Vorträge in der Woche 07.04.2025 bis 13.04.2025

Donnerstag, 10.04.2025: Spectral measure for uniform d-regular digraphs

Dr. Arka Adhikari (University of Maryland)

Consider the matrix $\sfA_\GG$ chosen uniformly at random from the finite set of all $N$-dimensional matrices of zero main-diagonal and binary entries, having each row and column of $\sfA_\GG$ sum to $d$. That is, the adjacency matrix for the uniformly random $d$-regular simple digraph $\GG$. Fixing $d \ge 3$, it has long been conjectured that as $N \to \infty$ the corresponding empirical eigenvalue distributions converge weakly, in probability, to an explicit non-random limit, given by the Brown measure of the free sum of $d$ Haar unitary operators. We reduce this conjecture to bounding the decay in $N$ of the probability that the minimal singular value of the shifted matrix $\sfA(w) = \sfA_\GG - w \sfI$ is very small. While the latter remains a challenging task, the required bound is comparable to the recently established control on the singularity of $\sfA_\GG$. The reduction is achieved here by sharp estimates on the behavior at large $N$, near the real line, of the Green's function (aka resolvent) of the Hermitization of $\sfA(w)$, which is of independent interest. Joint w/ A. Dembo