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