[Paper Review] Bootstrap percolation on the Hamming torus with threshold 2

Erik Slivken|arXiv (Cornell University)|Jul 9, 2014

Stochastic processes and statistical mechanics3 citations

TL;DR

This paper studies bootstrap percolation on the $d$-dimensional Hamming torus with threshold 2, establishing critical exponents for the emergence of $d'$-dimensional subtori and showing that for even $d'$, the number of such fully open subtori converges to a Poisson distribution under critical scaling. The work extends phase transition analysis to higher-dimensional structured graphs with non-nearest-neighbor interactions.

ABSTRACT

This paper analyzes various questions pertaining to bootstrap percolation on the $d$-dimensional Hamming torus where each node is open with probability $p$ and the percolation threshold is 2. For each $d'

Motivation & Objective

• To determine the critical exponent for the appearance of $d'$-dimensional subtori in $d$-dimensional Hamming torus under threshold-2 bootstrap percolation.
• To compute the limiting probability of a $d'$-dimensional subtorus becoming fully open under critical scaling.
• To establish Poisson approximation for the number of fully open $d'$-dimensional subtori when $d'$ is even, using the Chen-Stein method.
• To extend phase transition analysis beyond nearest-neighbor lattices to the Hamming torus, where vertex neighborhoods are larger and more symmetric.

Proposed method

• Uses the Chen-Stein method to prove Poisson convergence for the number of fully open $d'$-dimensional subtori when $d'$ is even.
• Applies induction and combinatorial estimates to bound the probability of joint activation of multiple subtori.
• Defines subtori as subsets where certain coordinates are fixed, and analyzes their dimensionality and intersection structure.
• Employs a recursive decomposition of configurations based on the dimension and intersection of subtori.
• Derives asymptotic bounds on probabilities using polynomial growth rates in $n$, the side length of the torus.
• Analyzes the critical scaling regime where $p = n^{- heta}$, and computes the exponent $\theta$ that governs the phase transition for subtorus formation.

Experimental results

Research questions

• RQ1 What is the critical exponent $\gamma$ such that the probability of a $d'$-dimensional subtorus becoming fully open transitions sharply from 0 to 1 at $p = n^{-\gamma}$?
• RQ2 For even $d'$, does the number of fully open $d'$-dimensional subtori converge in distribution to a Poisson random variable under critical scaling?
• RQ3 How does the critical probability for subtorus formation on the Hamming torus compare to that on the standard grid, given the larger neighborhood size?
• RQ4 What is the limiting probability of a $d'$-dimensional subtorus becoming fully open when $p$ is scaled as $n^{-\gamma}$ for the critical exponent $\gamma$?
• RQ5 How do the intersection patterns of subtori affect the probability of simultaneous activation in the bootstrap process?

Key findings

• For each $d' < d$, the paper identifies the critical exponent $\gamma$ such that the probability of a $d'$-dimensional subtorus becoming fully open tends to 0 if $p \ll n^{-\gamma}$ and to 1 if $p \gg n^{-\gamma}$.
• The limiting probability of a $d'$-dimensional subtorus becoming fully open under critical scaling is computed explicitly, showing a continuous transition from 0 to 1 as the scaling parameter increases.
• For even $d'$, the number of fully open $d'$-dimensional subtori converges in distribution to a Poisson random variable under the critical scaling, as shown via the Chen-Stein method.
• The critical exponent for $\mathcal{C}_{d'}$, the event that some $d'$-dimensional subtorus becomes fully open, is derived and shown to depend on the dimension $d'$ and the structure of the Hamming torus.
• The analysis reveals that the phase transition for subtorus formation is sharp, with a critical window of order $n^{-\gamma \pm \epsilon}$, consistent with the definition of sharp thresholds.
• The paper establishes that the probability of a subtorus becoming fully open is asymptotically independent of other subtori when $d'$ is even, justifying the Poisson approximation.