Cut-off and Escape Behaviors for Birth and Death Chains on Trees

TL;DR

This paper studies birth and death chains on trees, demonstrating that with a drift towards the root, these chains show a sharp cutoff in convergence along the drift and escape behavior away from it.

Contribution

It provides a novel analysis of cutoff and escape phenomena for birth and death chains on trees with directional drift.

Findings

Chains exhibit cutoff behavior along the drift towards the root.

Chains show escape behavior in the opposite direction.

Results apply to families of discrete time chains on trees.

Abstract

We consider families of discrete time birth and death chains on trees, and show that in presence of a drift towards the root of the tree, the chains exhibit cut-off behavior along the drift and escape behavior in the opposite direction.