First-order behavior of the time constant in non-isotropic continuous first-passage percolation
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866929704325349376 |
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| author | Basdevant, Anne-Laure Gouéré, Jean-Baptiste Théret, Marie |
| author_facet | Basdevant, Anne-Laure Gouéré, Jean-Baptiste Théret, Marie |
| contents | Consider $Ξ$ a homogeneous Poisson point process on $\mathbb{R}^d$ ($d\geq 2$) with unit intensity with respect to the Lebesgue measure. For $\varepsilon\geq 0$, we define the Boolean model $Σ_{p, \varepsilon}$ as the union of the balls of volume $\varepsilon$ for the $p$-norm ($p\in [1,\infty]$) and centered at the points of $Ξ$. We define a random pseudo-metric on $\mathbb{R}^d$ by associating with any path a travel time equal to its $p$-length outside $Σ_{p,\varepsilon}$. This defines a continuous model of first-passage percolation, that has been studied in \cite{GT17,GT22} for $p=2$, the Euclidean norm. For $p=1$, this model is expected to share common properties with the classical first-passage percolation on the graph $\mathbb{Z}^d$ with a distribution of passage times of the form $\varepsilon δ_0 + (1-\varepsilon) δ_1$. The exact calculation of the time constant of this model $\tilde μ_{p,\varepsilon} (x)$ is out of reach. We investigate here the behavior of $\varepsilon \mapsto \tilde μ_{p,\varepsilon} (x)$ near $0$, and enlighten how the speed at which $\| x \|_p - \tilde μ_{p,\varepsilon} (x) $ goes to $0$ depends on $x$ and $p$. For instance, for $p\in (1,\infty)$, we prove that $\| x \|_p - \tilde μ_{p,ε} (x)$ is of order $\varepsilon ^{κ_p(x)}$ with $$κ_p(x): = \frac{1}{d- \frac{d_1(x)-1}{2} - \frac{d-d_1 (x)}{p}}\,,$$where $d_1(x)$ is the number of non null coordinates of $x$. The exact order of $\| x \|_p - \tilde μ_{p,ε} (x)$ is also given for $p=1$ and $p=\infty$. Related results are also discussed, about properties of the geodesics, and analog properties on closely related models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07509 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | First-order behavior of the time constant in non-isotropic continuous first-passage percolation Basdevant, Anne-Laure Gouéré, Jean-Baptiste Théret, Marie Probability Consider $Ξ$ a homogeneous Poisson point process on $\mathbb{R}^d$ ($d\geq 2$) with unit intensity with respect to the Lebesgue measure. For $\varepsilon\geq 0$, we define the Boolean model $Σ_{p, \varepsilon}$ as the union of the balls of volume $\varepsilon$ for the $p$-norm ($p\in [1,\infty]$) and centered at the points of $Ξ$. We define a random pseudo-metric on $\mathbb{R}^d$ by associating with any path a travel time equal to its $p$-length outside $Σ_{p,\varepsilon}$. This defines a continuous model of first-passage percolation, that has been studied in \cite{GT17,GT22} for $p=2$, the Euclidean norm. For $p=1$, this model is expected to share common properties with the classical first-passage percolation on the graph $\mathbb{Z}^d$ with a distribution of passage times of the form $\varepsilon δ_0 + (1-\varepsilon) δ_1$. The exact calculation of the time constant of this model $\tilde μ_{p,\varepsilon} (x)$ is out of reach. We investigate here the behavior of $\varepsilon \mapsto \tilde μ_{p,\varepsilon} (x)$ near $0$, and enlighten how the speed at which $\| x \|_p - \tilde μ_{p,\varepsilon} (x) $ goes to $0$ depends on $x$ and $p$. For instance, for $p\in (1,\infty)$, we prove that $\| x \|_p - \tilde μ_{p,ε} (x)$ is of order $\varepsilon ^{κ_p(x)}$ with $$κ_p(x): = \frac{1}{d- \frac{d_1(x)-1}{2} - \frac{d-d_1 (x)}{p}}\,,$$where $d_1(x)$ is the number of non null coordinates of $x$. The exact order of $\| x \|_p - \tilde μ_{p,ε} (x)$ is also given for $p=1$ and $p=\infty$. Related results are also discussed, about properties of the geodesics, and analog properties on closely related models. |
| title | First-order behavior of the time constant in non-isotropic continuous first-passage percolation |
| topic | Probability |
| url | https://arxiv.org/abs/2402.07509 |