Strict inequality between the time constants of first-passage percolation and directed first-passage percolation - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2024

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). The first result of this short paper is the strict inequality µ(x) < \vec{µ}(x). To get this result, we use a lower bound on the number of edges of geodesics in first-passage percolation (where geodesics are paths with minimal passage time), which is the second result of this paper.
Fichier principal
Vignette du fichier
28.12.24 time constant directed fpp HAL.pdf (138.46 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04858044 , version 1 (30-12-2024)

Identifiants

Citer

Antonin Jacquet. Strict inequality between the time constants of first-passage percolation and directed first-passage percolation. 2024. ⟨hal-04858044⟩
0 Consultations
0 Téléchargements

Altmetric

Partager

More