Pré-Publication, Document De Travail Année : 2025

Strict inequality between the time constants of first-passage percolation and directed first-passage percolation

Antonin Jacquet
  • Fonction : Auteur
  • PersonId : 1131642

Résumé

In the models of first-passage percolation and directed first-passage percolation on 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 ∈ 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 ∈ R^d with nonnegative coordinates, t(0, ⌊nx⌋)/n converges to a constant µ(x) and that \vec{t}(0, ⌊nx⌋)/n converges to a constant \vec{µ}(x). With these definitions, we immediately get that µ(x) ≤ \vec{µ}(x). In this paper, we get the strict inequality µ(x) < \vec{µ}(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).

Fichier principal
Vignette du fichier
27.01.25 time constant directed fpp HAL v2.pdf (147.94 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Licence

Dates et versions

hal-04858044 , version 1 (30-12-2024)
hal-04858044 , version 2 (28-01-2025)

Licence

Identifiants

Citer

Antonin Jacquet. Strict inequality between the time constants of first-passage percolation and directed first-passage percolation. 2025. ⟨hal-04858044v2⟩
142 Consultations
127 Téléchargements

Altmetric

Partager

  • More