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

TL;DR

The paper investigates first-passage percolation on logarithmic wedge subgraphs of the 2D lattice, connecting percolation geometry to FPP growth through a gap condition near zero for edge-weights. By coupling FPP with a Bernoulli model and exploiting duality, the authors derive sharp asymptotics for the mean passage time $\mathbb{E}T_{a,b}(0,P(n))$ across regimes determined by the gap, the parameters $a,b$ of the logarithmic wedge, and the correlation length $\xi(1-p)$; they show sublinear, polylogarithmic, and even constant growth depending on these parameters. A central limit theorem and variance bounds are established for a broad class of wedges, using a Kesten–Zhang style martingale construction with detailed control of increments $\Delta_{i,i_0}$ and a robust concentration analysis. They also prove a discontinuity criterion for percolation in $\mathbb{G}_{a,b}$: the phase transition is discontinuous iff $b>a$, and they provide improved sponge-crossing dimension results for subcritical crossing probabilities. Collectively, the results illuminate how logarithmic-wedge geometry and gap conditions jointly govern FPP scaling, fluctuations, and phase transition behavior.

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 $(τ_e)$ satisfying $p = \mathbb{P}(τ_e=0) \in [1/2,1)$ and the "gap condition" $\mathbb{P}(τ_e \leq δ) = p$ for some $δ>0$. 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.

Figures (2)

• Figure 1: Illustration of $\mathbb{G}_{a,b}(n)$ for $n=10$. The curve is the graph of $f(u) = a \log (1+u) + b \log (1 + \log (1+u))$ and vertices of $\mathbb{G}_{a,b}(n)$ are the solid dots (black and red). The vertices of the dual graph are the non-filled circles, squares, and triangles. Those of the "highest path" $P_{a,b}(n)$ are displayed in red, the "top side" of $\mathbb{G}_{a,b}^\ast(n)$ is $H_{a,b}^\ast(n)$, whose vertices are displayed as squares, and the "bottom side" is $L_{a,b}^\ast(n)$, whose vertices are displayed as triangles. The values listed on the $x$-axis are $\ell_j$, the smallest value of $u$ for which $f(u) \geq j$. Because $f$ increases sharply near 0 in the figure, we have $f(0) = 0$ but $f(1) \geq 2$, so $\ell_1=\ell_2=1$.
• Figure 2: Illustration of the definitions involved in the martingale construction. In the figure, the top solid curve is the graph of $f$ and the region below the curve and between $r_{i-1}'=r_{2i-2}$ and $r_{2i-1}$ is $R_{i-1}'$. The dotted curves connecting the top solid curve to the bottom are open paths. There is no top-down open crossing of $R_{i-1}'$, so $m(i-1) > i-1$. The region to the right of $r_i'$ below the graph of $f$ is $R_i'$, and it has a top-down open crossing, so $m(i-1) = m(i)=i$ and the leftmost such crossing is labeled as $\Gamma_{i-1}$, which is also equal to $\Gamma_i$. The middle two open paths connecting the top solid curve to the bottom are not contained in any $R_j'$, so they do not affect the values of $m(i-1),m(i), \Gamma_{i-1},\Gamma_i$.

Theorems & Definitions (42)

• Theorem 1.1: Ahlberg
• Theorem 1.2
• Remark 1
• Remark 2
• Remark 3
• Theorem 1.3
• Remark 4
• Remark 5
• Corollary 1.4
• proof