[Paper Review] Two Phase Transitions in Two-way Bootstrap Percolation

Ahad N. Zehmakan|arXiv (Cornell University)|Sep 26, 2018

Stochastic processes and statistical mechanics10 references4 citations

TL;DR

This paper studies two-way r-bootstrap percolation on a d-dimensional torus, proving it exhibits two distinct phase transitions at critical thresholds p₁ and p₂. Using clustering and scaling techniques, it establishes that the process fully blackens the graph when p ≫ p₂, stabilizes in a mixed state when p₂ ≫ p ≫ p₁, and becomes fully white when p ≪ p₁, with precise threshold values derived for the d-dimensional torus.

ABSTRACT

Consider a graph $G$ and an initial random configuration, where each node is black with probability $p$ and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least $r$ black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a $d$-dimensional torus and identify the threshold values.

Motivation & Objective

• To analyze the threshold behavior of two-way r-bootstrap percolation on the d-dimensional torus.
• To identify two distinct phase transitions in the process's behavior as the initial infection probability p varies.
• To establish precise threshold values p₁ and p₂ that separate regimes of full blackening, mixed stabilization, and full whitening.
• To extend the analysis to a broader class of monotone models, including (r,r′)-bootstrap percolation with recovery.
• To provide a framework for proving similar threshold behaviors in more general models using eternal sets and structural properties.

Proposed method

• Uses a two-phase analysis: first proving that p ≪ p₂ leads to full whitening via clustering arguments.
• Applies Chernoff bounds to show that for p ≫ p₂, a.a.s. at least one b-eternal set is fully black initially, ensuring eventual full blackening.
• Employs a scaling technique by tiling the torus into r′-dimensional hyper-squares to reduce the problem to modified r-bootstrap percolation.
• Leverages the existence of constant-size b-eternal sets (of size s) as structural anchors for the analysis.
• Uses union bounds and symmetry arguments to handle boundary effects and dependency issues in the random configuration.
• Extends the proof framework to (r,r′)-bootstrap percolation models with recovery, where black nodes may revert if too few black neighbors.

Experimental results

Research questions

• RQ1 What are the critical thresholds p₁ and p₂ that separate the phase transitions in two-way r-bootstrap percolation on the d-dimensional torus?
• RQ2 How does the presence of b-eternal sets of size s influence the threshold behavior of the process?
• RQ3 Can the two-phase transition behavior be generalized to (r,r′)-bootstrap percolation models with recovery?
• RQ4 What is the role of structural properties like symmetry and dimensionality in determining the threshold values?
• RQ5 How do clustering and scaling techniques enable the derivation of sharp threshold behavior in non-monotone, reversible processes?

Key findings

• Two-way r-bootstrap percolation on the d-dimensional torus exhibits two phase transitions at thresholds p₁ = 𝒪(𝒫₁^{1/s}) and p₂ = 𝒪(𝒫₂^{1/s}), where s is the minimum size of a b-eternal set.
• When p ≪ p₂, the process a.a.s. becomes fully white due to insufficient initial black clusters.
• When p ≫ p₂, a.a.s. at least one b-eternal set is fully black initially, ensuring the entire graph turns black in finite time.
• For p₂ ≫ p ≫ p₁, the process stabilizes in a mixed state with both black and white nodes persisting, due to balanced dynamics.
• The threshold values are derived using clustering techniques for the whitening regime and scaling arguments for the blackening regime.
• The framework extends to (r,r′)-bootstrap percolation, suggesting two-phase transitions also occur when r′ < r, provided constant-size b-eternal sets exist.