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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.00450 |
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| _version_ | 1866913488169861120 |
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| author | Cai, Zhenhao Procaccia, Eviatar B. Zhang, Yuan |
| author_facet | Cai, Zhenhao Procaccia, Eviatar B. Zhang, Yuan |
| contents | 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 $\mathscr{M}_n(\mathscr{G})$, we prove in this paper that
$$[λ(\mathscr{G})]^{-n+c\sqrt{n}} \le \mathscr{M}_n(\mathscr{G})\le [λ(\mathscr{G})]^{-n+C\sqrt{n}},$$
where $λ(\mathbb{Z}^2)=(2+\sqrt{3})^2$, $λ(\mathscr{T})=3+2\sqrt{2}$ and $λ(\mathscr{H})=(\tfrac{3+\sqrt{5}}{2})^3$.
Our results confirm a stronger version of the conjecture proposed by Calvert, Ganguly and Hammond (2023) which predicts the asymptotic of the exponent of $\mathscr{M}_n(\mathbb{Z}^2)$. Moreover, these estimates also significantly extend the findings in our previous paper with Kozma (2023) that $\mathscr{M}_n(\mathscr{G})$ decays exponentially for a large family of graphs $\mathscr{G}$ including $\mathscr{T}$, $\mathscr{H}$ and $\mathbb{Z}^d$ for all $d\ge 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00450 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal harmonic measure on 2D lattices Cai, Zhenhao Procaccia, Eviatar B. Zhang, Yuan Probability 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 $\mathscr{M}_n(\mathscr{G})$, we prove in this paper that $$[λ(\mathscr{G})]^{-n+c\sqrt{n}} \le \mathscr{M}_n(\mathscr{G})\le [λ(\mathscr{G})]^{-n+C\sqrt{n}},$$ where $λ(\mathbb{Z}^2)=(2+\sqrt{3})^2$, $λ(\mathscr{T})=3+2\sqrt{2}$ and $λ(\mathscr{H})=(\tfrac{3+\sqrt{5}}{2})^3$. Our results confirm a stronger version of the conjecture proposed by Calvert, Ganguly and Hammond (2023) which predicts the asymptotic of the exponent of $\mathscr{M}_n(\mathbb{Z}^2)$. Moreover, these estimates also significantly extend the findings in our previous paper with Kozma (2023) that $\mathscr{M}_n(\mathscr{G})$ decays exponentially for a large family of graphs $\mathscr{G}$ including $\mathscr{T}$, $\mathscr{H}$ and $\mathbb{Z}^d$ for all $d\ge 2$. |
| title | Minimal harmonic measure on 2D lattices |
| topic | Probability |
| url | https://arxiv.org/abs/2409.00450 |