A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge\thanksref{T1
In a recent paper, Edelman, Guionnet and P\'{ech\'{e conjectured a particular $n^{-1$ correction term of the smallest eigenvalue distribution of the Laguerre unitary ensemble (LUE) of order $n$ in the hard-edge scaling limit: specifically, the derivative of the limit distribution, that is, the density, shows up in that correction term. We give a short proof by modifying the hard-edge scaling to achieve an optimal $O(n^{-2)$ rate of convergence of the smallest eigenvalue distribution. The appearance of the derivative follows then by a Taylor expansion of the less optimal, standard hard-edge scaling. We relate the $n^{-1$ correction term further to the logarithmic derivative of the Bessel kernel Fredholm determinant in the work of Tracy and Widom.
aap1121.tex
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% Institute of Mathematical Statistics (IMI)
% Journal "The Annals of Applied Probabability"

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\newcommand{\eqref}[1]{(\ref{#1})}
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\begin{document}
%
\begin{frontmatter}

%\dochead{}
\title{A note on the expansion of the smallest eigenvalue distribution
of the LUE at the hard edge\thanksref{T1}}
\runtitle{Smallest eigenvalue expansion of LUE}

\begin{aug}
% Corresponding author: Folkmar Bornemann - bornemann@tum.de% Updated by VTEXPTS2LaTeX.exe, 16.07.2015 15:27
%by VTEXPTS2LaTeX.exe, 16.07.2015 13:45
\author[A]{\fnms{Folkmar}~\snm{Bornemann}\corref{}\ead[label=e1]{bornemann@tum.de}}
\runauthor{F. Bornemann}
\affiliation{Technische Universit\"{a}t M\"{u}nchen}
%\dedicated{}
\address[A]{Zentrum Mathematik---M3\\
Technische Universit\"at M\"unchen\\
80290 M\"{u}nchen\\
Germany\\
\printead{e1}}
%\address[]{\\\printead{}}
\end{aug}
\thankstext{T1}{Supported by the DFG-Colla\-bo\-ra\-tive Research Center, TRR
109, ``Discretization in Geometry and Dynamics.''}

% HISTORY:
%
\received{\smonth{4} \syear{2015}}% Updated by VTEXPTS2LaTeX.exe,
%16.07.2015 13:45
%
\revised{\smonth{4} \syear{2015}}% Updated by VTEXPTS2LaTeX.exe,
%16.07.2015 13:45

% ABSTRACT
%
\begin{abstract}
In a recent paper, Edelman, Guionnet and P\'{e}ch\'{e} conjectured a
particular $n^{-1}$
correction term of the smallest eigenvalue distribution of the Laguerre
unitary ensemble
(LUE) of order $n$ in the hard-edge scaling limit:
specifically, the derivative of the limit distribution, that is, the
density, shows up
in that correction term.
We give a short proof by modifying the hard-edge scaling to achieve an
optimal $O(n^{-2})$
rate of convergence of the smallest eigenvalue distribution. The
appearance of the
derivative follows then by a Taylor expansion of the less optimal,
standard hard-edge
scaling. We relate the $n^{-1}$ correction term further to the logarithmic
derivative of the Bessel kernel Fredholm determinant in the work of
Tracy and Widom.
\end{abstract}

% KEYWORDS
% Pirmas kwd is didziosios raides
%
\begin{keyword}[class=AMS]
\kwd[Primary ]{60F05}
%\kwd{}
\kwd[; secondary ]{15B52}
\end{keyword}
%
\begin{keyword}
\kwd{Rate of convergence}
\kwd{random matrix}
\kwd{smallest eigenvalue}
\kwd{LUE}
\end{keyword}
\end{frontmatter}


