UNIVERSALITY OF THE TIME CONSTANT FOR 2D CRITICAL FIRST-PASSAGE PERCOLATION
Journal
Annals of Applied Probability
Journal Volume
33
Journal Issue
3
Date Issued
2023-06-01
Author(s)
Abstract
We consider first-passage percolation (FPP) on the triangular lattice with vertex weights (tv) whose common distribution function F satisfies F(0)= 1/2. This is known as the critical case of FPP because large (critical) zero-weight clusters allow travel between distant points in time which is sublinear in the distance. Denoting by T(0,∂B(n)) the first-passage time from 0 to {x : ⃦x⃦∞ = n}, we show existence of a “time constant” and find its exact value to be (Formula Presented) almost surely, where I = inf{x > 0: F(x) > 1/2} and F is any critical distribution for tv. 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.
Subjects
First-passage percolation | time constant | universality
Type
journal article