Transport through bootstrap percolation clusters

Abstract

In bootstrap percolation (BP) on lattices sites are initially occupied at random. Those occupied sites that do not have at least m occupied nearest-neighbors are then removed. For sufficiently large values of m (e.g., m≥4 for the cubic lattice) first-order phase transitions occur at the percolation threshold, pc, while for small values of m the phase transition is second-order. We study conductivity of BP clusters as a function of m, the dimensionality of the system and its linear size L. This is relevant to spin-wave stiffness of disordered magnetic systems, e.g., the dilute Blume-Capel model and, as we argue here, it may also be relevant to the behavior of disordered solids that undergo a brittle fracture process, and to flow through a porous medium. On a cubic lattice we find that the conductivity critical exponent t for m=3 is the same as that of random percolation (m=0). Since for m=0-3 the correlation length exponent also remains unchanged, but the critical exponent β of the strength of the infinite clusters is different for m=2 and 3, we argue that this indicates that for three-dimensional systems t cannot be related to β. For m≥4, the conductivity is discontinuous at pc, followed by a power-law jump, as the fraction of conducting material is increased, with a critical exponent that is apparently different from t.