Title: Permutation matrices, graph independence over the diagonal, and consequences

Abstract:

Often, one tries to understand the behaviour of non-commutative random variables or of von Neumann algebras through matricial approximations. In some cases, such as when appealing to the determinant conjecture or investigating the soficity of a group, it is important to find approximations by matrices with good algebraic conditions on their entries (e.g., being integers). On the other hand, the most common tool for generating asymptotic free independence — conjugating with random unitaries — often destroys such delicate structure.

In this talk, I will first give an introduction to the field of free probability and the study of non-commutative random variables. I will then speak on recent joint work with de Santiago, Hayes, Jekel, Kunnawalkam Elayavalli, and Nelson, where we investigate graph products (an interpolation between free and tensor products) and conjugation of matrix models by large structured random permutation matrices. We show that with careful control of how the permutations are chosen, we can achieve asymptotic graph independence with amalgamation over the diagonal matrices. We are able to use this fine structure to prove that strong $1$-boundedness for a large class of graph product von Neumann algebras follows from the vanishing of the corresponding first $L^2$-Betti number. The main idea here is to show that a version of the determinant conjecture holds as long as the individual algebras have generators with approximations by matrices with entries in the ring of integers of some finite extension of Q satisfying some conditions strongly reminiscent of soficity for groups.