Finite-Size Effects for Anisotropic Bootstrap Percolation: Logarithmic Corrections

Abstract In this note we analyse an anisotropic, two-dimensional bootstrap percolation model introduced by Gravner and Griffeath. We present upper and lower bounds on the finite-size effects. We discuss the similarities with the semi-oriented model introduced by Duarte.

[1]  E. F. Codd,et al.  Cellular automata , 1968 .

[2]  R. Lenormand Pattern growth and fluid displacements through porous media , 1986 .

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

[4]  Michael Aizenman,et al.  Metastability effects in bootstrap percolation , 1988 .

[5]  J.A.M.S. Duarte Simulation of a cellular automat with an oriented bootstrap rule , 1989 .

[6]  J. Adler,et al.  Finite-size effects for some bootstrap percolation models , 1990 .

[7]  R. Schonmann Critical points of two-dimensional bootstrap percolation-like cellular automata , 1990 .

[8]  Finite size effects for some bootstrap percolation models , 1991 .

[9]  R. Schonmann On the Behavior of Some Cellular Automata Related to Bootstrap Percolation , 1992 .

[10]  Comparison of Semi-Oriented Bootstrap Percolation Models with Modified Bootstrap Percolation , 1993 .

[11]  Thomas Mountford Critical length for semi-oriented bootstrap percolation , 1995 .

[12]  Janko Gravner,et al.  First passage times for threshold growth dynamics on ${\bf Z}\sp 2$ , 1996 .

[13]  Emilio N.M. Cirillo,et al.  Finite Size Scaling in Three-Dimensional Bootstrap Percolation , 1998 .

[14]  Janko Gravner,et al.  Scaling laws for a class of critical cellular automaton growth rules , 1999 .

[15]  In Ho Lee,et al.  Noisy Contagion Without Mutation , 2000 .

[16]  R. Connelly,et al.  Percolation of the Loss of Tension in an Infinite Triangular Lattice , 2001 .

[17]  A. Holroyd Sharp metastability threshold for two-dimensional bootstrap percolation , 2002, math/0206132.

[18]  Deepak Dhar,et al.  Hysteresis in the random-field Ising model and bootstrap percolation. , 2002, Physical review letters.

[19]  Winfried W. Wilcke,et al.  Percolation in dense storage arrays , 2002 .

[20]  Béla Bollobás,et al.  Sharp thresholds in Bootstrap percolation , 2003 .

[21]  Peter Sollich,et al.  Glassy dynamics of kinetically constrained models , 2002, cond-mat/0210382.

[22]  Joan Adler,et al.  Bootstrap Percolation: visualizations and applications , 2003 .

[23]  Clarification of the bootstrap percolation paradox. , 2004, Physical review letters.

[24]  Kenneth A. Dawson,et al.  Cellular Automata with Rare Events; Resolution of an Outstanding Problem in the Bootstrap Percolation Model , 2004, ACRI.

[25]  Béla Bollobás,et al.  Phase transitions in the neuropercolation model of neural populations with mixed local and non-local interactions , 2005, Biological Cybernetics.

[26]  G. Biroli,et al.  Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics , 2004, cond-mat/0410647.

[27]  Indranil Gupta,et al.  ContagAlert: Using Contagion Theory for Adaptive, Distributed Alert Propagation , 2006, Fifth IEEE International Symposium on Network Computing and Applications (NCA'06).

[28]  Daniel S Fisher,et al.  Jamming percolation and glass transitions in lattice models. , 2005, Physical review letters.

[29]  Funded in part by an Nserc Discovery Grant The Metastability Threshold for Modified Bootstrap Percolation in d Dimensions , 2006 .

[30]  Alexander E. Holroyd,et al.  Slow convergence in bootstrap percolation. , 2007 .

[31]  Kenneth A. Dawson,et al.  Bootstrap Percolation , 2009, Encyclopedia of Complexity and Systems Science.