First-passage percolation on Cartesian power graphs

Abstract

We consider first-passage percolation on the class of "high-dimensional" graphs that can be written as an iterated Cartesian product G G . . . G of some base graph G as the number of factors tends to infinity. We propose a natural asymptotic lower bound on the first-passage time between (v, v, . . . , v) and (w, w, . . . , w) as n, the number of factors, tends to infinity, which we call the critical time t * G (v, w). Our main result characterizes when this lower bound is sharp as n → ∞. As a corollary, we are able to determine the limit of the socalled diagonal time-constant in Z n as n → ∞ for a large class of distributions of passage times.2010 Mathematics Subject Classification. Primary 60C05; secondary 60K35, 82B43.

Random growth models: Shape and convergence rate

“…The following theorem is from Auffinger-Tang [7], which weakens various assumptions (widens the class of distributions in particular) of the work of previous authors. Some earlier work was done by Kesten [28], Dhar [16], Couronné-Enriquez-Gerin [11], and Martinsson [37]. For example, if B were the 1 -ball, one would have for e " p1, .…”

On the time constant of high dimensional first passage percolation

“…where α * is the non null solution of coth α = α. Recently, still under the assumption of exponential passage times, Martinsson [9] proved a matching upper bound establishing lim d→∞ √ dµ * = α 2 * − 1 2 .…”

First passage percolation in the mean field limit

“…The leading order in the mean field limit has been recently identified by Anders Martinsson, who settled a conjecture by Fill and Pemantle [29] in a series of papers: Theorem 8. [Martinsson,[43,44]] With E ≡ ln(1 + √ 2), it holds…”

“…Die führende Ordnung des Grundzustands wurde kürzlich von Anders Martinsson identifiziert, der eine Vermutung von Fill und Pemantle [29] in einer Reihe von Artikeln gelöst hat: Theorem 11. [Martinsson,[43,44]]…”

“…In the second proof of the above theorem, Martinsson proceeds through some ingenious use of the FKG inequality, and (related) subadditivity/monotonicity properties of paths with optimal energies, see [44] for details.…”