have that the eigenvalues
outside the bulk are asymptotically solution of the following equation in $z$
:

$\prod_{i=1}^{k}\det(I_{p}-\theta_{i}RD(\theta_{i},z))=0$

where $D(\theta,z)$ is defined as in [16, Section 5]. To conclude, it suffices
to prove that for any $z_{1}>...>z_{k}>r_{\sigma}$, there exists
$\theta_{1}\geq...\geq\theta_{k}\geq 0$ such that $z_{i}$ is the unique
solution of $\det(I_{p}-\theta RD(\theta,z))=0$ on $[z_{p},+\infty[$. We
already know thanks again to the proof of the large deviations lower bound in
[16] that there is for every $z$, a $\theta$ such that $z$ is the largest
solution. Let us prove that with Assumption 5, this solution is unique on
$]r_{\sigma},+\infty[$. First, one can notice that this assumption implies
that the quadratic form whose matrix is $R$ has signature $(1,p-1)$ and so it
is also true for the quadratic form whose matrix is
$\sqrt{D(\theta,z)}R\sqrt{D(\theta,z)}$. Therefore, if we denote
$\rho(\theta,z)$ the largest eigenvalue of
$\sqrt{D(\theta,z)}R\sqrt{D(\theta,z)}$, the equation
$\det(I_{p}-\theta_{i}RD(\theta_{i},z))=0$ is equivalent for $\theta>0$ and
$z>r_{\sigma}$ to $\theta\rho(\theta,z)=1$. Since $z\mapsto\rho(\theta,z)$ is
strictly decreasing, the result is then proved.

For the continuous case, we can as in [16, Section 6] approximate our
continuous variance profiles by piecewise constant ones. This approximation
step is in fact easier than in the more general case of [16] since if $\sigma$
satisfy Assumption 5 then we can approximate ${\bf X}_{N}^{\sigma}$ by the
${\bf X}_{N}^{(p)}$ defined as follows :

${\bf X}_{N}^{(p)}=\sigma^{(p)}_{N}(i,j)\frac{X_{i,j}}{\sqrt{N}}$

where
$\sigma^{(p)}_{N}(i,j)=\sum_{k,l=1}^{p}\sigma^{(p)}_{k,l}\mathds{1}_{I^{k}_{N}\times
I^{l}_{N}}(i,j)$ if $I_{N}^{1}=[0,N/p]$ and $I_{N}^{i}=]N(i-1)/p,Ni/p]$ for
$i=2...p$ and

$\sigma^{(p)}_{i,j}=\sqrt{p^{2}\int_{(i-1)/p}^{i/p}\int_{(j-1)/p}^{j/p}\sigma^{2}(x,y)dxdy}\,.$

Since $\sigma$ satisfy Assumption 5, it is easy to check that $\sigma^{(p)}$
also satisfies Assumption 5 for all $p$ and therefore if we denote
$\lambda^{(p),1}_{N},...,\lambda^{(p),k}_{N}$ its $k$ largest eigenvalues,
they satisfy a large deviations principle with rate function
$I^{(p)}(x_{1},...,x_{p})=\sum I^{(p)}(x_{i})$. If we denote
$\lambda^{1}_{N},...,\lambda^{k}_{N}$ the $k$ largest eigenvalues of ${\bf
X}_{N}$, we have for all $i=1,...,k$,
$|\lambda^{i}_{N}-\lambda^{(p),i}_{N}|\leq||{\bf X}_{N}-{\bf X}^{(p)}_{N}||$.
Using [16, Lemma 6.6], we have that $||{\bf X}_{N}-{\bf X}^{(p)}_{N}||$ can be
neglected at exponential scale once $p$ is large enough. And using again [16,
Lemma 6.4 and Lemma 6.5], we have that the rate function converges toward the
sum of the rate functions for one eigenvalue. Therefore,
$\lambda^{1}_{N},...,\lambda^{k}_{N}$ satisfy a large deviations principle and
the rate function is the sum of the rate functions for one eigenvalue.

