On Connected Diagrams and Cumulants of Erdős-Rényi Matrix Models

Abstract

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.

On asymptotic behavior of Bell polynomials and concentration of vertex degree of large random graphs

“…It would be interesting to study the asymptotic behavior of the moments L (n,ρ) k (q) (2.45) and its centered analogs in the limit k, n, ρ → ∞. It should be noted that k-th cumulant of Y (n,ρ) 1 (q) have been studied in the limit n, ρ → ∞ and given k ∈ N in paper [17].…”

Enumeration of tree-type diagrams assembled from oriented chains of edges

“…k . In the case of q = 2, this has been done in paper [5] with the help of the contour integration method of the Lagrange inversion theorem [6]. Here we give the answer in the general case q ≥ 2.…”

“…whose non-diagonal elements a ij are jointly independent Bernoulli random variables that take values 1 and 0 with probabilities p and 1 − p. Then the family {A} can be regarded as adjacency matrices of the Erdős-Rényi ensemble of random graphs with edge probability p [2]. In paper [5], the following sum has been considered,…”

“…k , where equal random variables a i l i l+1 are joined by an arc. It is shown in [5] that the leading term of the normalized cumulant of (4) in the limiting transition when p = c/n and n, c → ∞ is proportional to a number d…”

“…Equation ( 8) has been derived with the help of the recurrence procedure. For completeness, we repeat arguments of [5] to obtain recurrent relations for the numbers d k that are connected by arcs belong to the same color group. We also say that the edges of D (q) k that does not belong to any color group are grey edges.…”