Local large deviations: McMillian theorem for multitype Galton-Watson process

In this article, we prove a local large deviation principle (LLDP) for the critical multitype Galton-Watson process from spectral potential point. We define the so-called a spectral potential U_K(⋅, π) for the Galton-Watson process, where π is the normalized eigenvector corresponding to the leading Perron-Frobenius eigenvalue 1 of the transition matrix A(⋅, ⋅) defined from K, the transition kernel. We show that the Kullback action or the deviation function, J (π, ρ), with respect to an empirical offspring measure, ρ, is the Legendre dual of U_K(⋅, π). From the LLDP, we deduce a conditional large deviation principle and a weak variant of the classical McMillian theorem for the multitype Galton-Watson process. To be specific, given any empirical offspring measure ϖ, we show that the number of critical multitype Galton-Watson processes on n vertices is approximately (Formula presented), where H_ϖ is a suitably defined entropy.