Rate of convergence in first-passage percolation under low moments

Abstract

We consider first-passage percolation on the d dimensional cubic lattice for d ≥ 2; that is, we assign independently to each edge e a nonnegative random weight t e with a common distribution and consider the induced random graph distance (the passage time), T (x, y). It is known that for each x ∈ Z d , µ(x) = lim n T (0, nx)/n exists and that 0 ≤ ET (0, x)−µ(x) ≤ C x 1/2 1 log x 1 under the condition Ee αte < ∞ for some α > 0. By combining tools from concentration of measure with Alexander's methods, we show how such bounds can be extended to t e 's with distributions that have only low moments. For such edge-weights, we obtain an improved bound C( x 1 log x 1 ) 1/2 and bounds on the rate of convergence to the limit shape.

50 Years of First-Passage Percolation

“…This is not the main result of that paper, but an auxiliary one used to obtain the main one. The lower tail inequality comes from [61]. Our aim will be to prove the following result.…”

“…We finish this section with a summary of the current state of the art on (a) nonrandom fluctuation upper bounds and (b) convergence rate to the limit shape. For (a), Alexander's methods were used in the low moment case in [61] along with a concentration inequality for the lower tail of T (0, x) to obtain the following.…”

Random growth models: Shape and convergence rate

“…Specifically, if one has a concentration inequality of the form P pT p0, xq´ET p0, xq ă´λ}x} α q ď exp`´Cλ β˘, λ ě 0 for constants C, α, β ą 0, then one can show an upper bound of the type ET p0, xq´gpxq ď C}x} α plog }x}q δ , implying that nonrandom fluctuations should be no larger than random fluctuations. Indeed, using Gaussian concentration inequalities, one has the following version of Alexander's [1] result from Damron-Kubota [15].…”

“…Much more on the random fluctuation term and concentration estimates will be given in article [43] In summary, we have a strengthened shape theorem of the following type. Some improvements to the assumptions have been made by Tessera [44] and Damron-Kubota [15] more recently, in particular generally replacing the log with ? log.…”

Divergence of non-random fluctuation in First Passage Percolation

“…Related works. The non-random fluctuation ET(0, x) − g|x| is one of the central objects in FPP and there are several attempts to study this [1,3,6]. In particular, [7] and [4] obtained the sublinear upper bound in the Euclidean FPP.…”