Higher order corrections for anisotropic bootstrap percolation

We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with (1, 2)-neighbourhood and threshold r = 3 . The first order asymptotics for the critical probability were recently determined by the first and second authors. Here...

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Bibliographic Details
Published in:Probability theory and related fields 2018-10, Vol.172 (1-2), p.191-243
Main Authors: Duminil-Copin, Hugo, van Enter, Aernout C. D., Hulshof, Tim
Format: Article
Language:English
Subjects:
Anisotropy
Asymptotic properties
Economics
Finance
Insurance
Lattices
Management
Mathematical and Computational Biology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Metastable phases
Operations Research/Decision Theory
Percolation
Phase transitions
Probability
Probability Theory and Stochastic Processes
Quantitative Finance
Statistics for Business
Theoretical
Two dimensional models
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Summary:We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with (1, 2)-neighbourhood and threshold r = 3 . The first order asymptotics for the critical probability were recently determined by the first and second authors. Here we determine the following sharp second and third order asymptotics: p c ( [ L ] 2 , N ( 1 , 2 ) , 3 ) = ( log log L ) 2 12 log L - log log L log log log L 3 log L + log 9 2 + 1 ± o ( 1 ) log log L 6 log L . We note that the second and third order terms are so large that the first order asymptotics fail to approximate p c even for lattices of size well beyond 10 10 1000 .
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-017-0808-7