CLT for spectra of submatrices of Wigner random matrices, II: Stochastic evolution

Summary

We show that the global fluctuations of spectra of GOE and GUE matrices and their principal submatrices executing Dyson's Brownian motion are Gaussian in the limit of large matrix dimensions. For nested submatrices one obtains a limiting three-dimensional generalized Gaussian process; its restrictions to two-dimensional sections that are monotone in matrix sizes and time moments coincide with the two-dimensional Gaussian free field with zero boundary conditions. The proof is by moment convergence, and it extends to more general Wigner matrices and their stochastic evolution.

Introduction. The fact that the global spectral fluctuations of a GOE or a GUE random matrix evolving under Dyson's Brownian motion, are asymptotically Gaussian is well-known; see [Anderson et al. 2010, Section 4.3.3] and references therein, and also [Spohn 1998] for a generalβ analog. On the other hand, it was shown in [Borodin 2014] that the global fluctuations of spectra of various principal submatrices of a single GOE or GUE matrix are also Gaussian. The goal of this article is to put these two statements together.

We prove the asymptotic Gaussian behavior for submatrices of a class of stochastically evolving Wigner random matrices that includes Dyson's Brownian motion for GOE and GUE. The proof is by the method of moments, and the argument is slightly more general than the one presented in [Anderson et al. 2010] for a single Wigner matrix. We also compute the resulting covariance kernel explicitly. In the case of nesting submatrices, it represents a three-dimensional generalized Gaussian process, where one dimension comes from the position of the spectral variable, the second dimension reflects the size of the submatrix, and the third dimension is the time variable. When restricted to the two-dimensional sections that are monotone in matrix size and time variables, it reproduces the two-dimensional Gaussian free field (GFF) with zero boundary conditions.

In the case of GUE, the appearance of GFF on monotone sections could have been predicted from the determinantal structure of the correlation functions

[Ferrari and Frings 2010; Adler et al. 2010b], and from the analysis of [Borodin and Ferrari 2014] that showed how such a structure leads to GFF covariances in the global asymptotic regime. However, the complete three-dimensional covariance structure seems to be inaccessible via that approach for example because the spectra of the full set of submatrices evolve in a non-Markovian way [Adler et al. 2010a].