 
Summary: A Sharp Small Deviation Inequality for the
Largest Eigenvalue of a Random Matrix
Guillaume Aubrun
Universit’e de Paris 6, Institut de Math’ematiques, ’ Equipe d'Analyse Fonctionnelle,
Boite 186, 4 place Jussieu, 75005 PARIS
Summary. We prove that the convergence of the largest eigenvalue #1 of a n ×
n random matrix from the Gaussian Unitary Ensemble to its TracyWidom limit
holds in a strong sense, specifically with respect to an appropriate Wassersteinlike
distance. This unifying approach allows us both to recover the limiting behaviour and
to derive the inequality P(#1 # 2+t) # C exp(cnt 3/2 ), valid uniformly for all n and
t. This inequality is sharp for ``small deviations'' and complements the usual ``large
deviation'' inequality obtained from the Gaussian concentration principle. Following
the approach by Tracy and Widom, the proof analyses several integral operators,
which converge in the appropriate sense to an operator whose determinant can be
estimated.
Key words: Random matrices, largest eigenvalue, GUE, small deviations.
Introduction
Let Hn be the set of ndimensional (complex) Hermitian matrices. The general
element of Hn is denoted A (n) , and its entries are denoted (a (n)
ij ).
