Speaker: Benoît Collins
Affiliation: University of Ottawa & CNRS
Title: Convergence of unitary matrix integrals.
Time and Place: Thursday, February 22, 2:30-3:30pm, Milner 317.
Abstract: In this joint work with A. Guionnet and E. Maurel-Segala, we introduce the Schwinger-Dyson equation associated to a potential V on the set of tracial states of the free *-algebra generated by X_1=X_1^*, ..., X_n=X_n^*. If the potential is zero, the solutions of this equation are free product states. We prove that, for a prescribed spectral measure of the X_i's and for V small enough, the solution of this equation exists, is still unique and even analytic in V. Our techniques come from random matrix theory. As a by-product, we provide new examples of Connes-embeddable von Neumann algebras. Moreover, we solve the problem of convergence of unitary matrix integrals, and obtain new results on free entropy.