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The Time of Bootstrap Percolation in High Dimensions

Fengxing Zhu

TL;DR

This work analyzes the time to complete infection in the $d$-neighbor bootstrap percolation on the $d$-dimensional torus with threshold $r=d$, establishing that the percolation time $T$ is whp $\Theta\left(\log_{1/(1-p)} n\right)$ in the regime $p(n)\ge \frac{C}{\log^{(d)} n}$ and $1-p=n^{-o(1)}$. The authors extend the 2D result of Balister, Bollob\'as and Smith to higher dimensions by a cube-based renormalization approach: the upper bound comes from bounding the probability that a cube is not internally spanned, via an induction on dimension, while the lower bound leverages a Bollob\'as–Smith–Uzzell type analysis linking uninfected regions to empty rectangular configurations. The paper clarifies how higher-dimensional geometry affects percolation time and provides a scalable framework for time-to-percolation via coarse-graining into $[L]^d$ cubes. It also highlights open questions about the full distribution of $T$ in broader regimes and for $r<d$, suggesting directions for future research.

Abstract

We consider the $d$-neighbor bootstrap percolation process on the $d$-dimensional torus, with vertex set $V=\{1,\cdots,n\}^d$ and edge set $\{xy:\sum_{i=1}^d|x_i-y_i (\text{mod} \; n)|=1\}$. We determine the percolation time up to a constant factor with high probability when the initial infection probability is in a certain range and the infection threshold is $d$, extending one of the two main theorems from Balister,Bollob{á}s, and Smith (2016) about the percolation time with the infection threshold equal to $2$ on the two-dimensional torus.

The Time of Bootstrap Percolation in High Dimensions

TL;DR

This work analyzes the time to complete infection in the -neighbor bootstrap percolation on the -dimensional torus with threshold , establishing that the percolation time is whp in the regime and . The authors extend the 2D result of Balister, Bollob\'as and Smith to higher dimensions by a cube-based renormalization approach: the upper bound comes from bounding the probability that a cube is not internally spanned, via an induction on dimension, while the lower bound leverages a Bollob\'as–Smith–Uzzell type analysis linking uninfected regions to empty rectangular configurations. The paper clarifies how higher-dimensional geometry affects percolation time and provides a scalable framework for time-to-percolation via coarse-graining into cubes. It also highlights open questions about the full distribution of in broader regimes and for , suggesting directions for future research.

Abstract

We consider the -neighbor bootstrap percolation process on the -dimensional torus, with vertex set and edge set . We determine the percolation time up to a constant factor with high probability when the initial infection probability is in a certain range and the infection threshold is , extending one of the two main theorems from Balister,Bollob{á}s, and Smith (2016) about the percolation time with the infection threshold equal to on the two-dimensional torus.
Paper Structure (7 sections, 14 theorems, 170 equations, 6 figures)

This paper contains 7 sections, 14 theorems, 170 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Sketch of the proof for the upper bound
  3. Lower Bound
  4. Upper Bound
  5. $d=r=3$
  6. $d=r \geqslant 3$
  7. Open problems

Key Result

Theorem 1

Let $0<p=p(n) <1$ be such that $\liminf p \log \log n > 2\lambda$ and $1-p=n^{-o(1)}$ (that is $\log 1/(1-p) \ll \log n$). Let $T$ denotes the percolation time of a $p$-random subset of $[n]^2$ under the $2$-neighbor bootstrap percolation process and $\lambda=\frac{\pi^2}{18}$. Then we have with high probability as $n \rightarrow \infty$.

Figures (6)

  • Figure 1: Subcubes $C_1,C_2,\cdots,C_8$
  • Figure 2: Configuration on $[(1,2m),(1),(1,2m)]$ after $2m^3+Cm$ steps.
  • Figure 3: Bad subcubes $C_1$ and $C_2$.
  • Figure 4: Configuration on $[(1,2m),(1),(1,2m)]$ after $3m^3+Cm^2$ steps.
  • Figure 5: Bad subcubes $C_1$ and $C_4$.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Lemma 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 16 more