A phase transition for the biased tree-builder random walk

TL;DR

This paper analyzes a biased recursive tree-building random walk, revealing a phase transition at a critical bias level that determines whether the walk is transient or recurrent, extending previous unbiased results.

Contribution

It introduces a bias parameter into the Tree Builder Random Walk model and characterizes the phase transition for transience and recurrence, generalizing prior unbiased findings.

Findings

Identifies the critical bias threshold for phase transition.

Provides recursive analysis of local times to prove results.

Establishes law of large numbers and CLT with positive speed.

Abstract

We consider a recent model of random walk that recursively grows the network on which it evolves, namely the Tree Builder Random Walk (TBRW). We introduce a bias $\rho \in (0,\infty)$ towards the root, and exhibit a phase transition for transience/recurrence at a critical threshold $\rho_c =1+2\overline{\nu}$, where $\overline{\nu}$ is the (possibly infinite) expected number of new leaves attached to the walker's position at each step. This generalizes previously known results, which focused on the unbiased case $\rho=1$. The proofs rely on a recursive analysis of the local times of the walk at each vertex of the tree, after a given number of returns to the root. We moreover characterize the strength of the transience (law of large numbers and central limit theorem with positive speed) via standard arguments, establish recurrence at $\rho_c$, and show a condensation phenomenon in the non-critical recurrent case.