Minimal harmonic measure on 2D lattices

Abstract

We study the harmonic measure (i.e. the limit of the hitting distribution of a simple random walk starting from a distant point) on three canonical two-dimensional lattices: the square lattice ${\mathbb{Z}}^2$, the triangular lattice $\mathcal{T}$ and the hexagonal lattice $\mathcal{H}$. In particular, for the least positive value of the harmonic measure of any $n$-point set, denoted by ${\mathcal{M}}_n(\mathcal{G})$, we prove in this paper that $$[\hat{I}»(\mathcal{G}){]}^{-n+c\sqrt{n}}\le {\mathcal{M}}_n(\mathcal{G})\le [\hat{I}»(\mathcal{G}){]}^{-n+C\sqrt{n}},$$ where $\hat{I}»({\mathbb{Z}}^2)=(2+\sqrt{3}{)}^2$, $\hat{I}»(\mathcal{T})=3+2\sqrt{2}$ and $\hat{I}»(\mathcal{H})=(\frac{3+\sqrt{5}}{2}{)}^3$. Our results confirm a stronger version of the conjecture proposed by Calvert, Ganguly and Hammond (2023) which predicts the asymptotic of the exponent of ${\mathcal{M}}_n({\mathbb{Z}}^2)$. Moreover, these estimates also significantly extend the findings in our previous paper with Kozma (2023) that ${\mathcal{M}}_n(\mathcal{G})$ decays exponentially for a large family of graphs $\mathcal{G}$ including $\mathcal{T}$, $\mathcal{H}$ and ${\mathbb{Z}}^d$ for all $d\ge 2$.