A Sharp Small Deviation Inequality for the Largest Eigenvalue of a Random Matrix

Abstract

Summary. We prove that the convergence of the largest eigenvalue λ1 of a n × n random matrix from the Gaussian Unitary Ensemble to its Tracy-Widom limit holds in a strong sense, specifically with respect to an appropriate Wasserstein-like distance. This unifying approach allows us both to recover the limiting behaviour and to derive the inequality P(λ1 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.

Spectral Analysis of Large Dimensional Random Matrices

“…9. Suppose that W n = 1 √ n X n is a Wigner matrix and the entries above or on the diagonal of X n are independent but may be dependent on n and may not necessarily be identically distributed.…”

Hankel operators that commute with second-order differential operators

“…This result applies to an orthonormal sequence of functions which is used to analyse the Gaussian orthogonal ensemble as in [1]. Corollary 6.4.…”

“…(i) e −x L (1) n (2x); (ii) (x + s) −1 e −(x+s) ; or (iii) (x + s) −1/2 K ν (2 √ x + s ) where s > 0.…”

Operators associated with soft and hard spectral edges from unitary ensembles

“…(i) The operator W t = e itD 3 P − e −itD 3 = J t P + J t (5.6) on L 2 (R) is an orthogonal projection and the range of FW t F † equals e itξ 3 H 2 .…”

“…, n), see [28, p. 116, A7]. Aubrun [3] proved that the operator W α,β 1/3 = P (α,β) W 1/3 P (α,β) on L 2 (R + ) is of trace class for 0 < α < β ∞, and E σ (2) …”