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)

** 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.*,
for the cubic lattice) first-order phase transitions occur at the percolation threshold,
, 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
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
. For
, the conductivity is discontinuous at
, 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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*Les Editions de Physique 1991*