Benoît Collins: Strong convergence for random permutations
Given a non-commutative polynomial in non-commuting variables and their adjoints, we are interested in the sequence of random matrices obtained by replacing the variables by i.i.d random matrix permutations. The global spectral behavior of this sequence of random matrices in the limit of large dimension is very well understood thanks to free probability, however the behavior of the largest eigenvalues and the existence of non-trivial outliers was unknown until recently. This problem is closely related to (and actually almost equivalent to) an important problem in graph theory known as Alon’s generalized second eigenvalue conjecture. The purpose of this talk is to explain the relation between these problems and to describe the answer. The solution to these problems requires introducing new tools in random matrix theory, in particular, non-backtracking matrix valued operators, which we will try to describe. Time allowing, we will also mention additional applications of our tools to the study of random tensors. This talk is based on joint works with Charles Bordenave (to appear in Annals of Mathematics 2019, plus work in preparation).
Dr Benoît Collins, Kyoto University