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 \(\mathscr {T}\) and the hexagonal lattice \(\mathscr {H}\) . In particular, for the least positive value of the harmonic measure of any n-point set, denoted by \(\mathcal {M}_n(\mathscr {G})\) , we prove in this paper that \(\begin{aligned} [\lambda (\mathscr {G})]^{-n+c\sqrt{n}} \le \mathcal {M}_n(\mathscr {G}) \le [\lambda (\mathscr {G})]^{-n+C\sqrt{n}}, \end{aligned}\) where \(\lambda (\mathbb {Z}^2)=(2+\sqrt{3})^2\) , \(\lambda (\mathscr {T})=3+2\sqrt{2}\) and \(\lambda (\mathscr {H})=(\tfrac{3+\sqrt{5}}{2})^3\) . 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 \(\mathcal {M}_n(\mathscr {G})\) decays exponentially for a large family of graphs \(\mathscr {G}\) including \(\mathscr {T}\) , \(\mathscr {H}\) and \(\mathbb {Z}^d\) for all \(d\ge 2\) .
