Joint Large Deviation principle for empirical measures of the d-regular random graphs

TL;DR

This paper establishes a large deviation principle for empirical measures of spins and their co-operation in d-regular random graphs, providing a probabilistic framework for understanding their asymptotic behavior.

Contribution

It introduces a joint large deviation principle for empirical spin and co-operation measures in d-regular random graphs, extending the theoretical understanding of these models.

Findings

Established LDP for empirical measures in d-regular graphs

Characterized the asymptotic probabilities of empirical configurations

Extended large deviation theory to joint measures in graph models

Abstract

For a $d-$regular random model, we assign to vertices $q-$state spins. From this model, we define the \emph{empirical co-operate measure}, which enumerates the number of co-operation between a given couple of spins, and \emph{ empirical spin measure}, which enumerates the number of sites having a given spin on the $d-$regular random graph model. For these empirical measures we obtain large deviation principle(LDP) in the weak topology.