Large deviation results for Critical Multitype Galton-Watson trees

In this article, we prove a joint large deviation principle in $n$ for the \emph{empirical pair measure} and \emph{ empirical offspring measure} of critical multitype Galton-Watson trees conditioned to have exactly $n$ vertices in the weak topology. From this result we extend the large deviation principle for the empirical pair measures of Markov chains on simply generated trees to cover offspring laws which are not treated by \cite[Theorem~2.1]{DMS03}. For the case where the offspring law of the tree is a geometric distribution with parameter \sfrac{1}{2}, we get an exact rate function. All our rate functions are expressed in terms of relative entropies.

💡 Research Summary

The paper establishes a joint large‑deviation principle (LDP) for the empirical pair measure and the empirical offspring measure of critical multitype Galton–Watson trees conditioned to have exactly n vertices, working in the weak topology. The authors relax the strong‑topology assumptions of Dembo, Mörters and Sheffield (2003) (DMS03), which required all exponential moments of the offspring distribution to be finite, by introducing the notions of “sub‑consistency” and “weak shift‑invariance”. Under the weaker condition that only the second moment of the offspring law is finite, the pair ((\tilde L_Y, M_Y)) satisfies an LDP with speed n and a convex good rate function \