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arXiv · 1512.09065

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

Oleksiy Khorunzhiy
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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 $Γ_N^{(R)}$ of the long-range percolation radius model with the edge probability determined by a function $ϕ(t)$. 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 $Γ_N^{(R)}$ given by $ϕ_1 = \int ϕ(t) dt$. This measure converges in the limit of infinite $ϕ_1$ 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.

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Oleksiy Khorunzhiy. 2017-12-05. Limiting eigenvalue distribution of random matrices of Ihara zeta function of long-range percolation graphs. https://arxiv.org/abs/1512.09065

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