First Order Probabilities For Galton-Watson Trees

TL;DR

This paper explores the properties of Galton-Watson trees with Poisson offspring, showing that all first order logic statements depend on local neighborhoods and providing probabilities for these neighborhoods conditioned on the tree's infiniteness.

Contribution

It introduces the concept of universal trees and demonstrates that first order properties are determined by local neighborhoods, with explicit probability calculations.

Findings

Finite subtrees are almost surely present in infinite Poisson Galton-Watson trees.

First order logic properties depend only on local neighborhoods of the root.

Probabilities of local neighborhoods conditioned on infiniteness are computed.

Abstract

In the regime of Galton-Watson trees, first order logic statements are roughly equivalent to examining the presence of specific finite subtrees. We consider the space of all trees with Poisson offspring distribution and show that such finite subtrees will be almost surely present when the tree is infinite. Introducing the notion of universal trees, we show that all first order sentences of quantifier depth $k$ depend only on local neighbourhoods of the root of sufficiently large radius depending on $k$. We compute the probabilities of these neighbourhoods conditioned on the tree being infinite. We give an almost sure theory for infinite trees.