Joint large deviation result for empirical measures of the coloured random geometric graphs

We prove joint large deviation principle for the \emph{ empirical pair measure} and \emph{empirical locality measure} of the \emph{near intermediate} coloured random geometric graph models on $n$ points picked uniformly in a $d-$dimensional torus of a unit circumference.From this result we obtain large deviation principles for the \emph{number of edges per vertex}, the \emph{degree distribution and the proportion of isolated vertices } for the \emph{near intermediate} random geometric graph models.Comment: 13 pages. arXiv admin note: substantial text overlap with arXiv:1312.632

Asymptotics of the partition function of Ising model on inhomogeneous random graphs

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 empirical bond distribution, which enumerates the number of bonds between a given couple of spins, and 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.Comment: 14 page

Large deviation principles for empirical measures of colored random graphs

For any finite colored graph we define the empirical neighborhood measure, which counts the number of vertices of a given color connected to a given number of vertices of each color, and the empirical pair measure, which counts the number of edges connecting each pair of colors. For a class of models of sparse colored random graphs, we prove large deviation principles for these empirical measures in the weak topology. The rate functions governing our large deviation principles can be expressed explicitly in terms of relative entropies. We derive a large deviation principle for the degree distribution of Erd\H{o}s--R\'{e}nyi graphs near criticality.Comment: Published in at http://dx.doi.org/10.1214/09-AAP647 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org