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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.22434 |
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| _version_ | 1866916664938856448 |
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| author | Vernotte, David |
| author_facet | Vernotte, David |
| contents | Gaussian percolation can be seen as the generalization of standard Bernoulli percolation on $\mathbb{Z}^d$. Instead of a random discrete configuration on a lattice, we consider a continuous Gaussian field $f$ and we study the topological and geometric properties of the random excursion set $\mathcal{E}_\ell(f) := \{x\in \mathbb{R}^d\ |\ f(x)\geq -\ell\}$ where $\ell\in \mathbb{R}$ is called a level. It is known that for a wide variety of fields $f$, there exists a phase transition at some critical level $\ell_c$. When $\ell> \ell_c$, the excursion set $\mathcal{E}_\ell(f)$ presents a unique unbounded component while if $\ell<\ell_c$ there are only bounded components in $\mathcal{E}_\ell(f)$. In the supercritical regime, $\ell>\ell_c$, we study the geometry of the unbounded cluster. Inspired by the work of Peter Antal and Agoston Pisztora for the Bernoulli model \cite{Antal}, we introduce the chemical distance between two points $x$ and $y$ as the Euclidean length of the shortest path connecting these points and staying in $\mathcal{E}_\ell(f)$. In this paper, we show that when $\ell>-\ell_c$ then with high probability, the chemical distance between two points has a behavior close to the Euclidean distance between those two points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22434 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chemical distance for smooth Gaussian fields in higher dimension Vernotte, David Probability Gaussian percolation can be seen as the generalization of standard Bernoulli percolation on $\mathbb{Z}^d$. Instead of a random discrete configuration on a lattice, we consider a continuous Gaussian field $f$ and we study the topological and geometric properties of the random excursion set $\mathcal{E}_\ell(f) := \{x\in \mathbb{R}^d\ |\ f(x)\geq -\ell\}$ where $\ell\in \mathbb{R}$ is called a level. It is known that for a wide variety of fields $f$, there exists a phase transition at some critical level $\ell_c$. When $\ell> \ell_c$, the excursion set $\mathcal{E}_\ell(f)$ presents a unique unbounded component while if $\ell<\ell_c$ there are only bounded components in $\mathcal{E}_\ell(f)$. In the supercritical regime, $\ell>\ell_c$, we study the geometry of the unbounded cluster. Inspired by the work of Peter Antal and Agoston Pisztora for the Bernoulli model \cite{Antal}, we introduce the chemical distance between two points $x$ and $y$ as the Euclidean length of the shortest path connecting these points and staying in $\mathcal{E}_\ell(f)$. In this paper, we show that when $\ell>-\ell_c$ then with high probability, the chemical distance between two points has a behavior close to the Euclidean distance between those two points. |
| title | Chemical distance for smooth Gaussian fields in higher dimension |
| topic | Probability |
| url | https://arxiv.org/abs/2503.22434 |