Polluted Modified Bootstrap Percolation

TL;DR

This work analyzes polluted modified bootstrap percolation on the 2D lattice in the regime of vanishing $p$ and $q$, revealing a logarithmic correction to the critical scaling relative to the standard model. It establishes an upper bound via a safe-block blocking construction that, with high probability, prevents inward infection when $q \ge C p^2 / \log (1/p)$, and a lower bound via good-box percolation that ensures infection reaches the origin when $q \le p^2/(\log(1/p)(\log\log(1/p))^4)$. The results identify chimneys as a key obstruction mechanism and employ domination by independent percolation and duality to connect local block behavior to global infection outcomes. Together, they settle a long-standing conjecture by Gravner and McDonald (1997) and quantify a precise logarithmic correction to the critical scaling the final density follows.

Abstract

In the polluted modified bootstrap percolation model, sites in the square lattice are independently initially occupied with probability $p$ or closed with probability $q$. A site becomes occupied at a subsequent step if it is not closed and has at least one occupied nearest neighbor in each of the two coordinates. We study the final density of occupied sites when $p$ and $q$ are both small. We show that this density approaches $0$ if $q\ge Cp^2/\log p^{-1}$ and $1$ if $q\le p^2/(\log p^{-1})^{1+o(1)}$. Thus we establish a logarithmic correction in the critical scaling, which is known not to be present in the standard model, settling a conjecture of Gravner and McDonald from 1997.

Figures (5)

• Figure 1: The dynamics with $p=0.1$, $q=0.01$ attempts to fill in a square. Black sites are initially occupied, including all sites on the boundary, grey sites are eventually occupied, and red sites are closed.
• Figure 2: A safe block (outlined in blue) has a closed site marked by $\times$ and no occupied sites in the red protective region. Its core is outlined by thicker red lines. Dimensions of rectangles are given with integer parts omitted.
• Figure 3: Safe blocks are outlined in black, and red lines are the cores of their protective regions. Thicker red lines are the blocking structure. Even if all sites above the blocking structure are occupied, they cannot influence the configuration below the structure without help from additional occupied sites below.
• Figure 4: The box $S$ and its subregions. The name of each subregion is at its top right corner. The outline of $S_1$ is thicker. Integer parts are again omitted.
• Figure 5: Elimination of a closed site, marked by the red $\times$, at $x$. The dark grey region becomes occupied either by the assumption (the bottom portion) or because it is a part of $S_2$ (the right portion). The light grey region --- the dashed rectangle is $R(x-re_1)$ --- includes no closed site, and thus also becomes occupied. Each of the four arms of the cross has length $r$, includes an initially occupied site (a black circle) but no closed sites, and thus also becomes occupied.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof : Proof of Theorem 1.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof