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 (Formula presented.) where λ(Z2)=(2+3)2, λ(T)=3+22 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.