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Pré-Publication, Document De Travail Année : 2024

Disjoint finite geodesics in first-passage percolation

Olivier Durieu
Jean-Baptiste Gouéré
Antonin Jacquet
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Résumé

We investigate first-passage percolation on the lattice $\mathbb{Z}^d$ for dimensions $d \geq 2$. Each edge $e$ of the graph is assigned an independent copy of a non-negative random variable $\tau$. We only assume $\mathbb{P}[\tau=0] < p_c(\mathbb{Z}^d)$, the critical probability threshold for Bernoulli bond percolation on $\mathbb{Z}^d$. We obtain lower bounds of order $n^{-\alpha_d}$ (where $\alpha_d > 0$ is explicit) for the probability of having two disjoint geodesics between two pairs of neighbouring vertices at distance $n$. Additionally, under more specific assumptions on the distribution of $\tau$, we obtain similar lower bounds for the probability of having two disjoint geodesics (except for their starting and ending points) between the same two vertices.
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hal-04661565 , version 1 (24-07-2024)

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Olivier Durieu, Jean-Baptiste Gouéré, Antonin Jacquet. Disjoint finite geodesics in first-passage percolation. 2024. ⟨hal-04661565⟩
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