On the time constant of high dimensional first passage percolation, revisited
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909462704422912 |
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| author | Auffinger, Antonio Tang, Si |
| author_facet | Auffinger, Antonio Tang, Si |
| contents | In [2], it was claimed that the time constant $μ_{d}(e_{1})$ for the first-passage percolation model on $\mathbb Z^{d}$ is $μ_{d}(e_{1}) \sim \log d/(2ad)$ as $d\to \infty$, if the passage times $(τ_{e})_{e\in \mathbb E^{d}}$ are i.i.d., with a common c.d.f. $F$ satisfying $\left|\frac{F(x)}{x}-a\right| \le \frac{C}{|\log x|}$ for some constants $a, C$ and sufficiently small $x$.
However, the proof of the upper bound, namely, Equation (2.1) in [2] \begin{align} \limsup_{d\to\infty} \frac{μ_{d}(e_{1})ad}{\log d} \le \frac{1}{2} \end{align} is incorrect. In this article, we provide a different approach that establishes this inequality. As a side product of this new method, we also show that the variance of the non-backtracking passage time to the first hyperplane is of order $o\big((\log d/d)^{2}\big)$ as $d\to \infty$ in the case of the when the edge weights are exponentially distributed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11589 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the time constant of high dimensional first passage percolation, revisited Auffinger, Antonio Tang, Si Probability In [2], it was claimed that the time constant $μ_{d}(e_{1})$ for the first-passage percolation model on $\mathbb Z^{d}$ is $μ_{d}(e_{1}) \sim \log d/(2ad)$ as $d\to \infty$, if the passage times $(τ_{e})_{e\in \mathbb E^{d}}$ are i.i.d., with a common c.d.f. $F$ satisfying $\left|\frac{F(x)}{x}-a\right| \le \frac{C}{|\log x|}$ for some constants $a, C$ and sufficiently small $x$. However, the proof of the upper bound, namely, Equation (2.1) in [2] \begin{align} \limsup_{d\to\infty} \frac{μ_{d}(e_{1})ad}{\log d} \le \frac{1}{2} \end{align} is incorrect. In this article, we provide a different approach that establishes this inequality. As a side product of this new method, we also show that the variance of the non-backtracking passage time to the first hyperplane is of order $o\big((\log d/d)^{2}\big)$ as $d\to \infty$ in the case of the when the edge weights are exponentially distributed. |
| title | On the time constant of high dimensional first passage percolation, revisited |
| topic | Probability |
| url | https://arxiv.org/abs/2501.11589 |