Spectra of Overlapping Wishart Matrices and the Gaussian Free Field

TL;DR

This paper studies the spectral properties of overlapping Wishart matrices derived from a doubly-infinite array of iid variables, revealing their eigenstatistics converge to a Gaussian vector linked to the Gaussian Free Field.

Contribution

It introduces a novel analysis of eigenstatistics for overlapping Wishart matrices and connects their covariance structure to the Gaussian Free Field.

Findings

Eigenstatistics converge to a Gaussian vector with a specific covariance structure.

Connection established between eigenstatistics and the Gaussian Free Field.

Generalization from univariate polynomials to planar functions.

Abstract

Consider a doubly-infinite array of iid centered variables with moment conditions, from which one can extract a finite number of rectangular, overlapping submatrices, and form the corresponding Wishart matrices. We show that under basic smoothness assumptions, centered linear eigenstatistics of such matrices converge jointly to a Gaussian vector with an interesting covariance structure. This structure, which is similar to those appearing in work of Borodin, Borodin and Gorin, and Johnson and Pal can be described in terms of the height function, and leads to a connection with the Gaussian Free Field on the upper half-plane. Finally, we generalize our results from univariate polynomials to a special class of planar functions.