The spectral theory of random matrices provides powerful tools for analyzing high-dimensional data in statistical contexts. Seventy years ago, Nobel Prize-winning physicist Eugene Wigner observed that the spectra of heavy nuclei resemble those of random matrices. This observation not only advanced our understanding of the quantum chaos of these large atoms but also inspired significant mathematical development in random matrix theory through techniques adapted from physics.

 

This talk provides an overview of the historical evolution of this field and highlights recent advances, including work with applications to community detection and error-correction. I will focus on a result that gives the limiting distribution of the largest eigenvalues of the adjacency matrix of a randomly selected regular graph, which proves the existence of optimal spectral expanders of every degree. These findings offer new insights into the interplay between randomness and structure in discrete systems.