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Nucleation and growth in two dimensions
Author(s) -
Bollobás Béla,
Griffiths Simon,
Morris Robert,
Rolla Leonardo,
Smith Paul
Publication year - 2020
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20888
Subject(s) - sharpening , percolation (cognitive psychology) , nucleation , statistical physics , percolation threshold , graph , mathematics , constant (computer programming) , combinatorics , physics , computer science , thermodynamics , quantum mechanics , artificial intelligence , biology , neuroscience , programming language , electrical resistivity and conductivity
We consider a dynamical process on a graph G , in which vertices are infected (randomly) at a rate which depends on the number of their neighbors that are already infected. This model includes bootstrap percolation and first‐passage percolation as its extreme points. We give a precise description of the evolution of this process on the graphZ 2 , significantly sharpening results of Dehghanpour and Schonmann. In particular, we determine the typical infection time up to a constant factor for almost all natural values of the parameters, and in a large range we obtain a stronger, sharp threshold.