Limiting eigenvalue distribution of random matrices of Ihara zeta function of long-range percolation graphs

Abstract

We consider the ensemble of $N\times N$ real random symmetric matrices $H_N^{(R)}$ obtained from the determinant form of the Ihara zeta function associated to random graphs ${\Gamma }_N^{(R)}$ of the long-range percolation radius model with the edge probability determined by a function $\varphi (t)$. We show that the normalized eigenvalue counting function of $H_N^{(R)}$ weakly converges in average as $N,R\to \infty$, $R=o(N)$ to a unique measure that depends on the limiting average vertex degree of ${\Gamma }_N^{(R)}$ given by ${\varphi }_1=\int \varphi (t)dt$. This measure converges in the limit of infinite ${\varphi }_1$ to a shift of the Wigner semi-circle distribution. We discuss relations of these results with the properties of the Ihara zeta function and weak versions of the graph theory Riemann Hypothesis.

revised version: 38 pages, 2 figures