---
abstract: |
    Let $X$ be a $n\times p$ matrix and $l_1$ the largest eigenvalue of the covariance matrix $X^{*}X$. The “null case" where $X_{i,j}\sim {\cal N}(0,1)$ is of particular interest for principal component analysis.

    For this model, when $n, p{\rightarrow}\infty$ and $n/p {\rightarrow}\gamma \in \mathbb{R}_+^*$, it was shown in @imj that $l_1$, properly centered and scaled, converges to the Tracy-Widom law.

    We show that with the same centering and scaling, the result is true even when $p/n$ or $n/p{\rightarrow}\infty$, therefore extending the previous result to $\gamma \in \overline{\mathbb{R}}_+$. The derivation uses ideas and techniques quite similar to the ones presented in @imj. Following @sosh, we also show that the same is true for the joint distribution of the $k$ largest eigenvalues, where $k$ is a fixed integer.

    Numerical experiments illustrate the fact that the Tracy-Widom approximation is reasonable even when one of the dimension is small.
author:
- |
    Noureddine El Karoui[^1]\
    *Department of Statistics,*\
    *Stanford University*
bibliography:
- 'research.bib'
title: 'On the largest eigenvalue of Wishart matrices with identity covariance when $n$, $p$ and $p/n {\rightarrow}\infty$'
---

Introduction
============

Large scale principal component analysis (PCA) - concerning an $n\times p$ matrix $X$ where $n$ and $p$ are both large - is nowadays a widely used tools in many fields, such as image analysis, signal processing, functional data analysis and quantitative finance. Several examples come to mind, including Eigenfaces, subspace filtering, or @lalouxetal where PCA (as well as some random matrix theory) is used to try to improve on the naive solution to Markovitz’s portfolio optimization problem.

Important progress has been made recently in our understanding of the statistical properties of PCA in such settings. Emblematic of this is work of @imj, which explains the properties of the square of the largest singular value of a random matrix $X$ under the “null model" where its entries are iid ${\cal N}(0,1)$. Specifically, if we denote the sample eigenvalues of $X'X$ by $l_1\geq \ldots \geq l_p$, call $$\begin{aligned}
n_1&=\max{(n,p)}-1\;, \; \; \; \; p_1=\min{(n,p)} \;,\\
\mu_{np}&=(\sqrt{n_1}+\sqrt{p_1})^2 \; ,\\
\sigma_{np}&=(\sqrt{n_1}+\sqrt{p_1})\left(\frac{1}{\sqrt{n_1}}+\frac{1}{\sqrt{p_1}}\right)^{1/3} \; ,\end{aligned}$$ and $W_1$ the Tracy-Widom distribution (see **A0**), it was shown in @imj that

\[ThJ\] If $n,p {\rightarrow}\infty$ and $n/p{\rightarrow}\gamma \in (0,\infty)$, $$\frac{l_1-\mu_{np}}{\sigma_{np}} \overset{\cal L}{\rightarrow}W_1 \;.$$

Building on @imj and using properties of determinantal point processes, @sosh showed that the same result holds for the $k$ largest eigenvalues, where $k$ is a fixed integer: their joint distribution converges to their Tracy-Widom counterpart.

This is a very interesting development because the classical theory (e.g @anderson) was developed under the assumption that $p$ was fixed and $n$ grew to $\infty$, whereas in modern day applications both $p$ and $n$ are large. However, Johnstone’s assumption $n/p{\rightarrow}\gamma$ imposes a limit on the validity of his result which one would like to remove. In an actual data analysis, with given $p$ and $n$, $n={\mathrm{o}}(p)$ and $n\asymp p$ could be equally plausible. Furthermore, a specific $X$ of size $n\times p$ could arise in many triangular arrays settings, where we have $X_j$ of size $n_j\times p_j$, and the limitation $n_j/p_j{\rightarrow}\gamma$ finite might only hold in some triangular situations and not in others.

Accordingly in this paper we weaken the assumption that $n/p {\rightarrow}\gamma$ finite and show that

\[ThJextended\] If $n,p {\rightarrow}\infty$ and $n/p{\rightarrow}\infty$, $$\frac{l_1-\mu_{np}}{\sigma_{np}} \overset{\cal L}{\rightarrow}W_1 \;.$$

Moreover, with the same centering and scaling, the joint distribution of the $k$ largest eigenvalues converges in law to its Tracy-Widom counterpart.

Dually, the same result holds if $n/p{\rightarrow}0$.

Let us note that the remark we made about centering and scaling sequences after Theorem \[ThJ\] is still valid in this context.

There is clearly a mathematical motivation for dealing with this problem: the result completes the picture about the properties of $l_1$ with large $p$ and $n$ and, in a sense, closes Theorem \[ThJ\]. But is it interesting from a statistical standpoint?

The situation $p\gg n$ is indeed a fairly common one in modern statistics. Microarray data are a prototypical example: currently they usually have $p$ of the order of a few thousands and $n$ of the order of a few tens. One encounters $p\gg n$ or $n\gg p$ in many other instances: data collection mechanisms are now effective enough so as to, for example, collect and retain thousands of piece of information for millions of customers (transactional data), or millions of pieces of information for thousands of stocks (tick-by-tick data in Finance). Analyzing these very high dimensional datasets raises new challenges and is at the center of recent statistical work, both applied and theoretical.

Microarray analysis in particular is a very active field, and has contributed a flurry of activity in non classical situations (very high dimensional data), raising theoretical questions and sometimes revisiting classical techniques or results. As illustrated for instance in @wrr, PCA or PCA-related methods are used for various tasks in the microarray context, from traditional dimensionality reduction procedures to gene grouping. Having a good understanding of the behavior of the singular values of gaussian “white noise" matrices could provide valuable insights for these applications. Recent work of @bl03 about the properties of naive Bayes and Fisher’s linear discriminant function when $p\gg n$ illustrates the impetus these dimensionality assumptions are also gaining in theoretical studies. Our work is part of the larger effort to investigate the properties of high dimensional data structures. Here it is done in a simple, “null" situation.

We now present a few numerical experiments we realized to assess how big (or small) $n$ or $p$ should be for Theorems \[ThJ\] and \[ThJextended\] to be practically useful.

Numerical experiments
---------------------

@imj showed empirically that in that situation the Tracy-Widom approximation was reasonably satisfying, even for small matrices. Similarly, to try to assess its accuracy in our setup, we ran the following experiments in `Matlab`: we picked $n$ and $p$ and generated $10,000$ $n\times p$ matrices $X$ with entries iid ${\cal
N}(0,1)$. Then we used standard routines (`normest` in `Matlab`) to compute their spectral norms and squared them to obtain a dataset of $l_1$-s.

Following [@imj2], we adjust centering and scaling to $$\begin{aligned}
\tilde{\mu}_{np}&=\sqrt{n-1/2}+\sqrt{p-1/2}\; ,\\
\tilde{\sigma}_{np}&=(\sqrt{n-1/2}+\sqrt{p-1/2})\left(\frac{1}{\sqrt{n-1/2}}+\frac{1}{\sqrt{p-1/2}}\right)^{1/3}
\;.\end{aligned}$$ This leads to a very significant improvement in the quality of the Tracy-Widom approximation for our simulations. Simple manipulations (explained in section 2.2) show that we have some freedom in choosing the centering and scaling: if we replace $n$ by $n+a$ and $p$ by $p+b$ (where $a$ and $b$ are fixed real numbers) in the definitions of $\mu_{np}$ and $\sigma_{np}$, Theorem \[ThJ\] and Theorem \[ThJextended\] still hold. The particular choice used here is motivated by a careful theoretical analysis of the entries of $K_N$ mentioned in section 2.2.

Table 1 summarizes the “quantile" properties of the empirical distributions we obtained and compare them to the Tracy-Widom reference. We used the same reference points as @imj.

   TW Quantiles   TW    10$\times$1000   10$\times$ 4000   10$\times$ 10000   100$\times$4000   30$\times$5000
  -------------- ----- ---------------- ----------------- ------------------ ----------------- ----------------
       -3.9       .01       0.009             0.010             0.015              0.012            0.013
      -3.18       .05       0.047             0.050             0.060              0.053            0.055
      -2.78       .10       0.102             0.107             0.112              0.103            0.105
      -1.91       .30       0.303             0.308             0.316              0.304            0.303
      -1.27       .50       0.506             0.506             0.522              0.508            0.503
      -0.59       .70       0.705             0.704             0.723              0.706            0.702
       0.45       0.9       0.904             0.904             0.913              0.901            0.904
       0.98       .95       0.953             0.951             0.958              0.951            0.953
       2.02       .99       0.992             0.990             0.992              0.991            0.991

  : **Quality of the Tracy-Widom Approximation for some large matrices:** the leftmost columns displays certain quantiles of the Tracy-Widom distribution. The second column gives the corresponding value of its cdf. Other columns give the value of the empirical distribution functions obtained from simulations at these quantiles. $\tilde{\mu}_{np}$ and $\tilde{\sigma}_{np}$ are the centering and scaling sequences.

\

   TW Quantiles   TW    50$\times$5000   50$\times$20000   50$\times$50000   5$\times$200   5$\times$2000   5$\times$20000
  -------------- ----- ---------------- ----------------- ----------------- -------------- --------------- ----------------
       -3.9       .01       0.010             0.017             0.021           0.008           0.014           0.018
      -3.18       .05       0.053             0.067             0.079           0.047           0.057           0.069
      -2.78       .10       0.104             0.125             0.139           0.094           0.110           0.120
      -1.91       .30       0.309             0.331             0.345           0.293           0.314           0.320
      -1.27       .50       0.502             0.522             0.538           0.500           0.506           0.519
      -0.59       .70       0.705             0.718             0.727           0.714           0.712           0.710
       0.45       .90       0.899             0.905             0.911           0.911           0.906           0.907
       0.98       .95       0.949             0.955             0.957           0.959           0.951           0.954
       2.02       .99       0.991             0.992             0.992           0.994           0.992           0.992

  : **Quality of the Tracy-Widom Approximation for some large matrices:** the leftmost columns displays certain quantiles of the Tracy-Widom distribution. The second column gives the corresponding value of its cdf. Other columns give the value of the empirical distribution functions obtained from simulations at these quantiles. $\tilde{\mu}_{np}$ and $\tilde{\sigma}_{np}$ are the centering and scaling sequences.

We picked the dimensions according to two criteria: $100\times4000$, $30\times5000$, and $50\times5000$ were chosen to investigate “representative" microarray situations. We chose the other to have a range of ratios and estimate how valuable the Tracy-Widom approximation would be in situations that could be considered classical, i.e one small dimension (less than 10) and one large (several hundreds to several thousands). For the sake of completeness, we redid the simulations presented in @imj and present in Table 2 the results obtained with $\tilde{\mu}_{np}$ and $\tilde{\sigma}_{np}$ as centering and scaling.

   TW Quantiles    TW    5$\times$5   10$\times$ 10   100$\times$ 100   5$\times$20   10$\times$ 40   100$\times$400
  -------------- ------ ------------ --------------- ----------------- ------------- --------------- ----------------
       -3.9       .01        0            0.002            0.008           0.001          0.004           0.008
      -3.18       .05      0.003          0.018            0.043           0.019          0.032           0.044
      -2.78       0.10     0.022          0.054            0.090           0.056          0.077           0.095
      -1.91       .30      0.217          0.257            0.295           0.262          0.279           0.294
      -1.27       .50      0.464          0.486            0.497           0.490          0.494           0.489
       -.59       .70      0.702          0.703            0.700           0.702          0.707           0.702
       0.45       .90      0.903          0.903            0.901           0.905          0.906           0.899
       0.98       .95      0.949          0.950            0.950           0.952          0.953           0.949
       2.02       .99      0.988          0.990            0.991           0.989          0.990           0.990

  : **Quality of the Tracy-Widom Approximation (Continued):** the columns have the same meaning as in Table 1. The ratio $p/n$ is smaller than in Table 1 and the matrices are not as big, but the Tracy-Widom approximation is already acceptable for the upper quantiles. $\tilde{\mu}_{np}$ and $\tilde{\sigma}_{np}$ are the centering and scaling sequences.

