Strict inequality between the time constants of first-passage percolation and directed first-passage percolation
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.
Origine | Fichiers produits par l'(les) auteur(s) |
---|