Nicholas Cook, UCLA, EUA

Título: Random regular digraphs: singularity and discrepancy.

Resumen: We show that the adjacency matrix of a uniform random regular directed graph is invertible with high probability, assuming that the graph is sufficiently dense. This is an important step toward proving that this random matrix ensemble lies in the circular law universality class. The main challenge is to overcome the dependencies among the matrix entries. Our approach makes use of local symmetries of the matrix distribution to ``inject" independent random variables into the problem. We also prove some discrepancy properties for the distribution of edges in the graph, which may be of independent interest.