######  Remark 20.

Contrary to the Wigner case where we can see that asymptotically the positive
and negative eigenvalues deviate independently from one another, this is not
the case for matrices with variance profiles. An example is the linearization
of a Wishart matrix where the negative eigenvalues are always exactly the
opposite of the positive ones.

###  5.4.  Large deviations for the $k$ largest eigenvalues for the Gaussian
ensembles with a $k$-dimensional perturbation

We next prove proposition 7. We first observe that the result is well known
when $\bar{\theta}=0$, see e.g. Theorem 3. We next remark that the joint law
of the eigenvalues of ${\bf X}_{N}^{\theta}$ is given by

$d\mathbb{P}_{N}^{\theta}(\lambda)=\frac{1}{Z_{N}}\Delta(\lambda)^{\beta}\int\exp\\{-\frac{\beta}{4}N{\rm
Tr}|UD(\lambda)U^{*}-\sum_{i=1}^{k}\theta_{i}e_{i}e_{i}^{*}|^{2}\\}dU\prod_{1\leq
i\leq N}d\lambda_{i}$

where $U$ follows the Haar measure on the unitary group (resp. the orthogonal
group) when $\beta=2$ (resp, $\beta=1$).
$\Delta(\lambda)=\prod_{i<j}|x_{i}-x_{j}|$ is the Vandermonde determinant and
$D(\lambda)$ is a diagonal matrix with entries given by
$\lambda=(\lambda_{1},\ldots,\lambda_{N})$. Expanding the integral under the
unitary (or orthogonal) group, we find that

$d\mathbb{P}_{N}^{\theta}(\lambda)=\frac{1}{\tilde{Z}_{N}}\mathbb{E}_{e}[e^{\frac{\beta}{2}N\sum_{i=1}^{k}\theta_{i}\langle
e_{i},D(\lambda)e\rangle}]d\mathbb{P}_{N}^{0}(\lambda)\,,$

where $(e_{1},\ldots,e_{k})$ follows the uniform law on $k$ orthonormal
vectors in dimension $N$. Hence the density is exactly given by the spherical
integral. Using that Assumption 2 holds under $\mathbb{P}_{N}^{0}$ (see e.g
[11]), we see that the empirical measure of $\lambda$ is close to the semi-
circle law with overwhelming probability. Assume that
$\theta_{1}\geq\theta_{2}\cdots\geq\theta_{p}\geq
0\geq\theta_{p+1}\cdots\geq\theta_{k}$. Then, on the set where the extreme
eigenvalues $\lambda_{N}^{N}\geq\cdots\lambda_{N-p}^{N}$ and
$\lambda^{N}_{1}\leq\cdots\leq\lambda^{N}_{k-p+1}$ are close to $x_{1}\geq
x_{2}\geq\cdots\geq x_{p}\geq 2\geq-2\geq x_{-k+p}\geq\cdots\geq x_{-1}$,
Theorem 16 and Varadhan’s Lemma give the result.

###  5.5.  Large deviations for $k$ extreme eigenvalues for Gaussian Wishart
matrices with a $k$-dimensional perturbation

The proof of Proposition 8 is similar to the previous one. Again the proof is
based on the explicit joint law of
$\lambda_{1}^{N,\gamma}\geq\lambda_{2}^{N,\gamma}\geq\cdots\lambda_{M}^{N,\gamma}$
given by the law on $(\mathbb{R}^{+})^{M}$

$d\mathbb{P}_{M,N}^{\bar{\gamma}}(d\lambda)=\frac{1}{Z_{N}}\Delta(\lambda)^{\beta}\int
e^{-\frac{\beta}{2}N{\rm Tr}(UD(\lambda)U^{*}\Sigma^{-1})}dU\prod_{1\leq i\leq
M}\lambda_{i}^{\frac{\beta}{2}(N-M+1)}d\lambda_{i}$

Noticing that

$\Sigma^{-1}=I+\sum_{i=1}^{k}\frac{\gamma_{i}}{1-\gamma_{i}}e_{i}e_{i}^{*}$

we recognize again that the density with respect to the case $\gamma=0$ is
given by a spherical integral. The result follows as for the Wigner case.