\setcounter{footnote}{1}


%s1 #&#
\section{Introduction} We recall from random matrix theory that the
smallest eigenvalue distribution of the LUE of order $n$ with parameter
$a>-1$ is given by the Fredholm determinant \cite{ForrBook}, Chapter~9,
%
\[
\prob (\lambda_{\min} \geq s ) = \det \bigl(I- K_n^{a}
\projected{L^2(0,s)} \bigr),
\]
%
induced by the Laguerre projection kernel
%
\[
K_n^a(x,y) = \sum_{k=0}^{n-1}
\phi_k^a(x)\phi_k^a(y),\qquad\phi
_k^a(x) = \sqrt{\frac{k!}{\Gamma(k+a+1)}} e^{-x/2}
x^{a/2} L_k^{a}(x).
\]
%
Here, $L^a_k$ denotes the generalized Laguerre polynomial of degree $k$.
By Christoffel--Darboux and the relation $L^{a-1}_n(x) + L_{n-1}^{a}(x)
= L_n^{a}(x)$ (cf. \cite{MR0167642}, equation~(22.7.30)),
one gets the closed form
%
\[
K_n^a(x,y) = \frac{n!e^{-(x+y)/2}(xy)^{a/2}}{\Gamma(n+a)} \cdot\frac
{L_n^a(x)L_{n}^{a-1}(y)-L_{n}^{a-1}(x)L_n^a(y)}{x-y}.
\]
%
Using the Mehler--Heine type asymptotics (\cite{MR0372517}, Theorem~8.1.3) of the Laguerre polynomials, which holds uniformly
for bounded $z$ in the complex plane,\footnote{Hence,
for all $a\in\R$,
%
\[
z^{-a/2}J_a(2\sqrt{z}) = \sum_{k=0}^\infty
\frac{(-1)^k}{k! \Gamma
(a+k+1)}z^k
\]
%
constitutes a uniquely defined {\em entire} function of the complex
variable $z$.}
%
\[
n^{-a} L_n^{a}(z/n) = z^{-a/2}J_a(2
\sqrt{z}) + o(1) \qquad(n\to \infty)
\]
%
one immediately obtains, as first done by Forrester (\cite{MR1236195}, equation~(2.6); see also \cite{ForrBook}, Section 7.2.1), that in
the hard-edge scaling
%
\[
X = \frac{x}{4n},
\]
%
likewise for $Y$ and $y$, there is the limit, locally uniform for
positive $x$ and $y$,
%
\[
K_n^a(X,Y) \,dX = \bigl(K^a_\infty(x,y)
+ o(1) \bigr) \,dx \qquad (n\to\infty)
\]
%
with the Bessel kernel
%
\begin{eqnarray*}
K_\infty^a(x,y) &=& \frac{\sqrt{y} J_a(\sqrt{x})J_{a-1}(\sqrt
{y})-\sqrt{x} J_{a-1}(\sqrt{x})J_{a}(\sqrt{y})}{2(x-y)}
\\
&=& \frac{\sqrt{y} J_a(\sqrt{x}) J_a'(\sqrt{y}) -\sqrt{x}
J_a'(\sqrt{x}) J_a(\sqrt{y}) } {2(x-y)}.
\end{eqnarray*}
%
Lifted to the convergence of the induced Fredholm determinants (see the
next section for details),
one thus gets the hard-edge scaling limit of the LUE as a limit of
distributions, namely as $n\to\infty$
%
%e1 #&#
\begin{equation}
\label{eq:hard} F_n^a(s) = \prob \biggl(
\lambda_{\min} \geq\frac{s}{4n} \biggr) \to F_\infty^a(s)
= \det \bigl( I - K_\infty^a\projected {L^2(0,s)}
\bigr).
\end{equation}

Based on an identity of finite-dimensional Bessel function determinants
obtained from symbolical and numerical computer experiments, Edelman,
Guionnet and P\'{e}ch\'{e} \cite{1405.7590}, page~14, \emph{conjectured}
the following refinement of (\ref{eq:hard}):
%
%e2 #&#
\begin{equation}
\label{eq:conj} F_n^a(s) = F_\infty^a(s)
+ \frac{a}{2n} s f_\infty^a(s) + O\bigl(n^{-2}
\bigr),\qquad f_\infty^a(s) =\frac{d}{ds}F_\infty^a(s).
\end{equation}
%
In this note, we will give a short proof that this expansion, in fact,
holds true.

