The asymptotic distribution of a single eigenvalue gap of a Wigner matrix

Abstract

We show that the distribution of (a suitable rescaling of) a single eigenvalue gap ${\lambda }_{i+1}(M_n)-{\lambda }_i(M_n)$ of a random Wigner matrix ensemble in the bulk is asymptotically given by the Gaudin-Mehta distribution, if the Wigner ensemble obeys a finite moment condition and matches moments with the GUE ensemble to fourth order. This is new even in the GUE case, as prior results establishing the Gaudin-Mehta law required either an averaging in the eigenvalue index parameter $i$, or fixing the energy level $u$ instead of the eigenvalue index. The extension from the GUE case to the Wigner case is a routine application of the Four Moment Theorem. The main difficulty is to establish the approximate independence of the eigenvalue counting function $N_{(-\infty ,x)}({\tilde{M}}_n)$ (where ${\tilde{M}}_n$ is a suitably rescaled version of $M_n$) with the event that there is no spectrum in an interval $[x,x+s]$, in the case of a GUE matrix. This will be done through some general considerations regarding determinantal processes given by a projection kernel.

21 pages, no figures, submitted, Prob. Thy. and Related Fields. This is the final version, incorporating the referee comments

References in corpus (1)

• Bulk Universality for Wigner Matrices

Cited by in corpus (1)

• Random matrices, log-gases and Holder regularity