Analyticity for rapidly determined properties of Poisson Galton–Watson trees

Abstract

Let $T_{\lambda }$ be a Galton–Watson tree with Poisson($\lambda $) offspring, and let $A$ be a tree property. In this paper, we are concerned with the regularity of the function $\mathbb {P}_{\lambda }(A)\coloneqq \mathbb {P}(T_{\lambda }\models A)$. We show that if a property $A$ can be uniformly approximated by a sequence of properties $\{A_{k}\}$, depending only on the first $k$ vertices in the breadth first exploration of the tree, with a bound in probability of $\mathbb {P}_{\lambda }(A\triangle A_{k}) \le Ce^{-ck}$ over an interval $I = (\lambda _{0}, \lambda _{1})$, then $\mathbb {P}_{\lambda }(A)$ is real analytic in $\lambda $ for $\lambda \in I$. We also present some applications of our results, particularly to properties that are not expressible in first order logic on trees.