%re1 #&#
\begin{rem*} At FoCM'14, Gr\'{e}gory Schehr \cite{Schehr} presented
yet another proof (joint work with Anthony Perret) of
this expansion which
he had obtained as a byproduct of a new approach to the Painlev\'
{e}~III representation \cite{MR1266485} of the Bessel
kernel determinant.
\end{rem*}

%s2 #&#
\section{A short proof of the expansion} Lemma 4.1 of \cite
{1405.7590} easily implies a refinement of the Mehler--Heine type
asymptotics in the
form of an expansion (see also \cite{MR1509498}, page~29, for $a=0$ and
\cite{MR0029451}, page~156, for the
general case), namely,
%
%e3 #&#
%e4 #&#
\begin{eqnarray}
\label{eq:MehlerHeine}
&& (n+a)^{-a} L_n^{a}\bigl(z/(n+a)
\bigr)
\nonumber\\[-8pt]\\[-8pt]\nonumber
&&\qquad = z^{-a/2} J_a(2\sqrt{z}) -\frac{1}{2n}
z^{-(a-2)/2}J_{a-2}(2\sqrt {z})+ O\bigl(n^{-2}\bigr),
\end{eqnarray}
%
uniformly for bounded complex $z$. Subject to the following modified
hard-edge scaling, likewise for $Y$ and $y$,
%
\[
X = \frac{x}{4n} \biggl(1-\frac{a+c}{2n} \biggr),
\]
%
which transforms the Laguerre kernel according to
%
\[
K_n^a(X,Y) \,dX = \tilde K_n^a(x,y)
\,dx,
\]
%
this gives after some routine calculation\footnote{A {\em Mathematica}
notebook checking this result can be found at \arxivurl{arXiv:1504.00235}.}
the expansion
%
%e5 #&#
%e6 #&#
\begin{eqnarray}
\label{eq:kernelexp}
&& (xy)^{-a/2} \tilde K_n^a(x,y)
\nonumber\\[-8pt]\\[-8pt]\nonumber
&&\qquad = (x y)^{-a/2} K_\infty^a(x,y) -\frac
{c}{8n}
x^{-a/2} J_a(\sqrt{x}) y^{-a/2}J_a(
\sqrt{y}) + O\bigl(n^{-2}\bigr),
\end{eqnarray}
%
uniformly for bounded complex $x$ and $y$; in particular, uniformly for
$x,y \in[0,s]$. Because of the
pre-factor $(xy)^{-a/2},$ all the terms in \eqref{eq:kernelexp}
represent {\em entire} kernels; see the footnote in the last section.
Now, such a pre-multiplication of a kernel~$K$ by $(xy)^{-a/2}$ leaves
its Fredholm determinant invariant if we transform the measure defining
the underlying $L^2$-space accordingly:
%
\[
\det (I-K|_{L^2(0,s)} )= \det \bigl(I-(xy)^{-a/2}K|_{L^2((0,s);\nu_a)}
\bigr),\qquad\nu_a(dx) = x^{a} \,dx,
\]
%
which follows simply from conjugating $K$ with the \emph{unitary}
transformation
%
\[
U: L^2(0,s) \to L^2\bigl((0,s),\nu_a\bigr),
\qquad f(x) \mapsto x^{-a/2}f(x).
\]
%
Because of $a>-1$, the transformed measure $\nu_a$ on $[0,s]$ has
finite mass. Since the Fredholm determinant, when
defined with respect
to a measure $\nu$ of finite mass, is locally Lipschitz continuous
with respect to the \emph{uniform} convergence of the kernels (see
\cite{AGZ}, Lemma~3.4.5), the \emph{uniform} kernel expansion (\ref
{eq:kernelexp}) on $(0,s)^2$ immediately lifts to an expansion of the
induced Fredholm
determinants. For the particular choice $c=0$, which eliminates the
$O(n^{-1})$ term, we thus get
%
%e7 #&#
\begin{equation}
\label{eq:new} F_n^a \biggl( \biggl(1-\frac{a}{2n}
\biggr)s \biggr) = F_\infty^a(s) + O\bigl(n^{-2}
\bigr).
\end{equation}
%
Now, a simple Taylor expansion readily establishes \eqref{eq:conj}:
%
\[
F_n^a(s) = F_\infty^a \biggl(
\biggl(1-\frac{a}{2n} \biggr)^{-1}s \biggr) + O
\bigl(n^{-2}\bigr) = F_\infty^a(s) +
\frac{a}{2n} s f^a_\infty(s) + O\bigl(n^{-2}
\bigr).
\]
%
%re2 #&#
\begin{rem*}
The choice $c=-a$ of the additional scaling parameter which is
implicitly used in \eqref{eq:conj} is not the best
possible one; the optimally modified hard-edge scaling is given by
\eqref{eq:new}. Obtaining such a second-order convergence
rate by appropriately modifying the scaling was stimulated by the
corresponding work of Johnstone and Ma \cite{fast} for the \emph
{largest} eigenvalue distributions of the Gaussian unitary ensemble
(GUE) at the soft edge; cf. also the work
of Choup \cite{Choup} on the largest eigenvalue distributions of GUE
and LUE.
\end{rem*}

