On the time constant of high dimensional first passage percolation

Abstract

We study the time constant µ(e 1 ) in first passage percolation on Z d as a function of the dimension. We prove that if the passage times have finite mean,where a ∈ [0, ∞] is a constant that depends only on the behavior of the distribution of the passage times at 0. For the same class of distributions, we also prove that the limit shape is not an Euclidean ball, nor a d-dimensional cube or diamond, provided that d is large enough. * tuca@northwestern.edu.

First-passage percolation on Cartesian power graphs

“…In the case of Exp(1) passage times more precise estimates were given by [6,8,9]. A recent paper [3] extends this to more general distributions. In [10], Fill and Pemantle proposed the n-dimensional hypercube as an alternative high-dimensional graph, and this was subsequently studied in [1,5,10,15].…”

“…where α * is the unique positive solution to coth α = α. In a recent paper by Auffinger and Tang [3] this was generalized to other distributions. Suppose F ∈ C(ρ) for some ρ ∈ [0, ∞] exists.…”

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].…”

“…Although our selection rule appears not to bias any direction, the limiting shape is expected not to be rotationally invariant (that is, it is expected not to be a Euclidean ball). This statement is proved for high dimensions (see [7,28] and the references in Section 3.1.3 below).…”

“…For the statement, we define the balls D, B, C to be the 1 , 2 , and 8 -balls of radius gpe 1 q ´1. We will also make the following assumptions from [7]:…”