Topological expansion in isomorphisms with random walks for matrix valued fields

Titus Lupu (1, 2, 3)

1 CNRS - Centre National de la Recherche Scientifique
2 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
3 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We consider Gaussian fields of real symmetric, complex Hermitian or quaternionic Hermitian matrices over an electrical network, and describe how the isomorphisms with random walks for these fields make appear 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 make appear traces of holonomies of the connection along random walk loops parametrized by border cycles of ribbon graphs.

Mots clés: random walks; topological expansion; random matrices; Gaussian unitary ensemble; matrix integrals; holonomy; isomorphism theorems; matrix models; Gaussian free field; Gaussian orthogonal ensemble; discrete gauge theory; ribbon graphs; Wilson loops; Gaussian symplectic ensemble

Domaines: Probabilités [math.PR]

Titus Lupu. Topological expansion in isomorphisms with random walks for matrix valued fields. 2019. ⟨hal-02266669v2⟩