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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 $\phi(t)$. 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 $\phi_1 = \int \phi(t) dt$. This measure converges in the limit of infinite $\phi_1$ 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.

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