%s3 #&#
\section{Relation to the Tracy--Widom theory} Writing $\phi_a(x) =
J_a(\sqrt{x})$ for short and, as integral operators
acting on $L^2(0,s)$,
%
\begin{eqnarray*}
\hat K_n^a &=& K_\infty^a +
\frac{a}{8n}\phi_a\otimes\phi_a,
\\
I-\hat K_n^a &=& \bigl(I-K_\infty^a
\bigr) \biggl(I -\frac{a}{8n}\bigl(I-K_n^a
\bigr)^{-1} \phi_a \otimes\phi_a \biggr),
\end{eqnarray*}
%
we get from \eqref{eq:kernelexp} with $c=-a$, by the same reasoning as
in the last section,
%
\begin{eqnarray*}
F_n^a(s) &=& \det\bigl(I-\hat K_n^a
\bigr) + O\bigl(n^{-2}\bigr)
\\
&=& \det\bigl(I-K_\infty^a\bigr) \cdot\det \biggl(I -
\frac{a}{8n}\bigl(I-K_n^a\bigr)^{-1}
\phi_a \otimes\phi_a \biggr) + O\bigl(n^{-2}
\bigr)
\\
&=& F_\infty^a(s) \cdot \biggl(1- \frac{a}{8n}\bigl
\langle\bigl(I-K_n^a\bigr)^{-1}
\phi_a,\phi_a \bigr\rangle_{L^2(0,s)} \biggr) + O
\bigl(n^{-2}\bigr).
\end{eqnarray*}
%
Comparing the $n^{-1}$ term of the expansion with \eqref{eq:conj}
establishes the relation
%
%e8 #&#
\begin{equation}
\label{eq:u} -\frac{1}4 \bigl\langle\bigl(I-K_n^a
\bigr)^{-1} \phi_a,\phi_a \bigr
\rangle_{L^2(0,s)} = s \frac{f^a_\infty(s)}{F^a_\infty(s)} = s \frac{d}{ds} \log
F^a_\infty(s),
\end{equation}
%
which can already be found in the work of Tracy and Widom.
In fact, from the relation $\log\det= \tr\log,$ one gets that the
logarithmic derivative of
a Fredholm determinant $F(s)$ of a trace class operator $K$ acting on
$L^2(0,s)$ is generally given by \cite{MR1266485}, equation~(1.5),
%
\[
\frac{d}{ds}F(s) = \frac{d}{ds} \log\det\bigl(I-K
\projected{L^2(0,s)}\bigr) = -R(s,s),
\]
%
where $R(x,y)$ is the kernel of the operator $K(I-K)^{-1}$.
Specifically, in the case of the Bessel kernel, Tracy and Widom
calculated \cite{MR1266485}, equations~(2.5) and (2.21), that
%
\[
s R(s,s) = \tfrac{1}4 \bigl\langle\bigl(I-K_n^a
\bigr)^{-1} \phi_a,\phi_a \bigr
\rangle_{L^2(0,s)},
\]
%
which finally reproves \eqref{eq:u}.

