[Paper Review] Asymptotics for $2D$ Critical First Passage Percolation

Michael Damron, Wai‐Kit Lam+1 more|arXiv (Cornell University)|May 28, 2015

Stochastic processes and statistical mechanics16 references3 citations

TL;DR

This paper establishes sharp conditions for the asymptotic behavior of passage times in 2D critical first-passage percolation with i.i.d. edge weights satisfying $F(0) = p_c = 1/2$. It introduces a novel connection between first-passage percolation and invasion percolation, showing that passage time growth mirrors that of optimal paths constrained within an invasion cluster. The key result resolves long-standing conjectures by Zhang and Kesten, proving a central limit theorem under minimal moment assumptions and characterizing when mean or variance diverges.

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 $[-n,n] imes [-n,n]$. 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 '90s and, in particular, disprove a conjecture of Zhang from '99. In the case when both the mean and the variance go to infinity as $n o \infty$, 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.

Motivation & Objective

• To resolve open questions about passage time behavior in 2D critical first-passage percolation where the time constant vanishes due to $F(0) = p_c = 1/2$.
• To determine exact conditions under which the mean or variance of passage time $T(\mathbf{0}, \partial B(n))$ remains bounded or diverges as $n \to \infty$.
• To disprove Zhang's conjecture that $\sup\{a > 0 : \rho(F_a) < \infty\}$ is finite, showing instead that $\rho(F_a) < \infty$ for arbitrarily small $a > 0$.
• To establish a central limit theorem for passage times when variance diverges, under minimal moment assumptions.

Proposed method

• Introduce a coupling between first-passage percolation and invasion percolation, showing that passage time growth matches that of optimal paths within the invasion cluster up to a constant factor.
• Use a dyadic decomposition of space, defining $Y(x)$ as the sum of first-passage times to the boundaries of two disjoint boxes of radius $2^{q-1}$ centered at $\mathbf{0}$ and $x$, respectively.
• Bound the difference $|T(\mathbf{0},x) - Y(x)|$ using estimates on passage times to a random cluster $\mathcal{C}_{q+2}$, which captures the invasion cluster's influence.
• Apply moment bounds on the error term via Lemma 5.10 and Lemma 5.14, which control the $L^r$-norms of the difference between geodesic times and boundary times.
• Leverage independence of passage times in disjoint boxes to derive convergence in distribution and establish the central limit theorem under the condition $\sum_{k=2}^\infty [F^{-1}(p_c + 2^{-k})]^2 = \infty$.
• Use the inverse distribution function $F^{-1}(p_c + 2^{-k})$ as a key scale parameter to characterize the tail behavior of edge weights and derive necessary and sufficient conditions for divergence of mean and variance.

Experimental results

Research questions

• RQ1 Under what conditions on the edge-weight distribution $F$ does the passage time $T(\mathbf{0}, \partial B(n))$ have uniformly bounded mean or variance as $n \to \infty$?
• RQ2 Is Zhang's conjecture that $\sup\{a > 0 : \rho(F_a) < \infty\}$ is finite true, or can $\rho(F_a) < \infty$ almost surely for arbitrarily small $a > 0$?
• RQ3 When does the passage time $T(\mathbf{0}, \partial B(n))$ satisfy a central limit theorem, and what minimal moment assumptions are required?
• RQ4 How does the growth of passage times in critical first-passage percolation relate to the invasion percolation process?

Key findings

• The passage time $T(\mathbf{0}, \partial B(n))$ has uniformly bounded mean if and only if $\sum_{k=2}^\infty [F^{-1}(p_c + 2^{-k})] < \infty$, providing a sharp condition for bounded mean.
• The passage time has uniformly bounded variance if and only if $\sum_{k=2}^\infty [F^{-1}(p_c + 2^{-k})]^2 < \infty$, establishing a precise criterion for bounded variance.
• Zhang's conjecture that $\sup\{a > 0 : \rho(F_a) < \infty\}$ is finite is disproven; for any $a > 0$, $\rho(F_a) < \infty$ almost surely, even as $a \to 0^+$.
• When $\sum_{k=2}^\infty [F^{-1}(p_c + 2^{-k})]^2 = \infty$, the passage time satisfies a central limit theorem: $\frac{T(\mathbf{0},x) - \mathbb{E}[T(\mathbf{0},x)]}{\operatorname{Var}(T(\mathbf{0},x))^{1/2}} \Rightarrow N(0,1)$ as $\|x\|_\infty \to \infty$, under minimal moment assumptions.
• The asymptotic variance of $T(\mathbf{0},x)$ satisfies $\frac{\operatorname{Var}(T(\mathbf{0},x))}{\operatorname{Var}(T(\mathbf{0},\partial B(2^{q(x)})))} \to 2$ as $\|x\|_\infty \to \infty$, indicating a universal scaling behavior.
• The main technical advance is proving that passage time growth in critical first-passage percolation is governed by optimal paths constrained within an invasion cluster, up to a constant factor, enabling precise asymptotic analysis.