Entropy of Random Walk Range

TL;DR

This paper analyzes the entropy of the set traced by an n-step random walk on integer lattices, revealing different growth rates depending on the dimension, primarily influenced by the boundary size of the trace.

Contribution

It provides precise asymptotic estimates for the entropy of the random walk trace in different dimensions, highlighting the role of boundary size.

Findings

For d ≥ 3, entropy grows linearly with n.

For d = 2, entropy grows as n divided by log squared n.

Boundary size largely determines entropy behavior.

Abstract

We study the entropy of the set traced by an $n$-step random walk on $\Z^d$. We show that for $d \geq 3$, the entropy is of order $n$. For $d = 2$, the entropy is of order $n/\log^2 n$. These values are essentially governed by the size of the boundary of the trace.