Joint large deviation principle for some empirical measures of the d-regular random graphs

In this paper, we define a d–regular random model by perfect matching of vertices or paring of vertices. For each vertex, we assign a q–state spin. From this d–regular graph model, we define the empirical co-operate measure, which enumerates the number of co-operation between a given couple of spins, and 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.