J. Phys. I France
Volume 1, Numéro 5, May 1991
Page(s) 685 - 692
DOI: 10.1051/jp1:1991162
J. Phys. I France 1 (1991) 685-692

Transport through bootstrap percolation clusters

Muhammad Sahimi and Tane S. Ray

Supercomputer Center HLRZ, c/o KFA Jülich, P.O. Box 1913, D-5170 Jülich 1, Germany

(Received 2 January 1991, accepted 17 January 1991)

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., $m\geqslant 4$ for the cubic lattice) first-order phase transitions occur at the percolation threshold, $p_{\rm c}$, 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 $\beta$ 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 $\beta$. For $m\geqslant 4$, the conductivity is discontinuous at $p_{\rm c}$, followed by a power-law jump, as the fraction of conducting material is increased, with a critical exponent that is apparently different from t.

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