The Speed of a Biased Walk on a Galton-Watson Tree without Leaves is Monotonic with Respect to Progeny Distributions for High Values of Bias

Abstract

Consider biased random walks on two Galton-Watson trees without leaves having progeny distributions $P_1$ and $P_2$ (GW$(P_1)$ and GW$(P_2)$) where $P_1$ and $P_2$ are supported on positive integers and $P_1$ dominates $P_2$ stochastically. We prove that the speed of the walk on GW$(P_1)$ is bigger than the same on GW$(P_2)$ when the bias is larger than a threshold depending on $P_1$ and $P_2$. This partially answers a question raised by Ben Arous, Fribergh and Sidoravicius.

17 pages

References in corpus (2)

• A proof of the Lyons-Pemantle-Peres monotonicity conjecture for high biases
• Speed of the biased random walk on a Galton--Watson tree

Cited by in corpus (3)

• Nonlinear Hawkes Processes
• Biased random walks on random graphs
• The speed of a biased random walk on a Galton-Watson tree is analytic