PROBABILISTIC BOOTSTRAP PERCOLATION

In bootstrap percolation, sites are occupied with probability p, but those with less than m occupied first neighbors are removed. This culling process is repeated until a stable configuration (all occupied sites have at least m occupied first neighbors or the whole lattice is empty) is achieved. For m greater-than-or-equal-to m1, the transition is first order, while for m < m1 it is second order, with m-dependent exponents. In probabilistic bootstrap percolation, sites have probability r or (1 - r) of being m- or m'-sites, respectively (m-sites are those which need at least m occupied first neighbors to remain occupied). We have studied the model on Bethe lattices, where an exact solution is available. For m = 2 and m' = 3, the transition changes from second to first order at r1 = 1/2, and the exponent beta is different for r < 1/2, r = 1/2, and r > 1/2. The same qualitative behavior is found for m = 1 and m' = 3. On the other hand, for m = 1 and m' = 2 the transition is always second order, with the same exponents of m = 1, for any value of r > 0. We found, for m = z - 1 and m' = z, where z is the coordination number of the lattice, that p(c) = 1 for a value of r which depends on Z, but is always above zero. Finally, we argue that, for bootstrap percolation on real lattices, the exponents nu and beta for m = 2 and m = 1 are equal, for dimensions below 6.