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

TL;DR

This paper proves that the probability of certain properties in Poisson Galton--Watson trees varies analytically with the offspring rate, especially for properties approximable by finite-depth conditions, with applications to non-first-order properties.

Contribution

It establishes conditions under which the probability function of tree properties is real analytic in the offspring parameter, extending to properties beyond first-order expressibility.

Findings

Probability functions are real analytic under approximation conditions.

Analyticity holds over specific parameter intervals.

Applications include properties not definable in first-order logic.

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 $\mathbb{P}_\lambda(A):= \mathbb{P}(T_\lambda \vdash A)$. 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.