2009/04/30 by Marek Biskup, Roberto H. Schonmann · 1 citation
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:60K35 #msc:82C05 #msc:82C22 #msc:82C43
paper · pdf · doi:10.1007/s10955-009-9798-x
published as J. Statist. Phys. 136 (2009), no. 4, 667-676 · 10 pages, version to appear in J. Statist. Phys
arxiv created 2009/07/25 · openalex publication_date 2009/08/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least θ occupied neighbors, occupied sites remain occupied forever. It is known that, when b>θ≥2, the limiting density q=q(p) of occupied sites exhibits a jump at some p T=p T(b,θ)∈(0,1) from q T:=q(p T)<1 to q(p)=1 when p>p T. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p T+h with h>0 and show that, as h ↓0, the system lingers around the “critical” state for time order h −1/2 and then passes to fully occupied state in time O(1). The law of the entire configuration observed when the occupation density is q∈(q T,1) converges, as h ↓0, to a well-defined measure.