Local Large Deviations: McMillian Theorem for multitype Galton-Watson Processes

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 $U_{\text{skrik}}(\cdot ,\pi )$ for the Galton-Watson process, where $\pi$ is the normalized eigen vector corresponding to the leading \emph{Perron-Frobenius eigen value } $\text{1}$ of the transition matrix $\text{skria}(\cdot ,\cdot )$ defined from $\text{skrik},$ the transition kernel. We show that the Kullback action or the deviation function, $J(\pi ,\rho ),$ with respect to an empirical offspring measure, $\rho ,$ is the Legendre dual of $U_{\text{skrik}}(\cdot ,\pi )\mathrm{.}$ 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 $\varpi ,$ we show that the number of critical multitype Galton-Watson processes on $n$ vertices is approximately $e^{n\langle \skrih_{\varpi},,\pi\rangle},$where$\skrih_{\varpi}$ is a suitably defined entropy.