Noise sensitivity and variance lower bound for minimal left-right crossing of a square in first-passage percolation
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arXiv
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2025
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| author | Dembin, Barbara Elboim, Dor |
| author_facet | Dembin, Barbara Elboim, Dor |
| contents | We study first-passage percolation on $\mathbb Z ^2$ with independent and identically distributed weights, whose common distribution is uniform on $\{a,b\}$ with $0<a<b<\infty $.
Following Ahlberg and De la Riva, we consider the passage time $τ(n,k)$ of the minimal left-right crossing of the square $[0,n]^2$, whose vertical fluctuations are bounded by $k$. We prove that when $k\le n^{1/2-ε}$, the event that $τ(n,k)$ is larger than its median is noise sensitive. This improves the main result of Ahlberg and De la Riva which holds when $k\le n^{1/22-ε}$.
Under the additional assumption that the limit shape is not a polygon with a small number of sides, we extend the result to all $k\le n^{1-ε}$. This extension follows unconditionally when $a$ and $b$ are sufficiently close.
Under a stronger curvature assumption, we extend the result to all $k\le n$. This in particular captures the noise sensitivity of the event that the minimal left-right crossing $T_n=τ(n,n)$ is larger than its median.
Finally, under the curvature assumption, our methods give a lower bound of $n^{1/4-ε}$ for the variance of the passage time $T_n$ of the minimal left-right crossing of the square. We prove the last bound also for absolutely continuous weight distributions, generalizing a result of Damron--Houdré--Özdemir, which holds only for the exponential distribution.
Our approach differs from the previous works mentioned above; the key idea is to establish a small ball probability estimate in the tail by perturbing the weights for tail events using a Mermin--Wagner type estimate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03211 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Noise sensitivity and variance lower bound for minimal left-right crossing of a square in first-passage percolation Dembin, Barbara Elboim, Dor Probability We study first-passage percolation on $\mathbb Z ^2$ with independent and identically distributed weights, whose common distribution is uniform on $\{a,b\}$ with $0<a<b<\infty $. Following Ahlberg and De la Riva, we consider the passage time $τ(n,k)$ of the minimal left-right crossing of the square $[0,n]^2$, whose vertical fluctuations are bounded by $k$. We prove that when $k\le n^{1/2-ε}$, the event that $τ(n,k)$ is larger than its median is noise sensitive. This improves the main result of Ahlberg and De la Riva which holds when $k\le n^{1/22-ε}$. Under the additional assumption that the limit shape is not a polygon with a small number of sides, we extend the result to all $k\le n^{1-ε}$. This extension follows unconditionally when $a$ and $b$ are sufficiently close. Under a stronger curvature assumption, we extend the result to all $k\le n$. This in particular captures the noise sensitivity of the event that the minimal left-right crossing $T_n=τ(n,n)$ is larger than its median. Finally, under the curvature assumption, our methods give a lower bound of $n^{1/4-ε}$ for the variance of the passage time $T_n$ of the minimal left-right crossing of the square. We prove the last bound also for absolutely continuous weight distributions, generalizing a result of Damron--Houdré--Özdemir, which holds only for the exponential distribution. Our approach differs from the previous works mentioned above; the key idea is to establish a small ball probability estimate in the tail by perturbing the weights for tail events using a Mermin--Wagner type estimate. |
| title | Noise sensitivity and variance lower bound for minimal left-right crossing of a square in first-passage percolation |
| topic | Probability |
| url | https://arxiv.org/abs/2505.03211 |