Asymptotics for first-passage percolation on logarithmic subgraphs of $\mathbb{Z}^2$

TL;DR

This paper investigates first-passage percolation on specific logarithmic subgraphs of ^2, revealing how passage times grow and phase transition properties depend on parameters, with new limit theorems and improved percolation transition criteria.

Contribution

It provides new asymptotic growth rates for passage times, establishes a central limit theorem for general functions, and refines criteria for discontinuous phase transitions on these subgraphs.

Findings

Passage time growth rate is order n/(a log n) at p=1/2.

For p>1/2, passage times can grow as n^{c_1}/( ext{log} n)^{c_2}, ( ext{log} n)^{c_3}, ext{log} ext{log} n, or constant.

Percolation phase transition is discontinuous if and only if b > a.

Abstract

For $a>0$ and $b \geq 0$, let $\mathbb{G}_{a,b}$ be the subgraph of $\mathbb{Z}^2$ induced by the vertices between the first coordinate axis and the graph of the function $f = f_{a,b}(u) = a \log (1+u) + b \log(1+\log(1+u))$, $u \geq 0$. It is known that for $a>0$, the critical value for Bernoulli percolation on $\mathbb{G}_f = \mathbb{G}_{a,b}$ is strictly between $1/2$ and $1$, and that if $b>2a$ then the percolation phase transition is discontinuous. We study first-passage percolation (FPP) on $\mathbb{G}_{a,b}$ with i.i.d. edge-weights $(\tau_e)$ satisfying $p = \mathbb{P}(\tau_e=0) \in [1/2,1)$ and the "gap condition" $\mathbb{P}(\tau_e \leq \delta) = p$ for some $\delta>0$. We find the rate of growth of the expected passage time in $\mathbb{G}_f$ from the origin to the line $x=n$, and show that, while when $p=1/2$ it is of order $n/(a \log n)$, when $p>1/2$ it can be of order (a) $n^{c_1}/(\log n)^{c_2}$, (b) $(\log n)^{c_3}$, (c) $\log \log n$, or (d) constant, depending on the relationship between $a,b,$ and $p$. For more general functions $f$, we prove a central limit theorem for the passage time and show that its variance grows at the same rate as the mean. As a consequence of our methods, we improve the percolation transition result by showing that the phase transition on $\mathbb{G}_{a,b}$ is discontinuous if and only if $b > a$, and improve "sponge crossing dimensions" asymptotics from the '80s on subcritical percolation crossing probabilities for tall thin rectangles.