\section*{Acknowledgement}
The author thanks Jamal Najim for suggesting to publish this note.




%\begin{appendix}
%\section{}
%\end{appendix}

% zodis "Acknowledgments" paliekamas pagal autoriu
%\section*{Acknowledgments}

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\bauthor{\bsnm{Guionnet},~\bfnm{Alice}\binits{A.}} \AND
\bauthor{\bsnm{Zeitouni},~\bfnm{Ofer}\binits{O.}}
(\byear{2010}).
\btitle{An Introduction to Random Matrices}.
\bpublisher{Cambridge Univ. Press},
\blocation{Cambridge}.
\bid{mr={2760897}}
\end{bbook}
%
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G.~W. Anderson, A.~Guionnet, and O.~Zeitouni, \emph{An introduction to random
matrices}, Cambridge University Press, Cambridge, 2010.
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\begin{barticle}[mr]
\bauthor{\bsnm{Choup},~\bfnm{Leonard~N.}\binits{L.~N.}}
(\byear{2006}).
\btitle{Edgeworth expansion of the largest eigenvalue distribution
function of GUE and {LUE}}.
\bjournal{Int. Math. Res. Not.}
\bpages{Art. ID 61049, 32}.
\bid{doi={10.1155/IMRN/2006/61049}, issn={1073-7928}, mr={2233711}}
\bptnote{check pages}%
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L.~N. Choup, \emph{Edgeworth expansion of the largest eigenvalue distribution
function of {GUE} and {LUE}}, Int. Math. Res. Not. (2006), Art. ID 61049,
32pp.
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%b4 #&#
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%
\begin{bmisc}[auto:parserefs-M02]
\bauthor{\bsnm{Edelman},~\bfnm{A.}\binits{A.}},
\bauthor{\bsnm{Guionnet},~\bfnm{A.}\binits{A.}} \AND
\bauthor{\bsnm{P\'ech\'e},~\bfnm{S.}\binits{S.}}
(\byear{2014}).
\bhowpublished{Beyond universality in random matrix theory.
Preprint. Available at \arxivurl{arXiv:1405.7590}.}
\end{bmisc}
%
\iffalse\OrigBibText
A.~Edelman, A.~Guionnet, and S.~P{\'e}ch{\'e}, \emph{Beyond
universality in
random matrix theory}, 2014, e-print arXiv:1405.7590.
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\bibitem{MR1236195}
%
\begin{barticle}[mr]
\bauthor{\bsnm{Forrester},~\bfnm{P.~J.}\binits{P.~J.}}
(\byear{1993}).
\btitle{The spectrum edge of random matrix ensembles}.
\bjournal{Nuclear Phys. B}
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\bpages{709--728}.
\bid{doi={10.1016/0550-3213(93)90126-A}, issn={0550-3213}, mr={1236195}}
\end{barticle}
%
\iffalse\OrigBibText
P.~J. Forrester, \emph{The spectrum edge of random matrix ensembles}, Nuclear
Phys. B \textbf{402} (1993), no.~3, 709--728.
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\bibitem{ForrBook}
%
\begin{bbook}[mr]
\bauthor{\bsnm{Forrester},~\bfnm{P.~J.}\binits{P.~J.}}
(\byear{2010}).
\btitle{Log-Gases and Random Matrices}.
\bpublisher{Princeton Univ. Press},
\blocation{Princeton, NJ}.
\bid{doi={10.1515/9781400835416}, mr={2641363}}
\end{bbook}
%
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P.~J. Forrester, \emph{Log-gases and random matrices}, Princeton
University Press,
Princeton, 2010.
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%
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\bauthor{\bsnm{Johnstone},~\bfnm{Iain~M.}\binits{I.~M.}} \AND
\bauthor{\bsnm{Ma},~\bfnm{Zongming}\binits{Z.}}
(\byear{2012}).
