On the norm of a random jointly exchangeable matrix

TL;DR

This paper establishes a relationship between the spectral norm of a random jointly exchangeable matrix and its submatrix, with applications to spectral gap estimation in random regular graphs.

Contribution

It introduces a method to estimate the norm of such matrices via their submatrices, linking singular values and spectral gaps in random graph models.

Findings

Norm of exchangeable matrix estimated by submatrix norm

Relation between second largest singular values and submatrix

Spectral gap bounds for random regular graphs

Abstract

In this note, we show that the norm of an $n\times n$ random jointly exchangeable matrix with zero diagonal can be estimated in terms of the norm of its $n/2\times n/2$ submatrix located in the top right corner. As a consequence, we prove a relation between the second largest singular values of a random matrix with constant row and column sums and its top right $n/2\times n/2$ submatrix. The result has an application to estimating the spectral gap of random undirected $d$-regular graphs in terms of the second singular value of {\it directed} random graphs with predefined degree sequences.