  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.
