Random incidence matrices: moments of the spectral density

We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices : any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semi-circle of "small" eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit), we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e=2.72... is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix. Keywords: random graphs, random matrices, sparse matrices, incidence matrices spectrum, momentsComment: 39 pages, 9 figures, Latex2e, [v2: ref. added, Sect. 4 modified

Spectral exponential sums on hyperbolic surfaces I

We study an exponential sum over Laplace eigenvalues $\lambda_{j} = 1/4+t_{j}^{2}$ with $t_{j} \leqslant T$ for Maass cusp forms on $\Gamma \backslash \mathbb{H}$ as $T$ grows, where $\Gamma \subset PSL_{2}(\mathbb{R})$ is a cofinite Fuchsian group acting on the upper half-plane $\mathbb{H}$. Specifically, for the congruence subgroups $\Gamma_{0}(q), \, \Gamma_{1}(q)$ and $\Gamma(q)$, we explicitly describe each sum in terms of a certain oscillatory component, von Mangoldt-like functions and the Selberg zeta function. We also establish a new expression of the spectral exponential sum for a general cofinite group $\Gamma \subset PSL_{2}(\mathbb{R})$, and in particular we find that the behavior of the sum is decisively determined by whether $\Gamma$ is essentially cuspidal or not. We also work with certain moonshine groups for which our plotting of the spectral exponential sum alludes to the fact that the conjectural bound $O(X^{1/2+\epsilon})$ in the Prime Geodesic Theorem may be allowable. In view of our numerical evidence, the conjecture of Petridis and Risager is generalized.Comment: 20 pages, 3 figures, comments welcom