##  6\. Appendix

In this Appendix we investigate the continuity property of spherical
integrals. First we need to prove the continuity of the deterministic limit
itself :

######  Theorem 21.

Let $d$ be a distance compatible with the weak topology on the set
$\mathcal{P}(\mathbb{R})$ and $||.||$ any norm on $\mathbb{R}^{k+\ell}$, and
for $M>0$, $\mathcal{K}_{M}$ the subset of
$E=\mathbb{R}^{k+\ell}\times(\mathbb{R}^{+})^{k}\times(\mathbb{R}^{-})^{\ell}\times\mathcal{P}(\mathbb{R})$
defined by

$\displaystyle\mathcal{K}_{M}:=\\{(\bar{\lambda},\bar{\theta},\mu)\in
E|M\geq\theta_{1}\geq...\geq\theta_{k}\geq
0\geq\theta_{-\ell}\geq...\geq\theta{-1}\geq-M$ $\displaystyle
M\geq\lambda_{1}\geq...\geq\lambda_{k}\geq r_{\mu}\geq
l_{\mu}\geq\lambda_{-\ell}\geq...\theta_{-1}\geq-M\\}$

where $r_{\mu}$ and $l_{\mu}$ are respectively the rightmost and the leftmost
point of the support of $\mu$. We endow $\mathcal{K}_{M}$ with the distance
$D$ given by
$D((\bar{\lambda},\bar{\theta},\mu),(\bar{\lambda}^{\prime},\bar{\theta}^{\prime},\mu^{\prime}))=d(\mu,\mu^{\prime})+||\bar{\lambda}-\bar{\lambda}^{\prime}||+||\bar{\theta}-\bar{\theta}^{\prime}||$.
Then $\mathcal{K}_{M}$ is a compact set and the function $J$

$J(\mu,\bar{\theta},\bar{\lambda})=\sum_{i=-\ell,\neq
0}^{k}J(\mu,\theta_{i},\lambda_{i})$

extends continuously on $\mathcal{K}_{M}$.

###### Proof.

It is clear that we only need to prove the continuity of
$(\theta,\lambda,\mu)\mapsto J(\mu,\theta,\lambda)$ where either $\theta\geq
0$ and $\lambda\geq r_{\mu}$ or $\theta\leq 0$ and $\lambda\leq l_{\mu}$. We
assume without loss of generality that we are in the first case. Furthermore
since $J(\mu,\theta,\lambda)=J(\theta*\mu,\theta\lambda,1)$ we only need to
prove the continuity for the first two arguments with the third being fixed
equal to $1$. Let us take a sequence $(\mu_{n},\lambda_{n})$ such that
$\forall n\in\mathbb{N},l_{\mu}\geq-M,r_{\mu_{n}}\leq\lambda_{n}$ and
$\lim\lambda_{n}=\lambda$ and $\lim_{n}\mu_{n}=\mu$. First, since
$|J(\mu_{n},\lambda_{n},1)-J(\mu_{n},\lambda+\epsilon,1)|\leq|\lambda_{n}-\lambda|+\epsilon$
for $n$ large enough so that $\lambda+\epsilon\geq r_{\mu_{n}}$, and
$|J(\mu,\lambda,1)-J(\mu_{n},\lambda+\epsilon,1)|\leq\epsilon$ we can restrict
ourselves to proving $\lim
J(\mu_{n},\lambda+\epsilon,1)=J(\mu,\lambda+\epsilon,1)$. But, when we
differentiate $J(\mu,\lambda,1)$ on the variable $\lambda$, we find

