Addition to "Structured random matrices and cyclic cumulants: A free probability approach"

TL;DR

This work strengthens the framework of structured random matrix ensembles by refining axiom (iv) to control cumulants of disjoint cycles, enabling analysis of products of traces and other nonlinear transformations within a free-probability-based approach. It develops a diagrammatic and partition-based toolkit, leveraging local $U(1)$ invariance and non-crossing partition structure to establish precise $N$-scaling for cyclic cumulants and to show self-averaging of traces under nonlinear map operations. The authors show invariance of the axioms under polynomial and entrywise polynomial transformations, and provide detailed partition-based arguments to justify the leading-order scaling for cumulants of disjoint cycles, including Proposition 1 and Proposition 2. These results broaden the applicability of structured random matrix ensembles to nonlinear functionals while preserving stability properties, with example ensembles and discussions relevant to QSSEP and other unitary-invariant settings.

Abstract

We give a refined definition of the class of random matrix ensembles introduced in our paper "Structured random matrices and cyclic cumulants: A free probability approach" (arXiv:2309.14315) by extending the so-called fourth axiom to deal with cumulants of disjoint cycles. We argue that the theorems concerning the stability of such ensembles under non-linear transformations still hold with these refined axioms.

Theorems & Definitions (4)

• Definition : Axioms of structured random matrix ensembles
• Proposition 1: Invariance under polynomial operations
• Proposition 2: Invariance under entry-wise operations
• proof