We see that the fit is good to very good for the upper quantiles ($.9$ and beyond) across the range of dimensions we investigated. The practical interest of this remark is clear: these are the quantiles one would naturally use in a testing problem. We note that it appears empirically that the problem gets harder when the ratio $r$ of the larger dimension to the smaller one ($p_1$ in our notation) gets bigger: the larger $r$, the larger $p_1$ should be for the approximation to be acceptable.

Conclusions and Organization
----------------------------

From a technical standpoint, the method developed in @imj proves to be versatile, and, at least conceptually, relatively easy to adapt to the case where $n/p{\rightarrow}\infty$. Nevertheless, substantial technical work is needed to obtain Theorem \[ThJextended\]. Using the elementary fact (see e.g theorem 7.3.7 in @hj) that the largest eigenvalue of $X^{*}X$ is the same as the largest eigenvalue of $XX^{*}$, it will be sufficient to give the proof in the case $n/p{\rightarrow}\infty$.

From a practical point of view, we show that the Tracy-Widom limit law does not depend of how the sequence $(n,p)$ is embedded. As long as both dimensions go to infinity, the properly re-centered and re-scaled largest eigenvalue converges weakly to this law.\
We can compare this with the “classical" situation where $p$ is held fixed, in which case the limiting joint distribution is known, too (see e.g @anderson, corollary 13.3.2). In this case, the centering is done around $n$ and the scaling is $\sqrt{n}$; elementary computations show that $(l_1-\mu_{np})/\sigma_{np}$ also has a non-degenerate limiting distribution (possibly changing with each $p$). Nevertheless, even with the classical centering, it is hard to evaluate the marginals in this context and the results are therefore difficult to use in practice.

Our simulation results show that the Tracy-Widom approximation is reasonably good (for the upper quantiles) even when $p$ or $n$ are small. As remarked by @imj, Proposition 1.2, this implies that when doing PCA, one could develop (conservative) tests based on the Tracy-Widom distribution that could serve as alternatives to the scree plot or the Wachter plot.\
The paper is organized as follows: after presenting (Section 2) the main elements of the proof of Theorem \[ThJ\], we describe (Section 3) the strategy that will lead to the proof of Theorem \[ThJextended\]. We prove the two crucial points needed in Section 4. To make the paper self-contained, we give some background information about different aspects of the problem in the appendices. Several technical issues are also treated there in order to avoid obscuring the proof of the main result.

Outline of Johnstone’s proof
============================

Before describing the backbone of the proof presented in @imj, we need to introduce a few notational conventions. In what follows, we will use $N$ instead of $p$ to be consistent with the literature. We also denote by ${\mathbf{AB}}$ (for “asymptotic behavior") the situation where $n,N, \text{ and } n/N {\rightarrow}\infty$. We will frequently index functions that depend on both $N$ and $n$ with only $N$. The reason for this is that it will allow us to keep the notations relatively light, and that we think of $n$ as being a function of $N$. Notations like $\textbf{E}_N$ and $\textbf{P}_N$ will denote expectation and probability under the measure induced by the matrices (of size $n(N)\times N$) we are working with.\
Finally, it is technically simpler to work with a matrix $X$ whose entries are standard complex Gaussians (i.e the real and imaginary parts are independent, and they are both ${\cal N}(0,1/2)$), rather than with entries that are ${\cal N}(0,1)$. When we mention the complex case, we refer to this situation.

We now give a quick overview of the important points around which the proof of Theorem \[ThJ\] was articulated.

At the core of several random matrix theory results lie the fact that the joint distribution of the eigenvalues of the random matrices of interest is known and can be represented as the Fredholm determinant of a certain operator (or a totally explicit function of it).

Building on this, if we introduce a number $b$ that is $1$ in the real case and $2$ in the complex one, it turns out that one has the representation formula $$\label{freddet}
{\mathbf{E}_{N}\left(\prod_{i=1}^N (1+f(l_i))\right)}=\left[\det({\mathrm{Id}}+S_Nf)\right]^{b/2} \; ,$$ where $S_N$ is an explicit kernel, depending of course upon the kind of matrices in which one is interested. Here, $f$ treated as an operator means multiplication by this function. It is clear that if $\chi_t=-\mathbf{1}\{x:x\geq t\}$, we have $$\mathbf{P}_N(l_1\leq t)=\left[\det({\mathrm{Id}}+S_N\chi_t)\right]^{b/2}\;.$$

The interested reader can find background information on this in @mehta, chapters 5 and 6, @tw98 or @deift, chapter 5, which in turn (p.109) points to @simonreed, section 17, vol 4, for background on operator determinants. We stress the fact that all these formulas are finite dimensional.

From the last display, the strategy to show convergence in law in either Theorem \[ThJ\] or \[ThJextended\] is clear: fix $s_0$, show that under the relevant assumptions, $\mathbf{P}(l_{1,N}\leq s_0){\rightarrow}W_1(s_0)$, and use the fact that $W_1$ is continuous to conclude.

Complex case
------------

We just saw that to find the asymptotic behavior of $l_1$ is equivalent to showing the convergence of the determinant of a certain operator. This task can be reduced to showing convergence in trace class norm of this operator (see @simonreed for background on this, e.g, Lemma XIII.17.4 (p.323)). Through work from @widom99, @imj exhibits an integral representation formula for his operator, and the original problem is essentially transformed into showing that certain integrals have a predetermined limit. In somewhat more detail, if we call $\alpha=n-N$, and $L_k^{\alpha}$ the $k$-th Laguerre polynomial associated with $\alpha$ (as in @szego, p.100), let $$\phi_k(x)=\sqrt{\frac{k!}{(k+\alpha)!}}x^{\alpha/2}{\mathrm{e}}^{-x/2}L_k^{\alpha}(x) \; ,$$ $\xi_k(x)=\phi_k(x)/x$, $a_N=\sqrt{Nn}$, and finally $$\left\{
\begin{array}{cll}
\phi(x)&=(-1)^N\sqrt{\frac{a_N}{2}} (\sqrt{n}\xi_N(x)-\sqrt{N}\xi_{N-1}(x))  \; ,\\
\psi(x)&=(-1)^N\sqrt{\frac{a_N}{2}}(\sqrt{N}\xi_N(x)-\sqrt{n}\xi_{N-1}(x)) \; .
\end{array}
\right.$$ We note two things: first, there is a slight abuse of notation since $\phi$ and $\psi$ obviously depend on $n$ and $N$, but as in @imj, we choose to not carry these indices in the interest of readability. Also, $\phi$ and $\psi$ admit more “compact" representations, in terms of a single Laguerre polynomial, with a modified $\alpha$, or another degree. These are easy to derive using @szego, p.102, for instance. Nevertheless we choose to work (except in **A7**) with the previous representations because of the symmetries they present.

The kernel $S_N$ mentioned in (\[freddet\]) has the representation (@imj, equation (3.6)) $$S_N(x,y)=\int_{0}^{\infty} \phi(x+z)\psi(y+z)+ \psi(x+z)\phi(y+z) dz \; .$$

Now let $\bar{S}$ be the Airy operator. Its kernel is $$\bar{S}(x,y)=\frac{{\mathrm{Ai}}(x){\mathrm{Ai}}'(y)-{\mathrm{Ai}}(y){\mathrm{Ai}}'(x)}{x-y}=\int_0^{\infty}{\mathrm{Ai}}(x+u){\mathrm{Ai}}(y+u) du \; ,$$ where Ai denotes the Airy function. It was shown in @tw94 that, viewing $\bar{S}$ as an operator on $L^{2}[s,\infty)$, one had $$\det({\mathrm{Id}}-\bar{S})=W_2(s) \; ,$$ where $W_2$ is the Tracy-Widom law “emerging" in the complex case (see **A0**). So the complex analog of theorem \[ThJ\] follows from the fact that, after defining $S_{\tau}(x,y)={\sigma_N}S_N(\mu_N+{\sigma_N}x,\mu_N+{\sigma_N}y)$, Johnstone managed to show, for all $s$, that $$\det({\mathrm{Id}}-S_{\tau}) {\rightarrow}\det({\mathrm{Id}}-\bar{S}) \; .$$

To do this, he introduced $\phi_{\tau}(s)=\sigma_N \phi(\mu_N+s\sigma_N)$, and similarly $\psi_{\tau}$. Note that we have $$S_{\tau}(x,y)=\int_{0}^{\infty} \phi_{\tau}(x+z)\psi_{\tau}(y+z)+ \psi_{\tau}(x+z)\phi_{\tau}(y+z) dz \; .$$ Since what we are interested in is really $S_{\tau} \chi_s$, for some fixed $s$, we will view $S_{\tau}$ as an operator acting on $L^2[s,\infty)$ in what follows.\
So the problem becomes to show that, as $n,N {\rightarrow}\infty$ $$\label{whattoshow1}
\phi_{\tau}(s),\psi_{\tau}(s){\rightarrow}\frac{1}{\sqrt{2}}\mathrm{Ai}(s) \; ,$$ and that $\forall s_0 \in \mathbb{R}$, there exists $N_0(s_0)$ such that if $N>N_0$, we have on $[s_0,\infty)$, $$\label{whattoshow2}
\phi_{\tau}(s),\psi_{\tau}(s)=O({\mathrm{e}}^{-s/2}) \; .$$ Once this is shown (we give more details on this later), we can show that $S_{\tau} {\rightarrow}\bar{S}$ in the trace class norm of operators on $L^2[s,\infty)$. A classical way to do it is described in the remark at the end of section 3 of @imj, which bounds the trace class norm of the difference of $S_{\tau}-\bar{S}$ in terms of the Hilbert-Schmidt norm of operators whose kernels are related to $\phi_{\tau},
\psi_{\tau}$ and ${\mathrm{Ai}}$. This leads to the conclusion that $$\det({\mathrm{Id}}-S_{\tau}) {\rightarrow}\det({\mathrm{Id}}-\bar{S}) \;,$$ since $\det$ is continuous with respect to trace class norm. Therefore, the largest eigenvalue of $X^*X$ has the behavior it was claimed it has.

Real Case
---------

In the real case, using arguments from @tw96 and @widom99, @imj gets a representation similar to (\[freddet\]), this time involving an operator with kernel a $2\times 2$ matrix (instead of scalar in the complex case). He is then able to relate it to the complex case problem - the matrix operator determinant can be computed as the product of two scalar operator determinants - and shows that the “reduced" variable he works with ought to have the same limit as it had in the Gaussian Orthogonal Ensemble case, which was studied in depth by Tracy and Widom.

For the sake of completeness, we recall that in this situation $\alpha=n-1-N$ and $$\mathbf{P}_N(l_1\leq t)=\sqrt{\det({\mathrm{Id}}+K_N\chi_t)} \;.$$ $K_N$ has the representation (in the $N$ even case) $$K_N=
\begin{pmatrix}
S_N+ \psi\otimes \epsilon \phi & S_N D-\psi\otimes\phi \\
{\epsilon}S_N -{\epsilon}+{\epsilon}\psi \otimes {\epsilon}\phi & S_N + {\epsilon}\phi \otimes \psi
\end{pmatrix} \; ,$$ where $D$ is the differential operator, ${\epsilon}$ is convolution with the kernel ${\epsilon}(x-y)$, and ${\epsilon}(x)=\text{sgn}(x)/2$. We note the slight change in $\alpha$ and replace $n$ by $n-1$ when we need to use the results or formulas derived in the complex case (for instance, the $S_N$ we just mentioned is $S_{n-1,N}$, and not $S_{n,N}$). We refer the reader to @ggk for a complement of information on operator determinants and to the end of section VIII in @tw96 for details on the technical problems that $K_N$ poses.

