Topological expansion in Dynkin type isomorphisms 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 symmetric or Hermitian matrices over an electrical network, and describe how Dynkin type isomorphisms with random walks for these fields make appear topological expansions encoded by ribbon graphs. A particular case of this, in continuum, is that of a Dyson's Brownian motion for β equal to 1 or 2. We further consider matrix valued Gaussian fields twisted by an orthogonal or unitary connection. In this case the isomorphisms make appear traces of holonomies of the connection along random walk loops parametrized by cycles of ribbon graphs.

Mots clés: discrete gauge theory; Dyson's Brownian motion; Gaussian orthogonal ensemble; Gaussian free field; isomorphism theorems; Gaussian unitary ensemble; random matrices; random walks; matrix models; matrix integrals; local times; holonomy; ribbon graphs; topological expansion

Domaines: Probabilités [math.PR]

Titus Lupu. Topological expansion in Dynkin type isomorphisms for matrix valued fields. 2019. ⟨hal-02266669v1⟩