On the Distribution of Range for Tree-Indexed Random Walks

TL;DR

This paper investigates the distribution of the range in tree-indexed random walks, confirming a conjecture for lazy walks and providing partial results for standard walks, advancing understanding of labelings on trees.

Contribution

It proves the conjecture that the range distribution for lazy walks on trees is dominated by that of a path, and offers partial results for the standard walk case.

Findings

Confirmed the conjecture for lazy walks on all trees.

Provided partial results for the standard walk case.

Enhanced understanding of label difference distributions in tree-indexed walks.

Abstract

We study tree-indexed random walks as introduced by Benjamini, H\"aggstr\"om, and Mossel, i.e. labelings of a tree for which adjacent vertices have labels differing by 1. It is a conjecture of those authors that the distribution of the range for any such tree is dominated by that of a path on the same number of edges. The two main variants of this conjecture considered in the literature are the $\textit{standard}$ walks, in which adjacent vertices must have labels differing by $\textit{exactly}$ 1, and $\textit{lazy}$ walks, in which adjacent vertices must have labels differing by $\textit{at most}$ 1. We confirm this conjecture for all trees in the lazy case and provide some partial results in the standard case.