Diffusion percolation. I. Infinite time limit and bootstrap percolation

Several models for the dynamic growth of percolation clusters, or 'diffusion percolation' (DP), are introduced and analysed. In these models a random walker (an 'ant') walks on percolation clusters (which are occupied with initial site concentration, pi). The 'ant' is allowed to step off such clusters and add new sites to them if certain conditions are met. Some of these models are shown to have a one-to-one correspondence with models of bootstrap percolation (BP), in which sites which do not have a required number of neighbours are successively culled. Two new percolation thresholds have been calculated for two diffusion percolation models on the square lattice.

[1]  L. Paterson.  Simulations of fluid displacement in heterogeneous porous media , 1987 .

[2]  A. McKane,et al.  Critical exponents for the percolation problem and the Yang-Lee edge singularity , 1981 .

[3]  Harvey Gould,et al.  Kinetics of Aggregation and Gelation , 1985 .

[4]  Meyer,et al.  Transmission of order in some unusual dilute systems. , 1987, Physical review letters.

[5]  D. Rapaport.  Monte Carlo experiments on percolation: the influence of boundary conditions , 1985 .

[6]  David Wilkinson,et al.  Invasion percolation: a new form of percolation theory , 1983 .

[7]  A. Nihat Berker,et al.  Scaling for first-order phase transitions in thermodynamic and finite systems , 1982 .

[8]  H. Stanley,et al.  Tip splitting without interfacial tension and dendritic growth patterns arising from molecular anisotropy , 1986, Nature.

[9]  A. Enter.  Proof of Straley's argument for bootstrap percolation , 1987 .

[10]  Agudah ha-fisiḳalit le-Yiśraʾel,et al.  Fragmentation, form and flow in fractured media : proceedings of the F3-Conference held at Neve Ilan (Israel), 6-9 January, 1986 , 1986 .

[11]  P. Leath,et al.  Bootstrap percolation on a Bethe lattice , 1979 .

[12]  H. Gould,et al.  Monte Carlo renormalisation group study of bootstrap percolation , 1985 .

[13]  J. G. Zabolitzky,et al.  Re-examination of 3D percolation threshold estimates , 1986 .

[14]  P. Leath,et al.  Bootstrap percolation transitions on real lattices , 1981 .

[15]  M. Nauenberg,et al.  Renormalization-group calculation for the equation of state of an Ising ferromagnet , 1975 .

[16]  S. D. Queiroz,et al.  Critical exponents for high density and bootstrap percolation , 1986 .