Article,
Probabilistic bootstrap percolation
Journal of Statistical Physics, 70 (3): 1035--1044 (Feb 1, 1993)
DOI: 10.1007/bf01053606
Abstract
In bootstrap percolation, sites are occupied with probabilityp, but those with less thanm occupied first neighbors are removed. This culling process is repeated until a stable configuration (all occupied sites have at leastm occupied first neighbors or the whole lattice is empty) is achieved. Form?m1 the transition is first order, while formm1 it is second order, withm-dependent exponents. In probabilistic bootstrap percolation, sites have probabilityr or (1-r) of beingm- orm'-sites, respectively (m-sites are those which need at leastm occupied first neighbors to remain occupied). We have studied the model on Bethe lattices, where an exact solution is available. Form=2 andm'=3, the transition changes from second to first order atr1=1/2, and the exponent ß is different forrr=1/2, andr>1/2. The same qualitative behavior is found form=1 andm'=3. On the other hand, form=1 andm'=2 the transition is always second order, with the same exponents ofm=1, for any value ofr>0. We found, form=z-1 andm'=z, wherez is the coordination number of the lattice, thatpc=1 for a value ofr which depends onz, but is always above zero. Finally, we argue that, for bootstrap percolation on real lattices, the exponents ? and ß form=2 andm=1 are equal, for dimensions below 6.
