# Ramanujan Graphs and Free Probability

# Ramanujan Graphs and Free Probability

We use the method of interlacing polynomials and a finite dimensional analog of free probability to prove the existence of bipartite Ramanujan graphs of every degree and number of vertices. No prior knowledge of Ramanujan graphs or free probability will be assumed. Ramanujan graphs are defined in terms of the eigenvalues of their adjacency or Laplacian matrices. In this spectral perspective, they are the best possible expanders. Infinite families of Ramanujan graphs were first shown to exist by Margulis and Lubotzky, Phillips and Sarnak using Deligne's proof of the Ramanujan conjecture. These constructions were sporadic, only producing graphs of special degrees and numbers of vertices. In this talk, we outline an elementary proof of the existence of bipartite Ramanujan graphs of very degree and number of vertices. We do this by considering the expected characteristic polynomial of a random d-regular graph. We develop finite analogs of results in free probability to compute this polynomial and to bound its roots. By proving that this polynomial is the average of polynomials in an interlacing family, we then prove there exists a graph in the distribution whose eigenvalues satisfy the Ramanujan bound.