Stationary Harmonic Measure as the Scaling Limit of Truncated Harmonic Measure

TL;DR

This paper demonstrates that the stationary harmonic measure of an infinite set in the upper planar lattice can be obtained as the scaling limit of the harmonic measure of its finite truncations, linking finite and infinite measures.

Contribution

It establishes a rigorous connection between stationary harmonic measure and classical harmonic measure through a scaling limit, advancing understanding of harmonic measures in lattice structures.

Findings

Stationary harmonic measure equals the scaling limit of harmonic measure for truncations.

Provides a mathematical framework for relating finite and infinite harmonic measures.

Enhances theoretical understanding of harmonic measures in lattice models.

Abstract

In this paper we prove that the stationary harmonic measure of an infinite set in the upper planar lattice can be represented as the proper scaling limit of the classical harmonic measure of truncations of the infinite set.