## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 5 (2017), 2941-2970.

### Asymptotics for $2D$ critical first passage percolation

Michael Damron, Wai-Kit Lam, and Xuan Wang

#### Abstract

We consider first passage percolation on $\mathbb{Z}^{2}$ with i.i.d. weights, whose distribution function satisfies $F(0)=p_{c}=1/2$. This is sometimes known as the “critical case” because large clusters of zero-weight edges force passage times to grow at most logarithmically, giving zero time constant. Denote $T(\mathbf{0},\partial B(n))$ as the passage time from the origin to the boundary of the box $[-n,n]\times[-n,n]$. We characterize the limit behavior of $T(\mathbf{0},\partial B(n))$ by conditions on the distribution function $F$. We also give exact conditions under which $T(\mathbf{0},\partial B(n))$ will have uniformly bounded mean or variance. These results answer several questions of Kesten and Zhang from the 1990s and, in particular, disprove a conjecture of Zhang from 1999. In the case when both the mean and the variance go to infinity as $n\to\infty$, we prove a CLT under a minimal moment assumption. The main tool involves a new relation between first passage percolation and invasion percolation: up to a constant factor, the passage time in critical first passage percolation has the same first-order behavior as the passage time of an optimal path constrained to lie in an embedded invasion cluster.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 5 (2017), 2941-2970.

**Dates**

Received: August 2015

Revised: June 2016

First available in Project Euclid: 23 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1506132031

**Digital Object Identifier**

doi:10.1214/16-AOP1129

**Mathematical Reviews number (MathSciNet)**

MR3706736

**Zentralblatt MATH identifier**

1378.60115

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60F05: Central limit and other weak theorems 82B43: Percolation [See also 60K35]

**Keywords**

First passage percolation critical percolation correlation length invasion percolation central limit theorem

#### Citation

Damron, Michael; Lam, Wai-Kit; Wang, Xuan. Asymptotics for $2D$ critical first passage percolation. Ann. Probab. 45 (2017), no. 5, 2941--2970. doi:10.1214/16-AOP1129. https://projecteuclid.org/euclid.aop/1506132031