UNIVERSALITY OF THE TIME CONSTANT FOR 2D CRITICAL FIRST-PASSAGE PERCOLATION

We consider first-passage percolation (FPP) on the triangular lattice with vertex weights (t(v)) whose common distribution function F satisfies F(0) = 1/2. This is known as the critical case of FPP because large (critical) zeroweight clusters allow travel between distant points in time which is sublinear in the distance. Denoting by T (0, partial derivative B(n)) the first-passage time from 0 to {x : parallel to x parallel to(infinity) = n}, we show existence of a time constant and find its exact value to be lim(n ->infinity) T(0, partial derivative B(n))/log n = 2/2 root 3 pi al almost surely, where I = inf{x > 0 : F(x) > 1/2} and F is any critical distribution for t(v). This result shows that this time constant is universal and depends only on the value of I. Furthermore, we find the exact value of the limiting normalized variance, which is also only a function of I, under the optimal moment condition on F. The proof method also shows an analogous universality on other two-dimensional lattices, assuming the time constant exists.

Automated summary

This article discusses first-passage percolation on the triangular lattice with vertex weights, focusing on the critical case where large zero-weight clusters allow sublinear travel time between distant points. The existence of a universal time constant is proven, with its exact value depending on the value of I. The article also determines the exact value of the limiting normalized variance, which is solely a function of I. The proof method demonstrates similar universality on other two-dimensional lattices.