Metastable Behavior for Bootstrap Percolation on Regular Trees
Result description
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 ? occupied neighbors, occupied sites remain occupied forever. It is known that, when b > 8 >= 2, the limiting density q = q(p) of occupied sites exhibits a jump at some pT = pT(b, 8) (0, 1) from qT := q(pT) < 1 to q(p) = 1 when p > pT. We investigate the metastable behavior associated with this transition. Explicitly, we pick p = pT + 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 occupationdensity is q (qT, 1) converges, as h 0, to a well-defined measure.
Keywords
Bootstrap percolationBethe latticeMetastabilityCutoff phenomenon
The result's identifiers
Result code in IS VaVaI
Result on the web
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DOI - Digital Object Identifier
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Alternative languages
Result language
angličtina
Original language name
Metastable Behavior for Bootstrap Percolation on Regular Trees
Original language description
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 ? occupied neighbors, occupied sites remain occupied forever. It is known that, when b > 8 >= 2, the limiting density q = q(p) of occupied sites exhibits a jump at some pT = pT(b, 8) (0, 1) from qT := q(pT) < 1 to q(p) = 1 when p > pT. We investigate the metastable behavior associated with this transition. Explicitly, we pick p = pT + 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 occupationdensity is q (qT, 1) converges, as h 0, to a well-defined measure.
Czech name
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Czech description
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Classification
Type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BB - Applied statistics, operational research
OECD FORD branch
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Result continuities
Project
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Continuities
V - Vyzkumna aktivita podporovana z jinych verejnych zdroju
Others
Publication year
2009
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Journal of Statistical Physics
ISSN
0022-4715
e-ISSN
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Volume of the periodical
136
Issue of the periodical within the volume
4
Country of publishing house
US - UNITED STATES
Number of pages
10
Pages from-to
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UT code for WoS article
000269885000004
EID of the result in the Scopus database
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Basic information
Result type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP
BB - Applied statistics, operational research
Year of implementation
2009