Universality at the Edge of the Spectrum¶in Wigner Random Matrices

Abstract

We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit n → +∞. As a corollary, we show that, after proper rescaling, the 1st, 2nd, 3rd, etc. eigenvalues of Wigner random hermitian (or real symmetric) matrix weakly converge to the distributions established by Tracy and Widom in G.U.E.

The largest eigenvalue of small rank perturbations of Hermitian random matrices

“…It is shown in [12] that the above result actually holds for a wide class of random matrices M N (µ, µ ′ ) with centered distributions µ, µ ′ .…”

Universality results for the largest eigenvalues of some sample covariance matrix ensembles

“…By Proposition 2.3, (25) and the above, their contribution to E[TrM s N N ] is then at most of order (see also [29], p725)…”

“…One can then deduce from universality of moments of traces that the asymptotic joint distribution of the largest eigenvalues for any ensemble considered here is the same as for the corresponding Wishart ensemble. The detail of the derivation of such a result from formula (7), including the required asymptotics of correlation functions for Wishart ensembles, can be found in [29], [30] and [6]. The improvement we obtain with respect to [30] is actually that Formula (7) holds for any value γ.…”

“…The terminology we use is close to the one used in [29], [27], [28] and [30]. We recall the main definitions that will be needed here and also assume that the reader is acquainted with most of the techniques used in the above papers.…”

Definitions and Basic Properties

“…Our attention is restricted to global properties since they are of higher relevance for our further analysis. The reader interested in local properties of random matrix spectra is referred to Mehta [87], Tracy, Widom [107,108,109], where the properties of level spacings are derived, and to Tracy, Widom [108,110,112], Soshnikov [101,104], Johnstone [76], for the results related to the distribution of extremal eigenvalue from Hermite and Laguerre ensembles.…”