From a purely technical standpoint, one critical issue is to evaluate the large $n,N$ limit of $c_{\phi}=\int_{0}^{\infty}\phi(x)dx/2$. If one can show that it is $1/\sqrt{2}$ when $N{\rightarrow}\infty$ through even values, then Johnstone’s considerations hold true all the way and we have the same conclusion as in Theorem \[ThJ\].\
We note that using the interlacing properties of the singular values (as mentioned for instance in @sosh, Remark 5; see also @hj, theorem 7.3.9), as well as the estimates of the difference (resp. ratio) between two consecutive terms of the centering (resp. scaling) sequence, the $N$ odd case follows immediately from the $N$ even case. To be more precise, we use the fact that $$\frac{\mu_{n,N}-\mu_{n,N-1}}{{\sigma_N}}={\mathrm{O}}(N^{-1/3}) \; \; {\rightarrow}0 \text{ as } N {\rightarrow}\infty$$ to check that the $N$ even terms lower and upper bounding the $N$ odd probability have the same limit. Note that the same relationship holds for $\mu_{n+a,N+b}$ and $\mu_{n,N}$, if $a$ and $b$ are fixed real numbers. Therefore, after doing the proof with centering sequence $\mu_{n+3/2,N+1/2}$ (which is technically simpler), we will be able to conclude that the theorem holds true for $\mu_{n,N}$.

Last, to be able to use @sosh, Lemma 2, which gives the result we wish for the joint distribution of the $k$-largest eigenvalues, we will need to verify that the entries of the $2\times 2$ operator converge pointwise, and are bounded above in an exponential way. This is what is done in the proof of Lemma 1 of @sosh, and we will show in **A8** that the arguments given there can be extended to handle our situation.

Further Remarks and Agenda
==========================

Most of the work in @imj is done in closed form, and in the finite dimensional case. That has two advantages from our standpoint: as the limiting behavior is only investigated in the last “step", most of the arguments given there carry through for our problem, and the method certainly does.

Therefore, our contribution is mostly technical; it follows very closely the ideas of @imj, providing solutions to technical problems appearing in the case we consider. Only at a few points could we not use the approach developed in @imj. This led us to an analysis of the complex case that is slightly different from the original one, but the core reasons for which the result holds are the same.

In what follows, we first focus on showing that (\[whattoshow1\]) and (\[whattoshow2\]) hold true when $n,N$ and their ratio tend to infinity. This takes care of the complex case. We then turn to the problem of the asymptotic behavior of $c_{\phi}$, and the technical points we have to verify for @sosh results to hold.

The following remarks outline the differences between the analysis we present here and the one done in @imj.

Remarks on adaptation of the original proof
-------------------------------------------

### Complex case {#errorcontrolproblem}

To show that (\[whattoshow1\]) and (\[whattoshow2\]) held true, @imj essentially reduced his problem to studying the solution of a “perturbed" Airy equation and used tools from @olver to carefully study it. One point that was used repeatedly was that the turning points of the equation were bounded away from one another when $n,N$ were large. This is not true anymore in the case we consider, and we show how to get around this difficulty. So we do not work with a perturbed Airy equation anymore, but rather with Whittaker functions, which have a close relationship to Laguerre polynomials, and their expansion in terms of parabolic cylinder functions (see **A9** for some background information on special functions). In @olver80, the case we are interested in was studied in detail, giving us most of the tools we need to show (\[whattoshow2\]). Using @olver75, we reinterpret the [parabolic cylinder functions]{} results in terms of Airy functions and derive the elements we need to complete the proof of (\[whattoshow1\]) and (\[whattoshow2\]).

The reason for which we could not exactly follow the “original" method is related to the error control function called ${\cal V}(\zeta)$ in @imj. This function depends upon the parameter $\omega=2\lambda/\kappa$, which in the case $n/N{\rightarrow}\gamma \in \mathbb{R}$ is bounded away from 2. This essentially allows a uniform control over ${\cal V}$, and it is possible to show that this error control function is bounded as a function of $N$. Since the control is actually something like $\exp(\lambda_0 {\cal V}/\kappa)-1$, it tends to zero as $N{\rightarrow}\infty$. This gave @imj a way to get part of (\[whattoshow2\]).\
In our case, it seems that ${\cal V}$ would tend to $\infty$, at a rate that is nevertheless ${\mathrm{o}}(\kappa)$. As it seems easier and more promising to use @olver80 than to derive the growth of ${\cal V}$, we choose this approach. Nevertheless, this is the only (but crucial) technicality (in the complex case) that did not carry through by the method described in @imj under ${\mathbf{AB}}$.

### Real Case

For the $c_{\phi}$ problem, we provide a closed form expression at given $n,N$ and show that in the limit is the “right" one as long as $n$ and $N$ tend to $\infty$.. This does not use the saddlepoint method, but relies on the availability of a generating function formula for Laguerre polynomials. The proof is done in **A7**.\
A simple modification to @imj would give the same result: in the display preceding (6.13) there, we could write $$h(t)=\sum_{k=0}^{\infty}c_k t^k = 2^{\alpha/2}\Gamma(\alpha/2)(1+t)(1-t^2)^{-(\alpha/2+1)}$$ and expand $(1-t^2)^{-(\alpha/2+1)}$. Multiplying by $1+t$ has a very simple effect on the series, and so $c_k$ is known explicitly.

In **A8**, we show how to check that the conditions required for Soshnikov’s results to hold are indeed met. They are straightforward consequences of the analysis we will carry below.

Since the real case is derived from the complex one after analyzing a few technical points, we verify these in the appendices and present here the study of the complex case. We now turn to the main problem we solve in this note: showing (\[whattoshow1\]) and (\[whattoshow2\]) under our set of assumptions.

Complex case: study of asymptotics
==================================

In this section, we work on the problem of showing pointwise convergence and uniform boundedness, setting the problem in a way similar to section 5 of @imj. We recall his notations, slightly modified to avoid confusions: $N_+=N+1/2$, $n_+=n+1/2$, $z=\mu_N+\sigma_N s$, with $\mu_N=(\sqrt{(N+\alpha)_+}+\sqrt{N_+})^{1/2}$ and $\sigma_N=(\sqrt{(N+\alpha)_+}+\sqrt{N_+})(1/\sqrt{N_+}+1/\sqrt{(N+\alpha)_+})^{1/3}$. For reasons that will be transparent later on, our aim is to show that $$\label{pointwisecv}
F_N(z)=(-1)^N \sigma_N^{-1/2}\sqrt{N!/n!}\,z^{(\alpha+1)/2}{\mathrm{e}}^{-z/2}L_N^{\alpha_N}(z) {\rightarrow}{\mathrm{Ai}}(s), \;
\forall s \in \mathbb{R} \; ,$$ and $$\label{uniformcv}
F_N(z)={\mathrm{O}}({\mathrm{e}}^{-s})\; \text{uniformly in } \; [s_0,\infty), s_0 \in \mathbb{R} \; .$$

The scaling is slightly different from the original proof: $N^{-1/6}$ has been replaced by ${\sigma_N}^{-1/2}$. As in @imj, we focus on $w_N(z)=z^{(\alpha+1)/2}{\mathrm{e}}^{-z/2}L_N^{\alpha}(z)$, which satisfies $$\frac{d^2w}{dz^2}=\left(\frac{1}{4}-\frac{\kappa}{z}+\frac{\lambda^2-1/4}{z^2}\right)w \; ,$$ where $\kappa=N+(\alpha+1)/2$ and $\lambda=\alpha/2$. Remark that under ${\mathbf{AB}}{\overset{\text{def}}{\Leftrightarrow}}n,N,n/N {\rightarrow}\infty$, $\kappa \sim \lambda$. Our strategy is to reformulate the problem in terms of so-called Whittaker functions, denoted $W_{k,m}$, and to use the extensive available studies of these functions to show (\[whattoshow1\]) and (\[whattoshow2\]). @temme, formula (3.1) p.117 shows that $$w_N(z)=\frac{(-1)^N}{N!}W_{\kappa,\lambda}(z) \; .$$

From now on, we will closely follow @olver80. Let us remark that $$F_N(z)={\sigma_N}^{-1/2}\frac{1}{\sqrt{n!N!}}{W_{\kappa,\lambda}}(z) \;.$$ We fix $s_0 \in \mathbb{R}$, and we work only with $z=\mu_N+{\sigma_N}s$, where $s\geq s_0$.

#### Preliminaries

Following @olver80, we introduce $l=\kappa/\lambda$, $\beta=\sqrt{2(l-1)}$, and the turning points $x_1=2l-2\sqrt{l^2-1}$, $x_2=2l+2\sqrt{l^2-1}$, after the rescaling $x=z/\lambda$. We remark that the two turning points coalesce at 2 under the hypothesis ${\mathbf{AB}}$. In the new variable $x$, we have $$\frac{d^2W}{dx^2}=\left(\lambda^2 g(x) - \frac{1}{4x^2}\right)W \;,$$ where $g(x)=\frac{(x-x_1)(x-x_2)}{4x^2}$. Using the ideas explained in @imj, we shall be - eventually - interested in the asymptotics for $z=\mu_N+\sigma_N s$, or $x=z/\lambda=x_2+\sigma_N s/\lambda$ of $F_N(z)$. Let us now define an auxiliary variable $\upsilon$ by $$\begin{aligned}
\int_{\beta}^{\upsilon}(\tau^2-\beta^2)^{1/2}d\tau &= \int_{x_2}^x g^{1/2}(t)dt \hspace{1cm} \text{if } x_2 \leq x
< \infty \;,
\\
\int_{-\beta}^{\upsilon}(\beta^2-\tau^2)^{1/2}d\tau &= \int_{x_1}^x (-g)^{1/2}(t)dt \hspace{1cm} \text{if } x_1
\leq x \leq x_2 \;.\end{aligned}$$

We limit $x$ to this range because of the technically important following point: ${\sigma_N}/\lambda$ tends to zero faster than $x_2-x_1$ does, and so, when $s$ is bounded below, $x$ will stay in the range $(x_1,\infty)$ for all $N$ greater than a certain $N_0$. This is shown in **A2**, along with the closely related fact that we can focus on $\upsilon\geq 0$. Our analysis is based on section 3 of @olver80, where he builds on @olver75, in which he expands Whittaker functions in terms of parabolic cylinder functions. The condition ${\upsilon}\geq 0$ is critical, since Olver’s expansions depend on the sign of $\upsilon$. Therefore, **A2** entitles us to focus on only one specific form of these. From (3.10) p.219 in @olver80, one has $$\begin{aligned}
{W_{\kappa,\lambda}}(\lambda x)=
(2\lambda)^{1/4}&\{\lambda(2+\beta^2/2)/e\}^{\lambda(1+\beta^2/4)}
&\times\left(\frac{{\upsilon}^2-\beta^2}{x^2-4lx+4}\right)^{1/4}x^{1/2}
\{U(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})+{\epsilon}_1(\lambda^2,\beta^2,{\upsilon})\} \; ,\end{aligned}$$ where, if ${\mathbf{E}}$ and ${\mathbf{M}}$ are the weight and modulus functions associated with $U$ in @olver75 (p.156), we have, according to @olver80 (3.11) p.219, $$\label{error}
{\epsilon}_1(\lambda^2,\beta^2,{\upsilon})={\mathbf{E}}^{-1}(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})
{\mathbf{M}}(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})\;\mathrm{O}(\lambda^{-2/3})$$ **uniformly** with respect to $\beta\in [0,B]$ and ${\upsilon}\in [0,\infty)$, $B$ being an arbitrary positive constant. We recall that the main relationship between $U$, ${\mathbf{E}}$ and ${\mathbf{M}}$ : for $b \leq 0$ and $x\geq 0$, $|U(b,x)|\leq {\mathbf{E}}^{-1}(b,x){\mathbf{M}}(b,x)$.

