Critical exponents for site-bond-correlated percolation

Distribution of averages in a correlated Gaussian medium as a tool for estimation of the cluster size distribution

Calculation of the distribution of the average value of a Gaussian random field in a finite domain is carried out for different cases. The results of the calculation demonstrate a strong dependence of the width of the distribution on the spatial correlations of the field. Comparison to the simulation results for the distribution of the size of the cluster indicates that the distribution of an average field could serve as a useful tool for the estimation of the asymptotic behavior of the distribution of the size of the clusters for "deep" clusters where value of the field on each site is much greater than the rms disorder.

Finite-size scaling analysis of percolation in three-dimensional correlated binary Markov chain random fields

Percolation and finite-size scaling properties in three-dimensional binary correlated Markov-chain random fields on a cubic lattice are computed by extensive Monte Carlo simulation. At short correlation scales, the percolation threshold in correlated random fields decreases as the correlation scale increases. The rate of decrease rapidly diminishes for correlation lengths larger than 2-3 lattice sites. At correlation scales of 4-6 lattice sites, the percolation threshold is found to be 0.126 +/- 0.001 for the Markov chain random fields, similar to that for sequential Gaussian and indicator random fields, which are evaluated for comparison. The average percolation threshold in finite-size lattices is a function of both, the correlation length and the finite lattice size. The universal scaling constants for mean cluster size and backbone fraction are found to be consistent with results on uncorrelated lattices. But prefactors of scaling relationships vary with correlation length. The squared radius of gyration of non-percolating clusters is found to scale with gamma/nu and its scaling prefactors are independent of the correlation scale. Prefactors are similar between the three random field generators evaluated. The percolation properties derived here are useful to account for finite-size effects on percolation in natural or man-made correlated systems.