Convergence rate of the diagonal-valued Cauchy-transform for permutation invariant random matrices

TL;DR

This paper establishes quantitative bounds on the convergence rate of the diagonal-valued Cauchy-transform for permutation invariant random matrices, especially in sparse and bounded cases, by linking it to a graph adjacency operator.

Contribution

It provides explicit bounds on the convergence rate of the diagonal resolvent for permutation invariant matrices and constructs the free sum as a graph adjacency operator.

Findings

Bound on the difference between resolvent diagonals

Improved convergence rate for sparse matrices

Explicit construction of free sum as a graph adjacency operator

Abstract

Let $A$ be a permutation invariant random matrix and $B$ another random matrix. We give a quantitative bound on the difference between the diagonal of the resolvent of $A+B$ and the diagonal of the resolvent of the free sum with amalgamation over the diagonal of $A$ and $B$. Moreover, we improve the rate of convergence whenever the matrices $A$ and $B$ are sparse and bounded in operator norm. Doing so, we explicitly construct the free sum over the diagonal of $A$ and $B$ as an adjacency operator of a weighted locally finite graph.