We now show that we have uniform boundedness on $[s_0,\infty)$. The pointwise convergence result will be a straightforward consequence of the arguments we need to develop to solve this first problem.

Uniform Boundedness
-------------------

Following up on the previous displays, if $n,N$ are large enough so that ${\upsilon}\geq 0$, we have $$\begin{gathered}
\label{Wbound}
\left|{W_{\kappa,\lambda}}(\lambda x)\right|\leq (2\lambda)^{1/4}\{\lambda(2+\beta^2/2)/e\}^{\lambda(1+\beta^2/4)}
\times\left(\frac{{\upsilon}^2-\beta^2}{x^2-4lx+4}\right)^{1/4}x^{1/2}{\mathbf{M}}{\mathbf{E}^{-1}}(1+\mathrm{O}(\lambda^{-2/3})) \;,\end{gathered}$$ where we omitted the argument $(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})$ for readability purposes. Our plan is now to transform this upper bound into a somewhat similar one, involving the modulus and weight function associated with the Airy function, which have the advantage of having only one parameter and known asymptotics.

To carry out this program, we need to split the investigation into two parts: first $s\geq 0$ or ${\upsilon}\geq \beta$. This will allow us to find an $s_1 \geq 0$ such that $F_N(z)={\mathrm{O}}({\mathrm{e}}^{-s})$ on $[2s_1,\infty)$. In the second part, we will just have to consider the case $s\in [s_0,2s_1]$, and show that $F_N$ is merely uniformly bounded on this interval.

### Case $\mathbf{ s\geq 0} $ 

In order to use the results linking parabolic cylinder functions and the Airy function (proved in @olver59 and cited in @olver75), let us define yet another auxiliary variable, $\eta$, by $$\frac{2}{3}\eta^{3/2}\beta^2=\int_{x_2}^xg^{1/2}(t)dt \;.$$

Then, if we call ${{\cal E}}$ and ${{\cal M}}$ the weight and modulus functions associated with the Airy function, we have, as shown in **A3**: $$\begin{aligned}
{\mathbf{E}^{-1}}(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda}) &\leq
{{\cal E}^{-1}}(\lambda^{2/3}\beta^{4/3}\eta)(1+{\mathrm{O}}((\lambda\beta^2)^{-1})) \; ,\\
{\mathbf{M}}(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})&\leq
\frac{\sqrt{2}\pi^{1/4}\left(\Gamma((1+\lambda\beta^2)/2)\right)^{1/2}
\beta^{1/2}}{(\lambda\beta^2)^{1/12}}\\
&\times\left(\frac{\eta}{{\upsilon}^2-\beta^2}\right)^{1/4}{{\cal M}}(\lambda^{2/3}\beta^{4/3}\eta)
\left(1+{\mathrm{O}}((\lambda\beta^2)^{-1})\right) \; .\end{aligned}$$ Whence, if we call $\theta\triangleq \lambda^{2/3}\beta^{4/3}\eta$, $$|F_N(\lambda x)|\leq K_{n,N}x^{1/2}(\eta/(x^2-4lx+4))^{1/4}{{\cal E}^{-1}}(\theta){{\cal M}}(\theta)
\left(1+{\mathrm{O}}((\lambda\beta^2)^{-1}\vee \lambda^{-2/3})\right) \; .$$ In **A4**, we show that $K_{n,N}\sim 2^{2/3}(N/n)^{1/4}$ under ${\mathbf{AB}}$. From now on, $\Delta$ will denote a generic constant; its value may change from display to display. As long as $x\geq x_2$, or $s\geq 0$, we have $$|F_N(\lambda x)|\leq \Delta (N/n)^{1/4} x^{1/2}(\eta/(x^2-4lx+4))^{1/4}{{\cal E}^{-1}}(\theta){{\cal M}}(\theta)) \; .$$ Now using the fact that (see @olver, chap. 11) $x^{1/4}{{\cal M}}(x)\leq \Delta$, ${{\cal E}^{-1}}(x)\leq \Delta
\exp(-2x^{3/2}/3)$ for $x\geq 0$ and $\lambda\beta^2=2N+1$, we get the new inequality $$|F_N(\lambda x)|\leq \Delta
\left(\frac{N}{n}\right)^{1/4}N^{-1/6}\left(\frac{x^2}{x^2-4lx+4}\right)^{1/4}\exp(-(2\theta^{3/2})/3) \; .$$ In **A5.1**, we show that there exists $s_1$ such that if $s\geq 2 s_1$, $(2\theta^{3/2})/3 \geq s$. Also, as shown in **A6.1**, if $s\geq 0$, $g$ is positive and increasing in $x$ (or, equivalently, in $s$). Since the rational function of $x$ appearing in the previous display is just $(4g(x))^{-1/4}$, we can bound it by its value at $x(2s_1)$ on $[2s_1,\infty)$. In **A6.2**, we show that, at $s$ fixed, under ${\mathbf{AB}}$, we have $4g(x)\sim \beta{\sigma_N}s/\lambda$, and using the equivalents mentioned in **A1**, we have ${\sigma_N}\beta/\lambda\sim
4 N^{1/3}/n$, from which we conclude that $$\left(\frac{N}{n}\right)^{1/4}N^{-1/6}(4g(2s_1))^{-1/4}\sim N^{1/12}n^{-1/4}(8s_1N^{1/3}/n)^{-1/4}\sim
(8s_1)^{-1/4} \; .$$ Therefore, if $N$ is large enough, $$\forall s\in[2s_1,+\infty)\;\;|F_N(\lambda x)| \leq \Delta \exp(-s)$$

### Case $s\in[s_0,2s_1]$ {#casesinterval}

Our aim now is just to show that $F_N$ as a function of $s$ is bounded on this interval; from this we shall immediately have that $F_N={\mathrm{O}}(\exp(-s))$ on this interval, and we will have a proof of (\[uniformcv\]).\
This part is comparatively simpler: we use equation (\[Wbound\]), in which we have ${\mathbf{E}^{-1}}\leq 1$, by definition (@olver75, p.156, (5.22)). Now using the display between (6.12) and (6.13) p.159 of the same article, we have for ${\lambda\beta^2}\geq 1$ and ${\upsilon}\geq 0$, $$\frac{{\mathbf{M}}(-\lambda\beta^2/2,{\upsilon}\sqrt{2\lambda})}{\left(\Gamma((1+\lambda\beta^2)/2)\right)^{1/2}}\leq
\frac{\Delta\beta^{1/2}}{(\lambda\beta^2)^{1/12}}\left(\frac{\eta}{{\upsilon}^2-\beta^2}\right)^{1/4} \; .$$ Hence, $$|F_N(\lambda x)| \leq K_{n,N}\Delta\left(\frac{\eta}{x^2-4lx+4}\right)^{1/4}x^{1/2} \; .$$ However on this interval, $x{\rightarrow}2$, by **A5.2** $\eta=(\lambda\beta^2)^{-2/3}s+{\mathrm{o}}((\lambda\beta^2)^{-2/3})$, and by **A6.2** $(x^2-4lx+4)=4s{\sigma_N}\beta(1+{\mathrm{o}}(1))/\lambda$. Therefore, $$\frac{\eta}{x^2-4lx+4}\sim 2^{-2/3}2^{-4} n/N$$ on the whole interval, and, because of the asymptotic estimate of $K_{n,N}$ given in **A4**, $F_N$ is bounded uniformly in $N$ on the interval $[s_0,2s_1]$.

We can thus conclude that $$\forall s_0, \; \; \exists N_0(s_0) \; N>N_0(s_0) ,\; \; \; F_N(s)={\mathrm{O}}_{s_0}({\mathrm{e}}^{-s}) \; \; \text{ on }
[s_0,\infty) \; .
$$

Pointwise convergence
---------------------

Having studied in detail the uniform boundedness of $F_N$ makes the pointwise convergence problem easier. First, since we bounded above $F_N$ in terms of ${\mathbf{M}}$ and ${\mathbf{E}^{-1}}$, equation (\[error\]) shows that ${\epsilon}_1={\mathrm{O}}(\lambda^{-2/3}{\mathrm{e}}^{-s})$ on $[s_0,\infty)$. So for fixed $s$, it tends to zero as $N$ gets large. The pointwise limit of $F_N$ will be the pointwise limit of the parabolic cylinder function part of the expansion. We call this part $\wp F_N$, for “principal part".

Using the relationship between $U$ and ${\mathrm{Ai}}$ that we mention in **A3**, we have, with $\theta=(\lambda\beta^2)^{2/3}\eta$, $$\wp F_N(\lambda x)=K_{n,N}x^{1/2}\left(\frac{\eta}{x^2-4lx+4}\right)^{1/4}({\mathrm{Ai}}(\theta)+{{\cal E}^{-1}}(\theta){{\cal M}}(\theta) \,
{\mathrm{O}}((\lambda\beta^2)^{-1})) \; .$$ Since $x{\rightarrow}2$, $K_{n,N}\sim 2^{2/3}(N/n)^{1/4}$ and given the estimate we just mentioned for the ratio $\eta/(x^2-4lx+4)$, we have $$K_{n,N}\, x^{1/2}\left(\frac{\eta}{x^2-4lx+4}\right)^{1/4} \sim 1 \; .$$ In other respects, we show in **A5.2** that $\theta{\rightarrow}s$ under ${\mathbf{AB}}$. Finally, ${{\cal E}^{-1}}$ and ${{\cal M}}$ are bounded on $\mathbb{R}$, as shown in 11.2 (pp.394-397) of @olver. Hence ${{\cal E}^{-1}}(\theta){{\cal M}}(\theta) \,
(\lambda\beta^2)^{-1} {\rightarrow}0 $ under ${\mathbf{AB}}$, and we can conclude that $\wp F_N(\lambda x){\rightarrow}{\mathrm{Ai}}(s)$; combining all the elements gives $$\forall s \in \mathbb{R}, \; \; \; \;F_N(\lambda x){\rightarrow}{\mathrm{Ai}}(s) \; \text{ under }{\mathbf{AB}}\; .$$

Asymptotics for $\mathbf{\phi_{\tau}} \text{ and } \mathbf{\psi_{\tau}}$
------------------------------------------------------------------------

So far we have shown that $F_N(z)=(-1)^N
{\sigma_N}^{-1/2}\sqrt{z}\phi_N(z){\rightarrow}{\mathrm{Ai}}(s)$, and that ${\mathrm{e}}^s F_N$ was bounded when $N>N_0$ and $s \geq s_0$.\
Our aim is to show (\[whattoshow1\]) and (\[whattoshow2\]). Let us write, as in @imj, $$\phi_{\tau}=\phi_{I,N}+\phi_{I\!I,N} \; ,$$ where $$\phi_{I,N}(z)=(-1)^N{\sigma_N}\sqrt{a_N n}\phi_N(z)/(\sqrt{2} z)= F_N(z)d_N (z/\mu_N)^{-3/2}\;.$$

