Finite Free Probability and Ramanujan Graphs

Abstract: We introduce a finite analog of Voiculescu's Free Probability that allows us to compute the expected characteristic polynomials of certain random matrices and to prove bounds on the locations of the roots of those polynomials. We sketch how this theory may be used to prove that there exist bipartite Ramanujan graphs of every degree and number of vertices.

No prior knowledge of free probability or Ramanujan graphs will be assumed.

This is joint work with Adam Marcus and Nikhil Srivastava.

combinatorics

Audience: researchers in the topic

Warwick Combinatorics Seminar

Series comments: This is the online combinatorics seminar at Warwick.