The local relaxation flow approach to universality of the local statistics for random matrices

Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer; Yin, Jun

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 1-46.

Nous présentons une généralisation de la méthode du flot de relaxation locale servant à établir l'universalité des statistiques spectrales locales d'une vaste classe de grandes matrices aléatoires. Nous démontrons que la distribution locale des valeurs propres coïncide avec celle de l'ensemble gaussien pourvu que la loi des coefficients individuels de la matrice soit lisse et que les valeurs propres ${\left\{x_j\right\}}_{j=1}^N$ soient près de leurs quantiles classiques ${\left\{{\gamma }_j\right\}}_{j=1}^N$ determinées par la densité limite des valeurs propres. Dans la normalisation où la distance typique entre les valeurs propres voisines est d'ordre 1/N , la borne a priori nécessaire sur la position des valeurs propres nécessite uniquement l'établissement de $𝔼|x_j-{\gamma }_j{|}^2\le N^{-1-\epsilon }$ en moyenne. Cette information peut être obtenue par des méthodes bien établies pour divers ensembles de matrices. Nous illustrons la méthode en démontrant l'universalité spectrale locale pour des matrices de covariance.

We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the distribution of the individual matrix element is smooth and the eigenvalues ${\left\{x_j\right\}}_{j=1}^N$ are close to their classical location ${\left\{{\gamma }_j\right\}}_{j=1}^N$ determined by the limiting density of eigenvalues. Under the scaling where the typical distance between neighboring eigenvalues is of order 1/N , the necessary apriori estimate on the location of eigenvalues requires only to know that $𝔼|x_j-{\gamma }_j{|}^2\le N^{-1-\epsilon }$ on average. This information can be obtained by well established methods for various matrix ensembles. We demonstrate the method by proving local spectral universality for sample covariance matrices.

Classification : 15B52, 82B44
Mots clés : random matrix, sample covariance matrix, Wishart matrix, Wigner-Dyson statistics

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