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

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