2024/08/31 by Zhenhao Cai, Cai, Zhenhao, Eviatar B. Procaccia +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical methods in inverse problems #Probability (math.PR) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2409.00450
openalex publication_date 2024/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathscrT and the hexagonal lattice \mathscrH. In particular, for the least positive value of the harmonic measure of any n-point set, denoted by \mathscrMn(\mathscrG), we prove in this paper that [λ(\mathscrG)]-n+c√(n) ≤ \mathscrMn(\mathscrG)≤ [λ(\mathscrG)]-n+C√(n), where λ(ℤ2)=(2+√(3))2, λ(\mathscrT)=3+2√(2) and λ(\mathscrH)=(\tfrac3+√(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 \mathscrMn(ℤ2). Moreover, these estimates also significantly extend the findings in our previous paper with Kozma (2023) that \mathscrMn(\mathscrG) decays exponentially for a large family of graphs \mathscrG including \mathscrT, \mathscrH and ℤd for all d≥ 2.