Windings of planar random walks and averaged Dehn function

TL;DR

This paper establishes precise estimates for the expected index of planar random walks, leading to improved lower bounds on the averaged Dehn function, which quantifies the average area required to fill random curves.

Contribution

It provides a sharp estimate on the expected index of simple random walks and derives new lower bounds for the averaged Dehn function, connecting random walk behavior to geometric filling problems.

Findings

Sharp estimate on the expected index of random walks

New lower bounds on the averaged Dehn function

Enhanced understanding of filling random curves with minimal area

Abstract

We prove a sharp estimate on the expected value of the integral of the index of a simple random walk on the square or triangular lattice. This gives new lower bounds on the averaged Dehn function, which measures the expected area needed to fill a random curve with a disc.