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, are concerned with the regularity of the function ${\mathbb{P}}_{\lambda }(A):=\mathbb{P}(T_{\lambda }⊢A)$. 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 ${\mathbb{P}}_{\lambda }(A△A_k)\le C{e}^{-ck}$ 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 the first order language of trees.