Analyticity for rapidly determined properties of Poisson Galton--Watson trees

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 $\mathbb{P}_\lambda(A):= \mathbb{P}(T_\lambda \vdash A)$. 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 the first order language of trees.