2008/06/13 by Janko Gravner, Gravner, Janko, Alexander E. Holroyd +1 · 1 citation
Mathematics · Physics and Astronomy · #60K35 #82B43 #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:60K35 #msc:82B43
paper · pdf · doi:10.48550/arxiv.0806.2313
19 pages, 2 figures
arxiv created 2008/06/13 · openalex publication_date 2008/06/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single active site at the origin, while other sites of Z2 are independently occupied with small probability p, otherwise empty. Subsequently, an empty site becomes active by contact with 2 or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability exp[alpha(p)/p], where alpha(p) is between -pi2/9 + c sqrt p and pi2/9 + C sqrt p (-log p)3. This corrects previous numerical predictions for the scaling of the correction term.