Explosive appearance of cores and bootstrap percolation on lattices

TL;DR

The paper proves that, for two-dimensional bootstrap percolation on a torus with two-neighbour rules (and related large-neighbourhood variants), the time to form a nontrivial core is extremely sharp: just before the hitting time $\tau$, the infected set is $o(n^2)$, while at time $\tau$ it becomes all of $\mathbb{T}$, and $\tau=\Theta(n^2/\log n)$. This is shown via a hybrid approach: a probabilistic AL-style time analysis paired with a robust deterministic framework that builds large infected droplets and rectangles from sparsely infected regions, bypassing the fragility of the classical rectangles process for large neighbourhoods. The method yields a hitting-time result for a broad family of critical bootstrap rules, including $\ell^p$-neighborhoods, and provides a blueprint for extending sharp transition results beyond the classical AL setting. The findings address Benjamini’s question about geometry-driven transitions on planar lattices and highlight potential applications to higher dimensions and edge-infection variants. Overall, the work offers a versatile toolkit for sharp, instantaneous percolation phenomena across finite lattice approximations of planar domains.

Abstract

Consider the process where the $n$ vertices of a square $2$-dimensional torus appear consecutively in a random order. We show that typically the size of the $3$-core of the corresponding induced unit-distance graph transitions from $0$ to $n-o(n)$ within a single step. Equivalently, by infecting the vertices of the torus in a random order, under two-neighbour bootstrap percolation, the size of the infected set transitions instantaneously from $o(n)$ to $n$. This hitting time result answers a question of Benjamini. We also study the much more challenging and general setting of bootstrap percolation on two-dimensional lattices for a variety of finite-range infection rules. In this case, powerful but fragile bootstrap percolation tools such as the rectangles process and the Aizenman-Lebowitz lemma become unavailable. We develop a new method complementing and replacing these standard techniques, thus allowing us to prove the above hitting time result for a wide family of threshold bootstrap percolation rules on the $2$-dimensional square lattice, including neighbourhoods given by large $\ell^p$ balls for $p\in[1,\infty]$.

Figures (10)

• Figure 1: Counterexample for the droplet algorithm for $\square^4$.
• Figure 2: Illustration of the proof of Lemma \ref{['lem:big square']}. Points in the shaded semi-circle are closer to $u$ than $v$ is, their distance from $v$ is less than $s/50$ and they lie in the rectangle $R$.
• Figure 3: Illustration of the proof of Lemma \ref{['lem:scan']}. The translate $R_1-w$ of the shaded polygon $R_1$ is dashed.
• Figure 4: Figures accompanying the proof of Lemma \ref{['lem:blob']}. The thick contour is the boundary of $P$.
• Figure 5: Illustration of the proof of Lemma \ref{['lem:rectangle']}. The curvature of $\partial P$ being extremely small, the convex set $P'$ is much thinner than the rectangle $\Pi$. As a result, neither the upper boundary of $P'$ nor the point $z$ lying on it very close to $z'$ are depicted. Note that the rectangles depicted need not be axis-parallel.

Theorems & Definitions (65)

• Theorem 1
• Lemma 3
• proof
• Lemma 3
• Proposition 3
• proof : Proof of Theorem \ref{['th:main']}, assuming Lemma \ref{['lem:long:rectangle:suffices']} and Proposition \ref{['prop:long:rectangle']}
• Claim 4
• proof : Proof of Claim \ref{['cl:cases']}
• Lemma 5
• proof