#### Study of $\phi_{I,N}$

In the previous display, we have $d_N=({\sigma_N}/\mu_N)^{3/2}\sqrt{a_N n/2}$. As ${\sigma_N}\sim n^{1/2}N^{-1/6}$ and $\mu_N\sim n$, $({\sigma_N}/\mu_N)^{3/2}\sim n^{-3/4}N^{-1/4}$. Since $a_N=\sqrt{Nn}$, $a_Nn=n^{3/2}N^{1/2}$, and therefore $d_N{\rightarrow}1/\sqrt{2}$. But when $s$ is fixed, $z/\mu_N {\rightarrow}1$, so it follows that $$\text{Under }{\mathbf{AB}},\hspace{.5cm} \phi_{I,N}(\mu_N+{\sigma_N}s){\rightarrow}\frac{{\mathrm{Ai}}(s)}{\sqrt{2}} \;.$$ To bound $\phi_{I,N}$ for $N>N_0$ and $s\geq s_0$, we use, as in @imj, the uniform bound for $F_N$ and $(z/\mu_N)^{-3/2}\leq \exp(-3{\sigma_N}s/(2\mu_N))$, if $s\geq 0$. If $s\leq 0$, we have $(z/\mu_N)^{-3/2}\leq
(1+s_0{\sigma_N}/\mu_N)^{-3/2}$, and since this converges to 1 under ${\mathbf{AB}}$, it is bounded if $N$ is large enough. So we have shown that, $$\phi_{I,N}(\mu_N+s{\sigma_N})
\left\{
\begin{array}{clc} {\rightarrow}& 2^{-1/2}{\mathrm{Ai}}(s)\;,  \; \; N{\rightarrow}\infty \;, \\
\leq & M {\mathrm{e}}^{-s} \; \; \text{ on } [s_0,\infty) \text{ if } N>N_0(s_0) \; .
\end{array}
\right.$$

#### Study of $\phi_{I\!I,N}$

We use once again the same approach as in @imj. We have $$\phi_{I\!I,N}=u_N v_{N-1} \phi_{I,N-1} \; ,$$ where $u_N=({\sigma_N}/\sigma_{N-1})\sqrt{a_N/a_{N-1}}$ and $v_N=(N/n)^{1/2}$, and $n_{N-1}$ appearing in $\sigma_{N-1}$ is $n_N-1$ (for $\phi_{N-1}$ is defined in terms of $L_{N-1}^{\alpha_N}$ and we should therefore have the same $\alpha=n-N=(n-1)-(N-1)$). Remark that under ${\mathbf{AB}}$, $v_N {\rightarrow}0 $ and $u_N {\rightarrow}1$.\
Define $s'$ by $\mu_N+{\sigma_N}s=\mu_{N-1}+\sigma_{N-1}s'$. From $$s'=\frac{\mu_N-\mu_{N-1}}{\sigma_{N-1}}+\frac{{\sigma_N}}{\sigma_{N-1}}s \; ,$$ we deduce that $s'\geq s/2 $ on $[0,\infty)$, if $N$ is large enough: as a matter of fact, under ${\mathbf{AB}}$, $\mu_N
-\mu_{N-1 }={\mathrm{O}}(\sqrt{n/N})$, $\sigma_{N} \sim n^{1/2}N^{-1/6}$, and ${\sigma_N}/\sigma_{N-1}{\rightarrow}1$, so it is larger than $1/2$ when $N$ is large enough. To summarize, we just showed that $$\phi_{I\!I,N}(\mu_N+s{\sigma_N})\leq M v_N {\mathrm{e}}^{-s/2} \; \; \text{for } s\in [0,\infty) \; ,$$ by applying the bound we got for $\phi_{I,N}$ to $\phi_{I,N-1}$ and $s'$ as the dummy variable. Here, we are implicitly using the fact that since $n/N{\rightarrow}\infty$, $(n-1)/(N-1)$ does too, and we can apply all the results we derived before. On the other hand, when $s \in[s_0,0]$, we can use the fact that $(\mu_N-\mu_{N-1})\geq 0$ and ${\sigma_N}/\sigma_{N-1} \leq 2$ to show that $s'\geq 2 s $ and hence $$\phi_{I\!I,N}(\mu_N+s{\sigma_N})\leq M v_N {\mathrm{e}}^{-2s}\leq M' v_N {\mathrm{e}}^{-s/2} \; \; \text{for } s\in [s_0,0] \; .$$ The conclusion is therefore that $$\phi_{I\!I,N}(\mu_N+s{\sigma_N}) \left\{\begin{array}{clc} {\rightarrow}& 0 \;, \; \; N{\rightarrow}\infty ,\\
\leq & \Delta {\mathrm{e}}^{-s/2} \; \; \text{ on } [s_0,\infty)\;, \text{ if } N>N_0(s_0) \;.
\end{array}
\right.$$ Hence we have shown that (\[whattoshow1\]) and (\[whattoshow2\]) held for $\phi_{\tau}$. The analysis for $\psi_{\tau}$ is similar.

Appendices
==========

This section is devoted to giving background information needed to understand the problem and make the paper relatively self-contained. We also establish many of the properties needed in the course of the proofs of equations (\[whattoshow1\]) and (\[whattoshow2\]) here.\
Before we start, let us mention a notation issue: $\alpha$ changes value depending on whether we treat the complex case or the real one. For the complex case $\alpha+N=n$, whereas for the real one $\alpha+N=n-1$. We frequently replace $\alpha+N$ by $n$ in what follows; this is because the proof of equations (\[whattoshow1\]) and (\[whattoshow2\]) is done in the complex case and applies to the real one by just changing $n$ into $n-1$ everywhere. When dealing with problems which are real case specific, we keep the notation $N+\alpha$. The definition of ${\mu_N}$ and ${\sigma_N}$ are also given in terms of $N+\alpha$ to highlight the adjustments needed when dealing with the real or the complex case.

A0: Tracy-Widom distributions {#a0-tracy-widom-distributions .unnumbered}
-----------------------------

We recall here the definition of the Tracy-Widom distributions. We split the description according to whether the entries of the matrix we are considering are real or complex.\
We first need to introduce the function $q$, defined as $$\left\{
\begin{array}{l}
q''(x)=xq(x)+2q^3(x) \;,\\
q(x)\sim {\mathrm{Ai}}(x) \; \; \; \text{ as } x{\rightarrow}\infty \;.
\end{array}
\right.$$

**$\bullet$ Complex Case** The Tracy-Widom distribution appearing in the complex case, $W_2$, has cumulative distribution function $F_2$ given by $$F_2(s)=\exp\left(-\int_s^{\infty}(x-s)q^2(x) dx\right)\;.$$ The joint distribution is slightly more involved to define. Following @sosh, we do it through its $k$-point correlation functions, using its determinantal point process character (see e.g @soshDet).\
Let us first call $\bar{S}$ be the Airy operator. Its kernel is $$\bar{S}(x,y)=\frac{{\mathrm{Ai}}(x){\mathrm{Ai}}'(y)-{\mathrm{Ai}}(y){\mathrm{Ai}}'(x)}{x-y}=\int_0^{\infty}{\mathrm{Ai}}(x+u){\mathrm{Ai}}(y+u) du \;.$$ In the complex case, the $k$-point correlation functions have the property that $$\rho_k(x_1,\ldots,x_k)=\det_{1\leq i,j\leq k} \bar{S}(x_i,x_j) \;.$$

**$\bullet$ Real Case** The real counterpart of $W_2$, which is called $W_1$, has cdf $F_1$ with $$F_1(s)=\exp\left(-\frac{1}{2}\int_s^{\infty}q(x)+(x-s)q^2(x) dx\right) \;.$$ The $k$-point correlation functions satisfy $$\rho_k(x_1,\ldots,x_k)=\left(\det_{1\leq i,j\leq k} K(x_i,x_j)\right)^{1/2} \;,$$ where the $2\times 2$ matrix kernel of $K$ has entries (see @sosh, eq (2.18) to (2.21)) $$\begin{aligned}
K_{1,1}(x,y)&=\bar{S}(x,y)+\frac{1}{2}{\mathrm{Ai}}(x)\int_{-\infty}^y {\mathrm{Ai}}(u) du  \; ,\\
K_{2,2}(x,y)&=K_{1,1}(y,x)\; ,\\
K_{1,2}(x,y)&=-\frac{1}{2}{\mathrm{Ai}}(x){\mathrm{Ai}}(y)-\frac{\partial}{\partial y}\bar{S}(x,y)\;,\\
K_{2,1}(x,y)&=-\int_0^{\infty}dt\left(\int_{x+t}^{\infty}{\mathrm{Ai}}(v) dv \right){\mathrm{Ai}}(y+t)
-{\epsilon}(x-y)+\frac{1}{2}\int_y^x{\mathrm{Ai}}(u)du+\frac{1}{2}\int_x^{\infty}{\mathrm{Ai}}(u)du\int_{-\infty}^y{\mathrm{Ai}}(v)dv \; .\end{aligned}$$

A1: Asymptotic behavior of some simple functions {#a1-asymptotic-behavior-of-some-simple-functions .unnumbered}
------------------------------------------------

In this appendix, we present some basic facts and identities that we used throughout the proof.\
We will make repeated use of the following observations: since ${\sigma_N}=(\sqrt{(N+\alpha)_+}+\sqrt{N_+})(1/\sqrt{N_+}+1/\sqrt{(N+\alpha)_+})^{1/3} $ and $
\lambda =\alpha/2$, under **AB** we have $$\begin{aligned}
{\sigma_N}&\sim n^{1/2}N^{-1/6} \;,\\
\lambda&\sim n/2 \; .\end{aligned}$$

We also use several times the following identities:

\[factequal\] With $\lambda=\alpha/2$, $\kappa=N+(\alpha+1)/2$, and $l=\kappa/\lambda$, $\beta=\sqrt{2(l-1)}$, we have $$\begin{aligned}
\lambda\beta^2&=(2N+1) \;,\\
\beta &\sim 2\sqrt{N/n} \;.\end{aligned}$$

The first remark is simple algebra, and the second one comes from $\beta^2=2(l-1)=2(2N+1)/\alpha\sim 4N/n$ under ${\mathbf{AB}}$. We have the estimates:

\[factequiv\] $x_2-x_1 \sim
8 \sqrt{N/n}$ and $\sigma_N/\lambda \sim 2 n^{-1/2}N^{-1/6}$ .

The second one is obvious; the first one comes from the fact that $x_2-x_1 = 2\sqrt{2}\beta (l+1)^{1/2}$ as $x_{2,1}=2l \pm 2\sqrt{l^2-1}$. Using Fact \[factequal\] immediately gives the claimed result. Finally, we have the following estimates

\[factfunctions\] $\beta{\sigma_N}/\lambda \sim 4 N^{1/3}/n$ and ${\sigma_N}^3/(\lambda\beta^2)\sim(n/N)^{3/2}/2$ .

The result directly follows from the aforementioned estimates.

A2: Working with ${\upsilon}\geq 0$ {#a2-working-with-upsilongeq-0 .unnumbered}
-----------------------------------

Here we assume that $s\in[s_0,\infty)$. We also assume that $s<0$, for otherwise we can work with ${\upsilon}\geq
\beta>0$. From **A1**, we have $|x-x_2|=|s|{\sigma_N}/\lambda \leq |s_0|{\sigma_N}/\lambda \ll x_2-x_1 $ by Fact \[factequiv\]. Now ${\upsilon}=0$ corresponds to $x_0\leq \bar{x}=(x_1+x_2)/2$: as a matter of fact, since $(x_2-x)(x_1-x)$ is symmetric around $\bar{x}$ and $1/x$ is obviously larger on $[x_1,\bar{x}]$ than it is on $[\bar{x},x_2]$, we have $$\int_{-\beta}^{{\upsilon}_{\bar{x}}} (\beta^2 - \tau^2) d\tau = \int_{x_1}^{\bar{x}}(-g(t))^{1/2}dt \geq
\int_{\bar{x}}^{x_2}(-g(t))^{1/2}dt \; .$$ By symmetry, we also get $$\begin{aligned}
\int_{-\beta}^{0} (\beta^2 - \tau^2) d\tau = \int_{0}^{\beta} (\beta^2 - \tau^2) d\tau & = \frac{1}{2}
\int_{x_1}^{x_2}(-g(t))^{1/2}dt \\
&\leq \int_{x_1}^{\bar{x}}(-g(t))^{1/2}dt = \int_{-\beta}^{{\upsilon}_{\bar{x}}} (\beta^2 - \tau^2) d\tau \; ,\end{aligned}$$ and therefore, ${\upsilon}_{\bar{x}}>0$.\
However $\bar{x}$ is always smaller than $x(s_0)$ if $N$ is large enough. So we can limit our investigations to the case ${\upsilon}\geq 0$.

