The asymptotic determinant of the discrete Laplacian

TL;DR

This paper calculates the asymptotic determinant of the discrete Laplacian on certain regions and uses this to establish the growth exponent of loop-erased random walks in two dimensions as 5/4.

Contribution

It provides the first explicit asymptotic formula for the determinant of the discrete Laplacian on rectilinear regions and links it to the growth exponent of loop-erased random walks.

Findings

Asymptotic determinant formula for the discrete Laplacian on rectilinear regions

Proof that the loop-erased random walk growth exponent in Z^2 is 5/4

Connection between spectral properties and random walk behavior

Abstract

We compute the asymptotic determinant of the discrete Laplacian on a simply-connected rectilinear region in R^2. As an application of this result, we prove that the growth exponent of the loop-erased random walk in Z^2 is 5/4.