Bootstrap percolation in three dimensions

By bootstrap percolation we mean the following deterministic process on a graph G. Given a set A of vertices "infected" at time 0, new vertices are subsequently infected, at each time step, if they have at least ∈ N previously infected neighbors. When the set A is chosen at random, the main aim is to determine the critical probability p c (G, r) at which percolation (infection of the entire graph) becomes likely to occur. This bootstrap process has been extensively studied on the d-dimensional grid [n] d : with 2 ≤ r ≤ d fixed, it was proved by Cerf and Cirillo (for d = r = 3), and by Cerf and Manzo (in general), that p c ([n] d ,r)=Θ(1/log (r-1) n) d-r+1 , where log (r) is an r-times iterated logarithm. However, the exact threshold function is only known in the case d = r = 2, where it was shown by Holroyd to be (1 + o(1)) π 2 18 log n. In this paper we shall determine the exact threshold in the crucial case d = r = 3, and lay the groundwork for solving the problem for all fixed d and r.

[1]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[2]  J. Doob.  Stochastic processes , 1953 .

[3]  John von Neumann,et al.  Theory Of Self Reproducing Automata , 1967 .

[4]  J. Schwartz,et al.  Theory of Self-Reproducing Automata , 1967 .

[5]  C. Fortuin,et al.  Correlation inequalities on some partially ordered sets , 1971 .

[6]  J. Spencer,et al.  EVOLUTION OF THE n-CUBE , 1979 .

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

[8]  János Komlós,et al.  Largest random component of ak-cube , 1982, Comb..

[9]  H. Kesten,et al.  Inequalities with applications to percolation and reliability , 1985 .

[10]  Vojtech Rödl,et al.  On a Packing and Covering Problem , 1985, Eur. J. Comb..

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

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

[13]  W S McCulloch,et al.  A logical calculus of the ideas immanent in nervous activity , 1990, The Philosophy of Artificial Intelligence.

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

[15]  Roberto H. Schonmann,et al.  Finite size scaling behavior of a biased majority rule cellular automaton , 1990 .

[16]  W. Pitts,et al.  A Logical Calculus of the Ideas Immanent in Nervous Activity (1943) , 2021, Ideas That Created the Future.

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

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

[19]  G. Kalai,et al.  Every monotone graph property has a sharp threshold , 1996 .

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

[21]  József Balogh,et al.  Random disease on the square grid , 1998, Random Struct. Algorithms.

[22]  J. Balogh,et al.  Random disease on the square grid , 1998 .

[23]  David Reimer,et al.  Proof of the Van den Berg–Kesten Conjecture , 2000, Combinatorics, Probability and Computing.

[24]  D. L. Stein,et al.  Zero-temperature dynamics of Ising spin systems following a deep quench: results and open problems , 2000 .

[25]  Charles M. Newman,et al.  Dynamics of Ising spin systems at zero temperature , 2000 .

[26]  M. AizenmanS,et al.  Metastability effects in bootstrap percolation ? , 2001 .

[27]  F. Manzo,et al.  The Threshold Regime of Finite Volume Bootstrap Percolation , 2001 .

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

[29]  R. Schonmann,et al.  Stretched Exponential Fixation in Stochastic Ising Models at Zero Temperature , 2002 .

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

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

[32]  Remco van der Hofstad,et al.  Asymptotic expansions in n−1 for percolation critical values on the n‐Cube and ℤn , 2005, Random Struct. Algorithms.

[33]  Alexander E. Holroyd,et al.  The Metastability Threshold for Modified Bootstrap Percolation in $d$ Dimensions , 2006 .

[34]  Béla Bollobás,et al.  Bootstrap percolation on the hypercube , 2006 .

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

[36]  Yuval Peres,et al.  Bootstrap Percolation on Infinite Trees and Non-Amenable Groups , 2003, Combinatorics, Probability and Computing.

[37]  József Balogh,et al.  Bootstrap percolation on the random regular graph , 2007, Random Struct. Algorithms.

[38]  Svante Janson,et al.  On percolation in random graphs with given vertex degrees , 2008, 0804.1656.

[39]  R. Morris.  Zero-temperature Glauber dynamics on Z^d , 2008, 0809.0353.

[40]  R. Schonmann,et al.  Bootstrap Percolation on Homogeneous Trees Has 2 Phase Transitions , 2008 .

[41]  Béla Bollobás,et al.  Majority Bootstrap Percolation on the Hypercube , 2007, Combinatorics, Probability and Computing.

[42]  Robert Morris.  Glauber Dynamics in High Dimensions , 2009 .

[43]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[44]  H. Duminil-Copin,et al.  The sharp threshold for bootstrap percolation in all dimensions , 2010, 1010.3326.

[45]  J. Gravner,et al.  A sharper threshold for bootstrap percolation in two dimensions , 2010, 1002.3881.

[46]  Béla Bollobás,et al.  Random Graphs, Second Edition , 2001, Cambridge Studies in Advanced Mathematics.