A3: Relationship between ${\mathbf{E}^{-1}}$, ${{\cal E}^{-1}}$, ${\mathbf{M}}$ and ${{\cal M}}$ {#a3-relationship-between-mathbfe-1-cal-e-1-mathbfm-and-cal-m .unnumbered}
------------------------------------------------------------------------------------------------

We claim that if $s\geq 0$, and we define $\theta=(\lambda\beta^2)^{2/3}\eta$, the following inequalities hold true: $$\begin{aligned}
{\mathbf{E}^{-1}}(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda})&\leq {{\cal E}^{-1}}(\theta)(1+{\mathrm{O}}(({\lambda\beta^2})^{-1}))  \; ,\\
{\mathbf{M}}(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda})&\leq
\frac{(4\pi)^{1/4}}{({\lambda\beta^2})^{1/12}}[\Gamma((1+\lambda\beta^2)/2)]^{1/2}\beta^{1/2}\left(\frac{\eta}{{\upsilon}^2-\beta^2}\right)^{1/4}\\
& \times{{\cal M}}(\theta)(1+{\mathrm{O}}(({\lambda\beta^2})^{-1})) \;.\end{aligned}$$

For the sake of simplicity we call $\Xi$ the part that precedes the sign “$\times$" in the last inequality.\
According to @olver75, equations (5.12) and (5.13), we have $$\begin{aligned}
U(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda})&=\Xi\left\{{\mathrm{Ai}}(\theta)+{{\cal M}}(\theta){{\cal E}^{-1}}(\theta)\,{\mathrm{O}}(({\lambda\beta^2})^{-1})\right\} \; ,\\
\bar{U}(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda})&=\Xi\left\{{\mathrm{Bi}}(\theta)+{{\cal M}}(\theta){{\cal E}^{-1}}(\theta)\,{\mathrm{O}}(({\lambda\beta^2})^{-1})\right\} \;.\end{aligned}$$ We have, if $s\geq 0$, $x\geq x_2$, so $$\label{vardef}
2/3\beta^2\eta^{3/2}=\int_{x_2}^x g^{1/2}(t)dt=\int_{\beta}^{{\upsilon}}(\tau^2-\beta^2)^{1/2}d\tau \;.$$

For the Airy function, the weight and modulus functions had different definition depending on whether the argument was bigger than the largest root, $c$, of ${\mathrm{Ai}}(z)={\mathrm{Bi}}(z)$ or not. Likewise, the definition of ${\mathbf{E}^{-1}}$ and ${\mathbf{M}}$ depends on the position of the argument with respect to the largest root of the equation $\bar{U}(b,x)=U(b,x)$, which is called $\rho(b)$ in @olver75.

#### Where do the auxiliary variables lie when $\mathbf{s\geq 0}$?

We claim that the answer is that $\theta \geq 0 >
c$, and ${\upsilon}\sqrt{2\lambda}\geq \rho(-{\lambda\beta^2}/2)$.\
The first part of equation (\[vardef\]) implies that $\eta\geq 0$, so $\theta \geq c$, as $c<0$. This means that we can use the definition ${{\cal M}}^2=2{\mathrm{Ai}}{\mathrm{Bi}}$ and ${{\cal E}^{-1}}{{\cal M}}=2^{1/2}{\mathrm{Ai}}$. The second part implies that ${\upsilon}\geq
\beta$; therefore, $2\lambda{\upsilon}^2\geq 2{\lambda\beta^2}\geq \rho(-{\lambda\beta^2}/2)^2$, since by @olver75, equation (5.21), $\rho(b)\leq 2(-b)^{1/2}$ when $b{\rightarrow}-\infty$. This means that we have similar relationships between ${\mathbf{E}^{-1}}$, ${\mathbf{M}}$, $U$, and $\bar{U}$, to the one we had in the Airy case, $\bar{U}$ playing the role of ${\mathrm{Bi}}$, and $U$ playing the role of ${\mathrm{Ai}}$.

#### Consequences of their positions

The interesting consequence of the previous remarks is that we can write, if $N$ is large enough, for all $s\geq 0$ $$\mathbf{E}^{-2}(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda})=\frac{U}{\bar{U}}=\frac{{{\cal M}}(\theta){{\cal E}^{-1}}(\theta)}{{{\cal M}}(\theta){\cal
E}(\theta)} \frac{2^{-1/2}+{\mathrm{O}}(({\lambda\beta^2})^{-1})}{2^{-1/2}+{\mathrm{O}}(({\lambda\beta^2})^{-1})} \;.$$ In other words, we just proved that $\exists N_0$ such that $N>N_0$ implies, $\forall \, s\geq 0$ $${\mathbf{E}^{-1}}(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda}) \leq {{\cal E}^{-1}}(\theta)(1+{\mathrm{O}}(({\lambda\beta^2})^{-1}) \;.$$ By the same arguments, we derive that $${\mathbf{M}}(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda})\leq \Xi {{\cal M}}(\theta)(1+{\mathrm{O}}(({\lambda\beta^2})^{-1})\;.$$

A4: Asymptotic behavior of $K_{n,N}$ {#a4-asymptotic-behavior-of-k_nn .unnumbered}
------------------------------------

The aim here is to show that $$K_{n,N}\sim 2^{2/3} (N/n)^{1/4} \;.$$ $K_{n,N}$ has the following expression: $$K_{n,N}=\frac{(2\lambda)^{1/4}\{\lambda(2+1/2\beta^2)/e\}^{\lambda(1+\beta^2/4)}\sqrt{2}\pi^{1/4}[\Gamma((1+\lambda\beta^2)/2)]^{1/2}\beta^{1/2}}{(\lambda\beta^2)^{1/12}\sqrt{n!N!{\sigma_N}}}
\; .$$ Since $\lambda\beta^2=(2N+1)$, $\Gamma((1+\lambda\beta^2)/2)=\Gamma(N+1)=N!$ .\
In other respects, let $A_n=\{\lambda(2+1/2\beta^2)/e\}^{\lambda(1+\beta^2/4)}/\sqrt{n!}$ . Note that $2\lambda+\lambda\beta^2/2=n-N+(2N+1)/2=n+1/2=n_+$. So $A_n=(n_+/e)^{n_+/2}/\sqrt{n!}$. Using Stirling’s formula, we get that $A_n\sim (n_+/n)^{n/2}(n_+/n)^{1/4}(2\pi {\mathrm{e}})^{-1/4} \sim (2\pi)^{-1/4}$.\
Now rewriting $$K_{n,N}=\frac{A_n(\lambda\beta^2)^{1/4}(8\pi)^{1/4}}{(\lambda\beta^2)^{1/12}\sqrt{{\sigma_N}}} \;,$$ we get that $K_{n,N}\sim 2^{2/3} (N/n)^{1/4}$, from using $A_n(8\pi)^{1/4}\sim \sqrt{2}$ and the second estimates of Fact \[factfunctions\] in **A1**.

A5: Asymptotic properties of $\mathbf{\eta}$ {#a5-asymptotic-properties-of-mathbfeta .unnumbered}
--------------------------------------------

This appendix is divided into two parts. We first show that there exists $s_1$ such that, if $s\geq 2s_1$, $$\begin{gathered}
\frac{2}{3}{\lambda\beta^2}\eta^{3/2} \geq s \;.\tag{P1}\end{gathered}$$ Then we shall show: $$\begin{gathered}
\text{uniformly in }s \in [a,b], \; \; \;(2N+1)^{2/3}\eta=s+{\mathrm{o}}(1) \;.\tag{P2}\end{gathered}$$

### A5.1: Proof of P1 {#a5.1-proof-of-p1 .unnumbered}

This is the argument that was used in **A8** of @imj. We repeat it for the sake of completeness.\
Let us first suppose that $s$ is given. Since $g(x)=(x-x_1)(x-x_2)/(4x^2)$, we have $${\sigma_N}^2g(x) = s\frac{\sigma_N^3}{\lambda}\frac{(x_2-x_1)+s{\sigma_N}/\lambda}{4(x_2+s{\sigma_N}/\lambda)^2} \sim
s\frac{{\sigma_N}^32\sqrt{2}\beta(l+1)^{1/2}}{16 \lambda}\sim s\frac{\beta{\sigma_N}^3}{4\lambda}\;,$$ the first equivalent coming from the fact that when $s$ is fixed, $x_2-x_1 \gg s{\sigma_N}/\lambda$, and $x_2{\rightarrow}2$. The second is just $l{\rightarrow}1$ under ${\mathbf{AB}}$. Now using the first point of Fact \[factfunctions\] in **A1**, together with ${\sigma_N}^2\sim n N^{-1/3}$, we get that $(\beta{\sigma_N}^3)/(4\lambda){\rightarrow}1$. So at $s$ fixed, $${\sigma_N}^2 g(x){\rightarrow}s \;.$$ Having this information let us now pick $s_1=8$. If $N$ is large enough, we have ${\sigma_N}^2 g(x(s_1))\geq s_1/2=4$. For all (fixed) $N$ $g$ is an increasing function of $s$. Therefore for the same $N$ we will have $$\forall \, s\geq s_1 \; {\sigma_N}^2 g(x) \geq {\sigma_N}^2 g(x(s_1))\geq s_1/2=4 \; ,$$ and hence, since $s\geq s_1\geq 0$, $g$ is positive and we have $g^{1/2}(s)\geq 2/{\sigma_N}$. Therefore, $$\frac{2}{3}{\lambda\beta^2}\eta^{2/3}=\lambda \int_{x_2}^xg^{1/2}(t)dt \geq \int_{x(s_1)}^xg^{1/2}(t)dt \geq
\frac{2\lambda}{{\sigma_N}}\frac{{\sigma_N}}{\lambda}(s-s_1)=2(s-s_1)\;.$$ Consequently, if $s\geq 2s_1$, we have (P1).

### A5.2: Proof of P2 {#a5.2-proof-of-p2 .unnumbered}

Without loss of generality, we can suppose that $a$ and $b$ have the same sign, and $a\geq 0$. (If it is not the case, we can split $[a,b]=[a,0]\bigcup[0,b]$, apply the reasoning on each of these, and get the claimed result for the original interval.)\
The idea is that on $[a,b]$, we have $$\frac{(x-x_2)(x_2+b{\sigma_N}/\lambda -x_1)}{4(x_2+a{\sigma_N}/\lambda)^2} \geq g(x)\geq
\frac{(x-x_2)(x_2-x_1+a{\sigma_N}/\lambda)}{4(x_2+b{\sigma_N}/\lambda)^2} \; .$$ Now on both sides, the terms which are not $(x-x_2)$ are $(x_2-x_1)(1+{\mathrm{o}}(1))=4\beta(1+{\mathrm{o}}(1))$, again because ${\sigma_N}/\lambda \ll \beta$. So if we integrate the square root of the previous inequality between $x_2$ and $x(s)$, we get $$\begin{aligned}
2/3(s{\sigma_N}/\lambda)^{3/2}2\sqrt{\beta}(1+{\mathrm{o}}(1))/4 \geq 2/3 \eta^{3/2}\beta^2 \geq
2/3(s{\sigma_N}/\lambda)^{3/2}2\sqrt{\beta}(1+{\mathrm{o}}(1))/4 \;,\end{aligned}$$ or $$\frac{1}{2}s^{3/2}({\sigma_N}^3\beta/\lambda)^{1/2}(1+{\mathrm{o}}(1)) \geq \eta^{3/2}\lambda\beta^2 \geq
\frac{1}{2}s^{3/2}({\sigma_N}^3\beta/\lambda)^{1/2}(1+{\mathrm{o}}(1)) \;.$$ The conclusion follows from **A1**, Fact \[factfunctions\], whose first point, along with the estimate of ${\sigma_N}$ mentioned there, shows that ${\sigma_N}^3\beta/\lambda\sim 4$. We note that (P2) also gives us pointwise convergence of $({\lambda\beta^2})^{2/3}\eta$ to $s$.

