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On Connected Diagrams and Cumulants of Erdős-Rényi Matrix Models

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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 analogues 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.

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  1. Département de Mathématiques, Université de Versailles – Saint-Quentin, Versailles, 78035, France

    O. Khorunzhiy

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  1. O. Khorunzhiy
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Correspondence to O. Khorunzhiy.

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Communicated by M. Aizenman

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Khorunzhiy, O. On Connected Diagrams and Cumulants of Erdős-Rényi Matrix Models. Commun. Math. Phys. 282, 209–238 (2008). https://doi.org/10.1007/s00220-008-0533-2

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  • DOI: https://doi.org/10.1007/s00220-008-0533-2

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