Eigenvalue location in random regular graphs

Random regular graphs form a ubiquitous model for chaotic systems. However, the spectral properties of their adjacency matrices have proven difficult to analyze because of the strong dependence between different entries. In this talk, I will describe recent work that shows that despite this, the fluctuation of eigenvalues of the adjacency matrix are of the same order as for Gaussian matrices. This gives an optimal approximation of the second eigenvalue, which controls the spectral gap of the graph. We find this through tight analysis of the Green's function of the adjacency operator and an analysis of the change of the Green's function after a random edge switch.

This is based on joint work with Jiaoyang Huang and Horng-Tzer Yau.