$\frac{\partial}{\partial\lambda}J(\mu,\lambda,1)=1-\mathds{1}_{[G_{\mu}^{-1}(1),+\infty[}G_{\mu}(\lambda)-\mathds{1}_{]-\infty,G_{\mu}^{-1}(1)[}G_{\mu}^{-1}(1)$

On $[\lambda+\epsilon,+\infty[$, since $r_{\mu_{n}}\leq\lambda+\epsilon$ it is
in fact easy to see that the weak convergence of $\mu_{n}$ imply the uniform
convergence of $\partial/\partial\lambda J(\mu_{n},\lambda,1)$. The we
conclude by choosing $\Lambda>\lambda+\epsilon$ so that $G_{\mu}(\Lambda)\leq
1/2$ so that $v(\mu_{n},1,\Lambda)=\Lambda$ for $n$ large enough and then
using the weak convergence and the fact that $x\mapsto\log(\Lambda-x)$ is
bounded on $[-M,\lambda+\epsilon]$, we have that $J(\mu_{n},1,\Lambda)$
converges toward $J(\mu,1,\Lambda)$. ∎

With this continuity and the compactness of $\mathcal{K}_{M}$, we can prove
the following theorem of uniform continuity, which generalizes [18] :

######  Theorem 22.

Let $k,\ell\in\mathbb{N}$,
$\bar{\theta}\in(\mathbb{R}^{-})^{\ell}\times(\mathbb{R}^{+})^{k}$ and

$J_{N}(X_{N},\bar{\theta})=\frac{1}{N}\log\mathbb{E}\Big{[}\exp\Big{(}\frac{\beta
N}{2}\sum_{i=-\ell,i\neq 0}^{k}\theta_{i}\langle e_{i},{\bf
X}_{N}e_{i}\rangle\Big{)}\Big{]}$

Let us denote $\lambda^{N}_{-1}\leq...\leq\lambda^{N}_{-\ell}$ the smallest
outliers of ${\bf X}_{N}$ and $\lambda^{N}_{1}\geq...\geq\lambda^{N}_{k}$ the
largest outliers. We will denote this $k+\ell$-uplet $\bar{\lambda}_{N}$. Then
for every $M>0$ and $\epsilon>0$, there is $N_{0}\in\mathbb{N}$ so that for
every $N\geq N_{0}$, for any matrix ${\bf X}_{N}$ such that
$(\bar{\lambda}_{N},\bar{\theta},\mu_{{\bf X}_{N}})\in\mathcal{K}_{M}$

$|J_{N}({\bf X}_{N},\bar{\theta})-J(\mu_{{\bf
X}_{N}},\bar{\lambda}_{N},\bar{\theta})|\leq\epsilon$

###### Proof.

We first notice that in the proof of Proposition 16 we approximated
$J_{N}({\bf X}_{N},\bar{\theta})$ by $J_{N}({\bf
X}_{N}^{\varepsilon},\bar{\theta})$ with an error depending only on
$\varepsilon$. Hence we may and shall replace in the above statement ${\bf
X}_{N}$ by ${\bf X}_{N}^{\delta}$ for some small enough
$\delta=\delta(\epsilon)$. ${\bf X}_{N}^{\delta}$ has the same extreme
eigenvalues than ${\bf X}_{N}$ and otherwise eigenvalues
$\lambda_{-\ell}^{N}+j\delta$ with multiplicity $\lfloor
N\hat{\mu}^{N}([\lambda_{-\ell}^{N}+j\delta,\lambda_{-\ell}^{N}+(j+1)\delta])\rfloor$.
Therefore, we see that $J_{N}({\bf X}_{N},\bar{\theta})$ is a function of the
extreme eigenvalues and the empirical measure, hence a function on
$\mathcal{K}_{N}$. By the previous uniform approximation and the continuity of
the limit, we deduce that it is uniformly continuous on $\mathcal{K}_{N}$,
hence the result. ∎

