# Topological expansion in isomorphism theorems between matrix-valued fields and random walks

## TL;DR

This paper explores the relationship between matrix-valued Gaussian fields and random walks on electrical networks, revealing topological expansions via ribbon graphs and extending to fields twisted by various connections.

## Contribution

It introduces a topological expansion framework for matrix-valued Gaussian fields linked to random walks, including twisted cases with holonomies and boundary cycle considerations.

## Key findings

- Topological expansions are encoded by ribbon graphs.
- Isomorphisms involve traces of holonomies in twisted cases.
- Framework applies to real, complex, and quaternionic matrix fields.

## Abstract

We consider Gaussian fields of real symmetric, complex Hermitian or quaternionic Hermitian matrices over an electrical network, and describe how the isomorphisms between these fields and random walks give rise to topological expansions encoded by ribbon graphs. We further consider matrix-valued Gaussian fields twisted by an orthogonal, unitary or symplectic connection. In this case the isomorphisms involve traces of holonomies of the connection along random walk loops parametrized by boundary cycles of ribbon graphs.