Probabilistic bootstrap percolation

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⩾m ₁ the transition is first order, while form<m ₁ 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 atr ₁=1/2, and the exponent β is different forr<1/2,r=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, thatp _c=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 ν and β form=2 andm=1 are equal, for dimensions below 6.