Open Access
2020 Analyticity for rapidly determined properties of Poisson Galton–Watson trees
Yuval Peres, Andrew Swan
Electron. Commun. Probab. 25: 1-8 (2020). DOI: 10.1214/20-ECP320


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.


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Yuval Peres. Andrew Swan. "Analyticity for rapidly determined properties of Poisson Galton–Watson trees." Electron. Commun. Probab. 25 1 - 8, 2020.


Received: 8 October 2019; Accepted: 13 May 2020; Published: 2020
First available in Project Euclid: 19 June 2020

zbMATH: 07225536
MathSciNet: MR4116391
Digital Object Identifier: 10.1214/20-ECP320

Primary: 60J80

Keywords: analytic continuation , Galton–Watson tree , phase transition

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