Branching random walk in the presence of a hard wall

TL;DR

This paper analyzes the probability that a branching random walk on a tree remains positive at all vertices in the last generation, revealing that the conditional expectation at a typical vertex is significantly lower than the maximum, with implications for Gaussian processes with constraints.

Contribution

It provides new probabilistic estimates for the positivity event in branching random walks with a hard wall, connecting to Gaussian processes and their maxima.

Findings

Probability of positivity at all vertices in the last generation derived.

Conditional expectation at a typical vertex is less than the maximum by order log n.

Results have implications for Gaussian processes with long-range interactions.

Abstract

We consider a branching random walk on a $d$-ary tree of height $n$ ($n \in \mathbb{N}$), under the presence of a hard wall which restricts each value to be positive, where $d$ is a natural number satisfying $d\geqslant2$. The question of behaviour of Gaussian processes with long range interactions, for example the discrete Gaussian free field, under the condition that it is positive on a large subset of {\color{blue}vertices}, and a relation with the expected maximum of the processes has been observed. We find the probability of the event that the branching random {\color{blue}walk} is positive at every vertex in the $n^{th}$ generation, and show that the conditional expectation of the Gaussian variable at a typical vertex, under positivity, is less than the expected maximum by order of $\log n$.