Asymptotic Behavior of Partition Functions with Graph Laplacian
نویسنده
چکیده
We introduce the matrix sums that represent a discrete analog of the matrix integrals of random matrix theory. The summation runs over the set Γn of all possible n-vertex graphs γn weighted by exp{−β Tr∆n}, β > 0, where ∆n = ∆(γn) is the analog of the Laplace operator determined on γn. Corresponding probability measure on Γn reproduces the well-known Erdős-Rényi ensemble of random graphs. Here it plays the same role as that played by the Gaussian Unitary Invariant Ensemble (GUE) in matrix models. Regarding an analog of the matrix models with quartic potential, we study the cumulant expansion of related partition functions. We develop a diagram technique and describe the combinatorial structure of the coefficients of this expansion in two different asymptotic regimes β = O(1) and β = O(log n) as n → ∞.
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تاریخ انتشار 2008