On eigenvalue distribution of random matrices of Ihara zeta function of large random graphs

Abstract

We consider the ensemble of real symmetric random matrices $H^{(n,\rho )}$ obtained from the determinant form of the Ihara zeta function of random graphs that have $n$ vertices with the edge probability $\rho /n$. We prove that the normalized eigenvalue counting function of $H^{(n,\rho )}$ weakly converges in average as $n,\rho \to \infty$ and $\rho =o(n^{\alpha })$ for any $\alpha >0$ to a shift of the Wigner semi-circle distribution. Our results support a conjecture that the large Erdos-Rényi random graphs satisfy in average the weak graph theory Riemann Hypothesis.

arXiv commentversion 5: slightly corrected with respect to version 4