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

Titus Lupu (1, 2, 3, 4)

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
4 SU - Sorbonne Université

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.

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

Domaines: Probabilités [math.PR]

Titus Lupu. Topological expansion in isomorphism theorems between matrix-valued fields and random walks. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2022, 58 (2), pp.695-721. ⟨10.1214/21-AIHP1198⟩. ⟨hal-02266669v3⟩