A6: Properties of $\mathbf{g}$ {#a6-properties-of-mathbfg .unnumbered}
------------------------------

We first show that $g$ is increasing - at $N$ fixed - as a function of $s$, if $s\geq 0$. Then we give an estimate of $4x^2g(x)$ as $N{\rightarrow}\infty$ and $s\in[a,b]$.

### A6.1: $\mathbf{g}$ is increasing on $s\geq 0$ {#a6.1-mathbfg-is-increasing-on-sgeq-0 .unnumbered}

Since $g(t)=(t-x_2)(t-x_1)/(4t^2)=(t^2-4lt+4)/(4t^2)$, we have $$g'(t)=\frac{l}{t^2}-\frac{2}{t^3}=\frac{lt-2}{t^3} \;.$$ Now $lx_2=2l^2+2l\sqrt{l^2-1}\geq 2$, since $l=1+(2N+1)/\alpha\geq 1$. But $lx\geq lx_2$ when $s\geq 0$, and the assertion is proved.

### A6.2: On the asymptotic behavior of $\mathbf{4x^2g(x)}$ for $\mathbf{s\in [a,b]}$ {#a6.2-on-the-asymptotic-behavior-of-mathbf4x2gx-for-mathbfsin-ab .unnumbered}

This estimate is motivated by the fact that in the course of the proof of the main result, we have to deal with an expression of the form $$\frac{\eta}{x^2-4lx+4} \;.$$ We already studied in detail $\eta$ as a function of $s$ and $N$. We now focus on $x^2-4lx+4$.

Recalling that $x^2-4lx+4=(x-x_2)(x-x_1)$ and $x=x_2+s{\sigma_N}/\lambda$, we have $$x^2-4lx+4=s\frac{{\sigma_N}}{\lambda}(x_2-x_1+s\frac{{\sigma_N}}{\lambda})=s\frac{{\sigma_N}}{\lambda}(x_2-x_1+{\mathrm{o}}(\beta)) \;,$$ because the first estimate in Fact \[factfunctions\] shows that ${\sigma_N}/\lambda = {\mathrm{o}}(\beta)$, and since $s \in
[a,b]$, the previous statement holds true uniformly on this interval. Now $x_2-x_1\sim 4\beta$ under ${\mathbf{AB}}$, and therefore, uniformly on $[a,b]$, $$x^2-4lx+4=s\frac{{\sigma_N}}{\lambda}4\beta(1+{\mathrm{o}}(1)) \; ,$$ as was claimed in \[casesinterval\]. Also, since $x=x_2+s{\sigma_N}/\lambda$, and $x_2=2+(2l+2)^{1/2}\beta+\beta^2$, $$4g(x)=s\frac{{\sigma_N}}{\lambda}\beta(1+{\mathrm{o}}(1)) \;.$$

A7: Limit of $c_{\phi}$ {#a7-limit-of-c_phi .unnumbered}
-----------------------

Recall that under the notation of @imj, $$\sqrt{2} c_{\phi}=\frac{1}{2}\sqrt{a_N}\left(\sqrt{N+\alpha}\int \xi_N
  - \sqrt{N} \int \xi_{N-1}\right)$$ where $\xi_k(x)=x^{\alpha/2-1}e^{(-x/2)}L_k^{(\alpha)}(x)\sqrt{\frac{k!}{(k+\alpha)!}}$. We are interested in\
$$\begin{aligned}
v_{k,\alpha}&=\sqrt{k+\alpha}\int \xi_k - \sqrt{k} \int \xi_{k-1} \\
&=\sqrt{\frac{k!}{(k+\alpha-1)!}}\int_0^{\infty}x^{\alpha/2-1}e^{(-x/2)}
\left(L^{\alpha}_k(x)-L^{\alpha}_{k-1}(x)\right) dx \\
\text{(by \citet{szego} 5.1.13 p.102)} &=
  \sqrt{\frac{k!}{(k+\alpha-1)!}}\int_0^{\infty}x^{\alpha/2-1}e^{(-x/2)} L^{\alpha-1}_k(x) dx \\
&=\sqrt{\frac{k!}{(k+\alpha-1)!}} I_{k,\alpha} \;.\end{aligned}$$ Now using @szego 5.1.9 p.101, $$\sum_{k=0}^{\infty}w^k L_k^{\alpha-1}(x)=(1-w)^{-\alpha} \exp\left(-\frac{xw}{1-w}\right) \;.$$ So if $F(\alpha)=\int_0^{\infty} \left(\sum_{k=0}^{\infty}w^k
  L_k^{\alpha-1}(x)\right)x^{\alpha/2-1}e^{(-x/2)} dx$, we have: $$\begin{aligned}
F(\alpha)&=(1-w)^{-\alpha}\int_0^{\infty} x^{\alpha/2-1}e^{(-x/2)}
{\mathrm{e}}^{-xw/(1-w)}dx\\
&=(1-w)^{-\alpha}\Gamma(\alpha/2)\left(\frac{2(1-w)}{1+w}\right)^{\alpha/2}\\
&=2^{\alpha/2}(1-w^2)^{-\alpha/2}\Gamma(\alpha/2) \;.\end{aligned}$$ Now if $x\geq 0,\hspace{.3cm} |L_n^{\alpha-1}(x)| \leq L_n^{\alpha-1}(-x)$, by 5.1.6 in @szego, and hence $$\left|\sum_{k=0}^{+\infty}w^k L_k^{\alpha-1}(x)\right|\leq \sum_{k=0}^{+\infty}|w|^k
L_k^{\alpha-1}(-x)=(1-|w|)^{-\alpha}\exp\left(\frac{x|w|}{1-|w|}\right) \;.$$ Therefore, as long as $w \in(-1/3,1/3)$, we can switch orders of summation, and get $$\sum_{k=0}^{\infty}w^k I_{k,\alpha}=2^{\alpha/2}(1-w^2)^{-\alpha/2}\Gamma(\alpha/2) \;.$$ But $(1-w)^{-\alpha/2}\Gamma(\alpha/2)=\sum_{k=0}^{+\infty}\frac{\Gamma(\alpha/2+k)}{k!}w^k$, since the right-hand side converges without any difficulty on $(-1/3,1/3)$, and hence $$\sum_{k=0}^{\infty}w^kI_{k,\alpha}=\sum_{m=0}^{\infty}\frac{2^{\alpha/2}\Gamma(\alpha/2+m)}{m!}w^{2m}\;.$$ So we have $$\forall k\in 2\mathbb{N}, \hspace{.5cm} I_{k,\alpha}=\frac{2^{\alpha/2}\Gamma((\alpha+k)/2)}{(k/2)!} \;.$$ Now $v_{k,\alpha}=\sqrt{\frac{k!}{(k+\alpha-1)!}}I_{k,\alpha}=2^{\alpha/2}\frac{\Gamma((\alpha+k)/2)}{\sqrt{(k+\alpha-1)!}}\frac{\sqrt{k!}}{(k/2)!}$. Since $\Gamma(z)\sim (z/e)^z \sqrt{2\pi/z}$, we have $$\begin{aligned}
\frac{\Gamma((\alpha+k)/2)}{\sqrt{\Gamma(k+\alpha)}}&\sim
2^{-(\alpha+k)/2}(\alpha+k)^{-1/4}(2\pi)^{1/4} \sqrt{2} \;, \\
\frac{\sqrt{k!}}{(k/2)!}&\sim 2^{k/2}(\pi k)^{-1/4}2^{1/4} \;,\end{aligned}$$ which in turn leads to $$\begin{aligned}
v_{k,\alpha}&\sim
2^{\alpha/2}(k(\alpha+k))^{-1/4}2^{-(\alpha+k)/2}2^{k/2} \sqrt{2}
\sqrt{2}\\
&\sim 2 (k(\alpha+k))^{-1/4} \\
&\sim 2/\sqrt{a_k} \;.\end{aligned}$$ Hence, as $N$ is even, $\sqrt{2}c_{\phi}=v_{N,\alpha} \sqrt{a_N}/2 {\rightarrow}1 $.

A8: On @sosh Lemma 1 {#a8-on-lemma-1 .unnumbered}
--------------------

For @sosh Lemma 1 to hold true in our case, we have to check two things. First that not only does ${\sigma_N}\phi({\mu_N}+{\sigma_N}s){\rightarrow}{\mathrm{Ai}}(s)/\sqrt{2}$, but also that this is true for the derivative: $$\label{eq:sosh1}
{\sigma_N}^2 \phi'({\mu_N}+{\sigma_N}s) {\rightarrow}\frac{1}{\sqrt{2}} {\mathrm{Ai}}'(s) \; .\tag{S1}$$ We also have to verify that ${\sigma_N}^2 \phi'({\mu_N}+{\sigma_N}s)$ is bounded above by $\Delta(s_0) \exp(-\Delta s)$ on $[s_0,\infty)$, where $\Delta$ is a positive constant. We need to verify this for $\psi$ as well, but the techniques are similar, so we will verify it only for $\phi$.

The second point that we need to check is that $$\label{eq:sosh2}
\int_0^{\infty}\left(\int_0^{z}\phi(u) du \; \psi(y+z)\right) dz {\rightarrow}0 \; \text { as } N {\rightarrow}\infty \;
.\tag{S2}$$

### A8.1: Proof of (\[eq:sosh1\]) {#a8.1-proof-of-eqsosh1 .unnumbered}

It is easy to see that all we need to work on are the properties of $g_N(s)=F_N({\mu_N}+{\sigma_N}s)$; if we can show that ${\sigma_N}F_N'({\mu_N}+{\sigma_N}s) {\rightarrow}{\mathrm{Ai}}'(s)$, and that it is bounded by $\Delta(s_0) {\mathrm{e}}^{-\Delta s}$ on $[s_0,\infty)$, we will be done.\
We have very easily that $$-{\sigma_N}F_N'({\mu_N}+{\sigma_N}s_1)=\int_{s_1}^{\infty} {\sigma_N}^2 \left. \frac{d^2 F_N}{ds^2} \right|_{{\mu_N}+{\sigma_N}u} du \; .$$

So the strategy is clear: we want to show that the integrand in the right-hand side is bounded by an integrable function and that it converges pointwise to ${\mathrm{Ai}}''(u)=u{\mathrm{Ai}}(u)$.\
However, $$\left. \frac{d^2 F_N(x)}{dx^2} \right|_{{\mu_N}+{\sigma_N}u} = \left[ \frac{1}{4}-\frac{{\kappa_N}}{{\mu_N}+{\sigma_N}u}+\frac{\lambda^2-1/4}{({\mu_N}+{\sigma_N}u)^2}\right] F_N({\mu_N}+{\sigma_N}u) \;,$$ and since we already know that $F_N({\mu_N}+{\sigma_N}s){\rightarrow}{\mathrm{Ai}}(s)$, we first need to check that, pointwise, $${\sigma_N}^2 \left[ \frac{1}{4}-\frac{{\kappa_N}}{{\mu_N}+{\sigma_N}s}+\frac{\lambda^2-1/4}{({\mu_N}+{\sigma_N}s)^2}\right] {\rightarrow}s \;.$$ In turn, this reduces to showing that $$\begin{gathered}
{\sigma_N}^2 \left[ \frac{1}{4}-\frac{{\kappa_N}}{{\mu_N}}+\frac{\lambda^2-1/4}{{\mu_N}^2}\right] {\rightarrow}0  \;, \text{ and }\\
\frac{{\sigma_N}^3}{{\mu_N}}\left[\frac{{\kappa_N}}{{\mu_N}}-2 \frac{\lambda^2-1/4}{{\mu_N}^2}\right] {\rightarrow}1 \;.\end{gathered}$$

