Strict inequality between the time constants of first-passage percolation and directed first-passage percolation
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915129087492096 |
|---|---|
| author | Jacquet, Antonin |
| author_facet | Jacquet, Antonin |
| contents | In the models of first-passage percolation and directed first-passage percolation on $\mathbb{Z}^d$, we consider a family of i.i.d. random variables indexed by the set of edges of the graph, called passage times. For every vertex $x \in \mathbb{Z}^d$ with nonnegative coordinates, we denote by $t(0,x)$ the shortest passage time to go from $0$ to $x$ and by $\vec t(0,x)$ the shortest passage time to go from $0$ to $x$ following a directed path. Under some assumptions, it is known that for every $x \in \mathbb{R}^d$ with nonnegative coordinates, $t(0,\lfloor nx \rfloor)/n$ converges to a constant $μ(x)$ and that $\vec t(0,\lfloor nx \rfloor)/n$ converges to a constant $\vecμ(x)$. With these definitions, we immediately get that $μ(x) \le \vecμ(x)$. In this paper, we get the strict inequality $μ(x) < \vecμ(x)$ as a consequence of a new exponential bound for the comparison of $t(0,x)$ and $\vec{t}(0,x)$ when $\|x\|$ goes to $\infty$. This exponential bound is itself based on a lower bound on the number of edges of geodesics in first-passage percolation (where geodesics are paths with minimal passage time). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20779 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strict inequality between the time constants of first-passage percolation and directed first-passage percolation Jacquet, Antonin Probability In the models of first-passage percolation and directed first-passage percolation on $\mathbb{Z}^d$, we consider a family of i.i.d. random variables indexed by the set of edges of the graph, called passage times. For every vertex $x \in \mathbb{Z}^d$ with nonnegative coordinates, we denote by $t(0,x)$ the shortest passage time to go from $0$ to $x$ and by $\vec t(0,x)$ the shortest passage time to go from $0$ to $x$ following a directed path. Under some assumptions, it is known that for every $x \in \mathbb{R}^d$ with nonnegative coordinates, $t(0,\lfloor nx \rfloor)/n$ converges to a constant $μ(x)$ and that $\vec t(0,\lfloor nx \rfloor)/n$ converges to a constant $\vecμ(x)$. With these definitions, we immediately get that $μ(x) \le \vecμ(x)$. In this paper, we get the strict inequality $μ(x) < \vecμ(x)$ as a consequence of a new exponential bound for the comparison of $t(0,x)$ and $\vec{t}(0,x)$ when $\|x\|$ goes to $\infty$. This exponential bound is itself based on a lower bound on the number of edges of geodesics in first-passage percolation (where geodesics are paths with minimal passage time). |
| title | Strict inequality between the time constants of first-passage percolation and directed first-passage percolation |
| topic | Probability |
| url | https://arxiv.org/abs/2412.20779 |