Minimal harmonic measure on 2D lattices

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 Z², the triangular lattice T and the hexagonal lattice H. In particular, for the least positive value of the harmonic measure of any n-point set, denoted by Mₙ(G), we prove in this paper that [λ(G)]-n+c√n ≤ Mₙ(G)≤ [λ(G)]-n+C√n, where λ(Z²)=(2+√3)², λ(T)=3+2√2 and λ(H)=(3+√52)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 Mₙ(Z²). Moreover, these estimates also significantly extend the findings in our previous paper with Kozma (2023) that Mₙ(G) decays exponentially for a large family of graphs G including T, H and Zᵈ for all d≥ 2.