June 2023 Universality of the time constant for 2D critical first-passage percolation
Michael Damron, Jack Hanson, Wai-Kit Lam
Author Affiliations +
Ann. Appl. Probab. 33(3): 1701-1731 (June 2023). DOI: 10.1214/22-AAP1808

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:x=n}, we show existence of a “time constant” and find its exact value to be

limnT(0,B(n))logn=I23πalmost 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.

Funding Statement

The first author was supported by an NSF CAREER grant.
The second author funded in part by NSF Grant DMS-1612921.

Acknowledgments

The authors wish to thank an anonymous referee for many helpful suggestions that led to a more readable exposition.

Citation

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Michael Damron. Jack Hanson. Wai-Kit Lam. "Universality of the time constant for 2D critical first-passage percolation." Ann. Appl. Probab. 33 (3) 1701 - 1731, June 2023. https://doi.org/10.1214/22-AAP1808

Information

Received: 1 January 2021; Revised: 1 January 2022; Published: June 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583656
zbMATH: 07692303
Digital Object Identifier: 10.1214/22-AAP1808

Subjects:
Primary: 60K35
Secondary: 82B43

Keywords: First-passage percolation , Time constant , Universality

Rights: Copyright © 2023 Institute of Mathematical Statistics

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Vol.33 • No. 3 • June 2023
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