The first result comes from the remarkable equality ${\kappa_N}/{\mu_N}-\lambda^2/{\mu_N}^2=1/4$, which follows from the fact that if we call $x=\sqrt{N_+/(N+\alpha)_+}$, we have ${\kappa_N}/{\mu_N}=.5-x/(1+x)^2$ and $\lambda^2/{\mu_N}^2=.25-x/(1+x)^2$. Using these estimates, we see that ${\kappa_N}/{\mu_N}-2(\lambda/{\mu_N})^2 =x/(1+x)^2 \sim
\sqrt{N_+/n_+}$, from which we conclude that the second result holds.\
Note that if we changed the centering and scaling (replacing $n$ by $\tilde{n}=n+\alpha$ and $N$ by $\tilde{N}=N+\beta$), by studying the first expression in this case as a “perturbation" of the study we just did, and using the fact that $\mu_{\tilde{N}}-{\mu_N}={\mathrm{O}}(\sqrt{n/N})$, one could show that the first expression is then ${\mathrm{O}}(N^{-1/3})$, and so the result would hold. We also have corresponding results for the second expression. This shows that we have some freedom in the centering and scaling we pick. It is also needed to show that $${\sigma_N}^2 \phi'({\mu_N}+{\sigma_N}s){\rightarrow}{\mathrm{Ai}}(s) \;,$$ since in our splitting of $\phi$, the second part $\phi_{II,N}$ corresponds to parameters $(n-1,N-1)$, but is centered and scaled using ${\mu_N}$ and ${\sigma_N}$, defined with $(n,N)$.

To show that the sequence of functions we are interested in is bounded above by an integrable function, we split $[s_0,\infty)$ into $[s_0,\sqrt{n}]$ and $[\sqrt{n},\infty)$. On the first interval, we can apply the previous results since ${\sigma_N}s/{\mu_N}$ is small compared to 1. So in particular the whole integrand will be smaller that $\Delta(s_0)
(1+|s|)^2 \exp(-s/2)$, after taking into account the properties of $F_N$. On the other hand, on $[\sqrt{n},\infty)$, ${\sigma_N}^2 \leq s^2$, and the denominators involving $s$ are bigger than ${\mu_N}$ and ${\mu_N}^2$ respectively, which gives immediately that the integrand is less than $\Delta(s_0) s^2 \exp(-s/2)$. From this we conclude that the integrand is less than $\Delta(s_0) \exp(-s/4)$, for instance, and that therefore the derivative we are interested in is too.

It then follows easily that (\[eq:sosh1\]) is true, and we also showed that the left-hand side of (\[eq:sosh1\]) is dominated on $[s_0,\infty)$ and for $N>N_0(s_0)$ by $\Delta(s_0){\mathrm{e}}^{-s/4}$.

### A8.2: Proof of (\[eq:sosh2\]) {#a8.2-proof-of-eqsosh2 .unnumbered}

The approach laid out in @sosh p.1044 works after some modifications. We first write $$\int_0^{\infty}\left(\int_0^{z}\phi(u) du \; \psi(y+z)\right) dz = \int_0^{n^{5/8}}\left(\int_0^{z}\phi(u) du \;
\psi(y+z)\right) dz+ \int_{n^{5/8}}^{\infty}\left(\int_0^{z}\phi(u) du \; \psi(y+z)\right) dz \;.$$ Then we can check, via a third order asymptotic development in $x$ of the right-hand side of equation (2.10) in @olver80, that equation (2.18) therein is still true in our case, since, with his notations, $x_N\leq
n^{-3/8}$. Therefore, the analysis carried out after equation (3.21) of the same reference applies, and after integration of the expansion following (3.22) adapted to our situation, we can show that $$\int_0^{n^{5/8}} \phi(u) du = {\mathrm{O}}(n^{-n/16})$$ With this estimate and this splitting of $[0,\infty)$, the rest of Soshnikov’s argument holds true and therefore (\[eq:sosh2\]) can be verified.

A9: A quick look at special functions {#a9-a-quick-look-at-special-functions .unnumbered}
-------------------------------------

In this note, we mentioned three types of special functions, Airy, Whittaker, and parabolic cylinder functions. We recall their definition in this appendix, as well as the main ideas behind some of the transformations Olver used. To justify their introduction, let us say that they play a special role because it is possible, in the setting we were in, to write the functions we studied as a perturbation of the differential equations these functions satisfy.

### A9.1: Airy function {#a9.1-airy-function .unnumbered}

Let us consider the following second order differential equation: $$\frac{d^2w}{dx^2}=xw \;.$$

#### General remark: Recessive solutions

Since these functions are used to get asymptotic expansions, it makes sense to define the independent solutions with respect to their behavior at $+\infty$. Usually, independent solutions $w_1$ and $w_2$ are sought, so that $w_2={\mathrm{o}}(w_1)$ at a particular point of the (extended) real line. In our cases, it will be $\infty$. $w_2$ is called the *recessive* solution. That leaves the problem underdetermined, but with this in mind, one can then give enough constraints so the problem is fully determined, and solve in terms of recessive and dominant solutions. For a more precise definition of recessivity, see @olver, p.155.

In the case of the Airy function, we have for example: (from @olver, 11.1, p.392) $$\begin{aligned}
{\mathrm{Ai}}(x)&= \frac{1}{\pi}\int_0^{\infty}\cos(t^3/3+xt)dt \\
{\mathrm{Bi}}(x)&= \frac{1}{\pi}\int_0^{\infty}\{\exp(-t^3/3+xt)+\sin(t^3/3+xt)\}dt\end{aligned}$$

### A9.2: Whittaker functions {#a9.2-whittaker-functions .unnumbered}

These are solution of the following differential equation $$\label{eq:whittaker}
\frac{d^2 W}{dx^2}=\left(\frac{1}{4}-\frac{\kappa}{z}+\frac{\lambda^2 -1/4}{z^2}\right)W \;.$$ ${W_{\kappa,\lambda}}$, the recessive solution at $\infty$, is obtained by requiring $${W_{\kappa,\lambda}}(x) \sim e^{-x/2}x^{\kappa} \; \text{as } x{\rightarrow}\infty \;.$$ The other solution is $M_{\kappa,\lambda}$, which is required to satisfy $$M_{\kappa,\lambda}(x) \sim x^{\lambda+1/2} \; \text{as } x{\rightarrow}0^+ \;.$$ For more detail on these, see @olver, p.260, or @olver80.

### A9.3: Parabolic cylinder functions {#a9.3-parabolic-cylinder-functions .unnumbered}

According to @olver59, equation (2.9) p.133, [parabolic cylinder functions]{} satisfy (in the case we are interested in) $$\frac{d^2 W}{dx^2}=\left(\frac{1}{4}x^2+a\right)W \;.$$ $U(a,x)$ is chosen to satisfy $$U(a,x) \sim x^{-a-1/2}{\mathrm{e}}^{-x^2/4} \; \text{as } x{\rightarrow}+\infty \;.$$ On the other hand, $\bar{U}$ satisfies $$\bar{U}(a,x) \sim (2/\pi)^{1/2} \Gamma(1/2-a)x^{a-1/2}{\mathrm{e}}^{x^2/4} \; \text{as } x{\rightarrow}+\infty\;.$$ $\bar{U}$’s definition is actually fairly complicated, and can be found in @olver59, equation (2.12) or in @olver75, section 5.1.

### A9.4: On the usage of these functions {#a9.4-on-the-usage-of-these-functions .unnumbered}

As we mentioned earlier, these functions play a central role because it is relatively easy to transform the equations in which we are interested into one of the three mentioned above, or a perturbation of it. Then a range of techniques are available to study the effect of the perturbation, and one can sometimes, and obviously in the case we examine, get asymptotic expansions in terms of the “non-perturbed" solutions. Since these functions are quite well known, information can be gathered about the function of original interest this way.\
For example, in @imj, section 5, after the scaling $\xi=x/\kappa$, the Whittaker equation (\[eq:whittaker\]) becomes $$\frac{d^2W}{d\xi^2}=\left(\kappa^2\frac{(\xi-\xi_1)(\xi-\xi_2)}{4\xi^2}-\frac{1}{4\xi^2}\right)W \;.$$ Using the Liouville-Green transformation $\zeta(d\zeta/d\xi)^2=(\xi-\xi_1)(\xi-\xi_2)/(4\xi^2)$, with $w=(d\zeta/d\xi)^{-1/2} W$, one has $$\frac{d^2w}{d\zeta^2}=\{\kappa^2 \zeta + \psi(\zeta)\} w \;.$$ This is a perturbation of the (scaled) Airy equation, for ${\mathrm{Ai}}(\kappa^{2/3} \zeta)$ and ${\mathrm{Bi}}(\kappa^{2/3} \zeta)$ are solutions of $d^2w/d\zeta^2=\kappa^2 \zeta w$.\
$w$ is not $W$, but it can be related to it, and it is through this mean that Johnstone did his original analysis. As $\psi$ is a relatively involved function of $\xi$ and $\zeta$, we do not explicit it, but just mention that the understanding of $\psi$ is key to getting the uniform bound (\[whattoshow2\]). For more on this, see @imj or @olver, theorem 11.3.1 p.399.

The problem we encountered (and mentioned in \[errorcontrolproblem\]) about the error control function is exactly here: we could not get enough information about the behavior of $\psi$ under ${\mathbf{AB}}$, so we slightly changed approach and turned to other studies.

In @olver80, Olver starts with equation (\[eq:whittaker\]), where the dummy variable was $z$. Writing $x=z/\lambda$ and $l=\kappa/\lambda$, he gets $$\frac{d^2 W}{dx^2}=\left(\lambda^2g(x)-\frac{1}{4x^2}\right)W \;.$$ As he aims to expand the solution in terms of parabolic cylinder functions, he changes variables another time, by writing $$W=\left(\frac{dx}{d\zeta}\right)^{1/2} w \; \; \;, \;
\left(\frac{d\zeta}{dx}\right)^2=\frac{x^2-4lx+4}{4x^2(\zeta^2-\beta^2)} \;,$$ with $\beta=\{2(l-1)\}^{1/2}$. Hence, he gets $$\frac{d^2w}{d\zeta^2}=\{\kappa^2 (\zeta^2-\beta^2) + \psi(\kappa,\beta,\zeta)\} w \;,$$ with $\psi(\kappa,\beta,\zeta)=-\dot{x}^2/(4x^2)+\dot{x}^{1/2}d^2(\dot{x}^{-1/2})/d\zeta^2$. His @olver75 is a study of this type of equations, and in particular of the control of the deviation of the solution of the previous equation to the corresponding parabolic cylinder function. In @olver80, he studies very explicitly the abstract estimate he gets in @olver75 in the case of Whittaker functions. We use this repeatedly in our study, as it is essential to get the crucial property (\[whattoshow2\]).

[^1]: **Acknowledgements:** The author is grateful to Pr. Iain Johnstone for many discussions, key references, and guidance and to Pr. David Donoho for his helpful comments and support. Supported in part by NSF -0140698 and -008584 (ITR). **AMS 2000 SC:** Primary 62E20, Secondary 62H25. **Key words and Phrases :** Principal Component Analysis, largest singular value, Tracy-Widom distribution, Fredholm determinant, Random Matrix Theory, Wishart Matrices. **Contact :** `nkaroui@stanford.edu`
