2002/06/12 by Alexander E. Holroyd, Holroyd, Alexander E. · 4 citations
Computer Science · Mathematics · Physics and Astronomy · #60K35 #82B43 #Bayesian Methods and Mixture Models #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:60K35 #msc:82B43
paper · pdf · doi:10.48550/arxiv.math/0206132
arxiv created 2002/06/12 · openalex publication_date 2002/06/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the bootstrap percolation model, sites in an L by L square are initially independently declared active with probability p. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as p → 0 and L → ∞ simultaneously of the probability I(L,p) that the entire square is eventually active. We prove that I(L,p) → 1 if \liminf p log L > λ, and I(L,p) → 0 if \limsup p log L < λ, where λ= π2/18. We prove the same behaviour, with the same threshold λ, for the probability J(L,p) that a site is active by time L in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold λ' = π2/6. The existence of the thresholds λ,λ' settles a conjecture of Aizenman and Lebowitz, while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony.