Asymptotics of the Partition Function of Ising Model on Inhomogeneous Random Graphs

Abstract

For a finite random graph, we defined a simple model of statistical mechanics. We obtain an annealed asymptotic result for the random partition function for this model on finite random graphs as $n,$ the size of the graph is very large. To obtain this result, we define the \emph{ empirical bond distribution}, which enumerates the number of bonds between a given couple of spins, and \emph{ empirical spin distribution}, which enumerates the number of sites having a given spin on the spinned random graphs. For these empirical distributions we extend the large deviation principle(LDP) to cover random graphs with continuous colour laws. Applying Varandhan Lemma and this LDP to the Hamiltonian of the Ising model defined on Erdos-Renyi graphs, expressed as a function of the empirical distributions, we obtain our annealed asymptotic result.