Hitting times for random walks with restarts

TL;DR

This paper investigates the expected hitting times for random walks with restart strategies on Euclidean lattices, establishing asymptotic formulas and confirming conjectures about the 'grade' of a vertex, with implications for control problems.

Contribution

It proves conjectures on the asymptotics of the grade in Euclidean lattices and derives precise asymptotics for hitting times with and without restarts.

Findings

Expected hitting time in planar lattice asymptotic to 2|x|^2 log|x|

Confirmed conjectures on the grade's asymptotics in Euclidean lattices

Derived second order asymptotics for lattice disk hitting times

Abstract

The time it takes a random walker in a lattice to reach the origin from another vertex $x$, has infinite mean. If the walker can restart the walk at $x$ at will, then the minimum expected hitting time $T(x,0)$ (minimized over restarting strategies) is finite; it was called the ``grade'' of $x$ by Dumitriu, Tetali and Winkler. They showed that, in a more general setting, the grade (a variant of the ``Gittins index'') plays a crucial role in control problems involving several Markov chains. Here we establish several conjectures of Dumitriu et al on the asymptotics of the grade in Euclidean lattices. In particular, we show that in the planar square lattice, $T(x,0)$ is asymptotic to $2|x|^2\log|x|$ as $|x| \to \infty$. The proof hinges on the local variance of the potential kernel $h$ being almost constant on the level sets of $h$. We also show how the same method yields precise second order asymptotics for hitting times of a random walk (without restarts) in a lattice disk.