## References

  * [1] Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni, _An introduction to random matrices_ , Cambridge Studies in Advanced Mathematics, vol. 118, Cambridge University Press, Cambridge, 2010. MR 2760897
  * [2] Jinho Baik, Gérard Ben Arous, and Sandrine Péché, _Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices_ , Ann. Probab. 33 (2005), no. 5, 1643–1697. MR 2165575 (2006g:15046)
  * [3] G. Ben Arous, A. Dembo, and A. Guionnet, _Aging of spherical spin glasses_ , Probab. Theory Rel. Fields 120 (2001), 1–67.
  * [4] Giulio Biroli and Alice Guionnet, _Large deviations of spiked random matrices_ , To appear in ECP (2020).
  * [5] Benoît Collins and Piotr Śniady, _New scaling of Itzykson-Zuber integrals_ , Ann. Inst. H. Poincaré Probab. Statist. 43 (2007), no. 2, 139–146. MR 2303115
  * [6] Benoît Collins, _Product of random projections, jacobi ensembles and universality problems arising from free probability_ , Probability Theory and Related Fields 133 (2005), no. 3, 315–344.
  * [7] Robert Coquereaux, Colin McSwiggen, and Jean-Bernard Zuber, _On Horn’s problem and its volume function_ , Comm. Math. Phys. 376 (2020), no. 3, 2409–2439. MR 4104554
  * [8] D. Dean and S. Majumdar, _Large deviations of extreme eigenvalues of random matrices_ , Phys. Rev. Lett. 97 (2006), no. 16, 160201, 4. MR 2274338
  * [9] Vadim Gorin and Greta Panova, _Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory_ , Ann. Probab. 43 (2015), no. 6, 3052–3132. MR 3433577
  * [10] A. Guionnet and M. Maï da, _A Fourier view on the $R$-transform and related asymptotics of spherical integrals_, J. Funct. Anal. 222 (2005), no. 2, 435–490. MR 2132396
  * [11] A. Guionnet and O. Zeitouni, _Concentration of the spectral measure for large matrices_ , Electron. Commun. Prob. 5 (2000), 119–136 (electronic).
  * [12] by same author, _Large deviations asymptotics for spherical integrals_ , J. Funct. Anal. 188 (2002), 461–515.
  * [13] Alice Guionnet and Jonathan Husson, _Large deviations for the largest eigenvalue of Rademacher matrices_ , Ann. Probab. 48 (2020), no. 3, 1436–1465. MR 4112720
  * [14] Harish-Chandra, _Invariant differential operators on a semisimple Lie algebra_ , Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 252–253.
  * [15] Jiaoyang Huang, _Asymptotic expansion of spherical integral_ , J. Theoret. Probab. 32 (2019), no. 2, 1051–1075. MR 3959637
  * [16] Jonathan Husson, _Large deviations for the largest eigenvalue of matrices with variance profiles_ , (2020).
  * [17] C. Itzykson and J. B. Zuber, _The planar approximation. II_ , J. Math. Phys. 21 (1980), 411–421.
  * [18] Mylène Maï da, _Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles_ , Electron. J. Probab. 12 (2007), 1131–1150. MR 2336602
  * [19] M. Maïda, J. Najim, and S. Péché, _Large deviations for weighted empirical mean with outliers_ , Stochastic Process. Appl. 117 (2007), no. 10, 1373–1403. MR 2353032
  * [20] Alessandro Pizzo, David Renfrew, and Alexander Soshnikov, _On finite rank deformations of wigner matrices_ , Annales de l’I.H.P. Probabilités et statistiques 49 (2013), no. 1, 64–94 (en). MR 3060148
  * [21] Marc Potters and Pierre Mergny, _Asymptotic behavior of the multiplicative counterpart of the harish-chandra integral and the s-transform_ , arXiv 2007.09421 (2020).
  * [22] Jean-Bernard Zuber, _Horn’s problem and Harish-Chandra’s integrals. Probability density functions_ , Ann. Inst. Henri Poincaré D 5 (2018), no. 3, 309–338. MR 3835548