We study the asymptotic behavior of single variable Bell polynomials B k (x) and polynomials Bk (x) related with restricted Bell numbers in the limit of infinite k and x > 0. We discuss our results in connection with the spectral theory of random matrices and properties of the vertex degree of large random graphs.

Regarding the adjacency matrices of n-vertex graphs and related graph Laplacian, we introduce two families of discrete matrix models constructed both with the help of the Erdős-Rényi ensemble of random graphs. Corresponding matrix sums represent the characteristic functions of the average number of walks and closed walks over the random graph. These sums can be considered as discrete analogs of the matrix integrals of random matrix theory.We study the diagram structure of the cumulant expansions of logarithms of these matrix sums and analyze the limiting expressions as n → ∞ in the cases of constant and vanishing edge probabilities.

We consider a dilute version of the Wigner ensemble of n × n random real symmetric matrices H (n,ρ) , where ρ denotes the average number of non-zero elements per row. We study the asymptotic properties of the moments M (n,ρ) 2s = E Tr(H (n,ρ) ) 2s in the limit when n, s and ρ tend to infinity.Our main result is that the sequence M (n,ρn) 2snwith sn = ⌊χρn⌋, χ > 0, ρn → ∞ and ρn = o(n 1/5 ) is asymptotically close to a sequence of numbers nmsn , where {ms } s≥0 are determined by an explicit recurrence that involves the second and the fourth moments of the random variables (H (n,ρ) )ij , V2 and V4, respectively. This recurrent relation generalizes the one that determines the moments of the Wigner's semicircle law given by ms = limρ→∞ms(ρ), s ∈ N. It shows that the spectral properties of random matrices at the edge of the limiting spectrum regarded in the asymptotic regime of the strong dilution essentially differ from those observed in the cases of moderate and weak dilution, where the dependence on the fourth moment V4 does not intervene.

We consider the ensemble of N ×N real random symmetric matrices H (R) N * MSC: 05C50, 05C80, 15B52, 60F99 arXiv:1512.09065v3 [math-ph]

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