\btitle{Fast approach to the {T}racy--{W}idom law at the edge of GOE
and {GUE}}.
\bjournal{Ann. Appl. Probab.}
\bvolume{22}
\bpages{1962--1988}.
\bid{doi={10.1214/11-AAP819}, issn={1050-5164}, mr={3025686}}
\end{barticle}
%
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I.~M. Johnstone and Z.~Ma, \emph{Fast approach to the {T}racy-{W}idom
law at
the edge of {GOE} and {GUE}}, Ann. Appl. Probab. \textbf{22} (2012), no.~5,
1962--1988.
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%
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\bauthor{\bsnm{Moecklin},~\bfnm{E.}\binits{E.}}
(\byear{1934}).
\btitle{Asymptotische {E}ntwicklungen der {L}aguerreschen {P}olynome}.
\bjournal{Comment. Math. Helv.}
\bvolume{7}
\bpages{24--46}.
\bid{doi={10.1007/BF01292706}, issn={0010-2571}, mr={1509498}}
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%
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\bibitem{Schehr}
%
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\bauthor{\bsnm{Schehr},~\bfnm{G.}\binits{G.}}
(\byear{2014}).
\bhowpublished{On the smallest eigenvalue at the hard edge of the Laguerre
ensemble of complex random matrices: {$1/N$} corrections, talk at FOCM'14,
December 16, 2014, Montevideo, joint work with Anthony Perret,
available at
the \surl{http://lptms.u-psud.fr/gregory-schehr/files/2014/12/Talk\_FOCM.pdf}.}
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%
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G.~Schehr, \emph{On the smallest eigenvalue at the hard edge of the {Laguerre}
ensemble of complex random matrices: {$1/N$} corrections}, talk at FOCM'14,
December 16, 2014, Montevideo, joint work with Anthony Perret,
available at
the URL {\tt
http://lptms.u-psud.fr/gregory-schehr/files/2014/12/Talk\_FOCM.pdf}, 2014.
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%b10 ###
%b10 #&#
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%
\begin{bbook}[mr]
\bauthor{\bsnm{Szeg{\H{o}}},~\bfnm{G.}\binits{G.}}
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\bpublisher{Amer. Math. Soc.},
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%
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G.~Szeg{\H o}, \emph{Orthogonal polynomials}, 4th ed., American Mathematical
Society, Providence, 1975.
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%b11 ###
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(\byear{1994}).
\btitle{Level spacing distributions and the {B}essel kernel}.
\bjournal{Comm. Math. Phys.}
\bvolume{161}
\bpages{289--309}.
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C.~A. Tracy and H.~Widom, \emph{Level spacing distributions and the {B}essel
kernel}, Comm. Math. Phys. \textbf{161} (1994), 289--309.
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%b12 ###
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%
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\bjournal{Ann. Mat. Pura Appl. (4)}
\bvolume{26}
\bpages{141--175}.
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%
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F. Tricomi, \emph{Sulle funzioni ipergeometriche confluenti}, Ann. Mat.
Pura Appl. \textbf{26} (1947), 141--175.
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