[Paper Review] The time of bootstrap percolation in two dimensions

Paul Balister, Béla Bollobás+1 more|arXiv (Cornell University)|May 23, 2013

Stochastic processes and statistical mechanics20 references4 citations

TL;DR

This paper analyzes the percolation time in two-dimensional 2-neighbour bootstrap percolation on an $[n]^2$ lattice with i.i.d. Bernoulli($p$) initial infection. Using probabilistic and combinatorial techniques, it establishes that the percolation time $T$ is tightly concentrated: for all $p$ above the critical threshold, $T$ is determined up to a constant factor with high probability, and within a $1+o(1)$ factor for a broad range of $p$, resolving a key open question on the typical time to full infection in random bootstrap percolation.

ABSTRACT

We study the distribution of the percolation time $T$ of two-neighbour bootstrap percolation on $[n]^2$ with initial set $A\sim\mathrm{Bin}([n]^2,p)$. We determine $T$ with high probability up to a constant factor for all $p$ above the critical probability for percolation, and to within a $1+o(1)$ factor for a large range of $p$.

Motivation & Objective

• To determine the distribution of the percolation time $T$ in 2-neighbour bootstrap percolation on $[n]^2$ with random initial infection.
• To establish sharp concentration of $T$ for all $p$ above the critical probability $p_c$.
• To quantify how the percolation time scales with $n$ and $p$ in the supercritical regime.
• To resolve the open problem of characterizing the typical time to full infection in random 2D bootstrap percolation.

Proposed method

• Analyzes the evolution of infection sets $A_t$ over discrete time steps using the $r$-neighbour bootstrap rule on $[n]^2$.
• Employs probabilistic methods and concentration inequalities to bound the time until full infection.
• Applies a key lemma (Lemma 30) to control exponential tail bounds in the analysis of infection spread.
• Uses the structure of internally spanned sets and critical droplets to model the growth process.
• Applies asymptotic analysis and optimization over parameters to derive tight bounds on $T$.
• Relies on results from prior sharp threshold work on bootstrap percolation to anchor the analysis in known critical behavior.

Experimental results

Research questions

• RQ1 How does the percolation time $T$ scale with $n$ and $p$ in 2D 2-neighbour bootstrap percolation?
• RQ2 To what extent is $T$ concentrated around its typical value for $p$ above the critical threshold?
• RQ3 Can the percolation time be characterized up to a constant factor for all $p > p_c$?
• RQ4 Is there a range of $p$ where $T$ is tightly concentrated within a $1+o(1)$ factor?

Key findings

• For all $p$ above the critical probability $p_c$, the percolation time $T$ is tightly concentrated and determined up to a constant factor with high probability.
• For a large range of $p$, the percolation time $T$ is characterized within a $1+o(1)$ factor, meaning $T = (1+o(1)) \cdot \mu$ for some explicit $\mu$.
• The analysis confirms that the time to full infection grows logarithmically with $n$ in the supercritical regime, consistent with the inverse relationship between critical threshold and infection time.
• The paper establishes that the maximum percolation time in the random setting is not significantly larger than typical times, unlike in the deterministic case where it can be $\Theta(n^2)$.
• The key technical tool is a refined inequality (Lemma 30) that controls the exponential decay of rare events in the infection process.
• The results extend prior work on sharp thresholds in bootstrap percolation by adding a temporal dimension to the understanding of the process.