k-core percolation in four dimensions.

The k-core percolation on the Bethe lattice has been proposed as a simple model of the jamming transition because of its hybrid first-order-second-order nature. We investigate numerically k-core percolation on the four-dimensional regular lattice. For k=4 , the presence of a discontinuous transition is clearly established but its nature is strictly first-order. In particular, the k-core density displays no singular behavior before the jump and its correlation length remains finite. For k=3, the transition is continuous.

[1]  A. J. Liu,et al.  The onset of jamming as the sudden emergence of an infinite k-core cluster , 2004 .

[2]  A. B. Harris,et al.  1/d expansion for k-core percolation. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[4]  T. Lubensky,et al.  Field theoretic approaches to biconnectedness in percolating systems , 1983 .

[5]  Sergey N. Dorogovtsev,et al.  k-core (bootstrap) percolation on complex networks: Critical phenomena and nonlocal effects , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[7]  C. M. Chaves,et al.  Universality in bootstrap and diffusion percolation , 1997 .

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

[9]  Andrea J. Liu,et al.  Jamming at zero temperature and zero applied stress: the epitome of disorder. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Andrea J Liu,et al.  Vibrations and diverging length scales near the unjamming transition. , 2005, Physical review letters.

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

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

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

[14]  UNIVERSALITY CLASS FOR BOOTSTRAP PERCOLATION WITH m = 3 ON THE CUBIC LATTICE , 1999, cond-mat/9904239.

[15]  Threshold Value Of Three-Dimensional Bootstrap Percolation , 2002, cond-mat/0210579.

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

[17]  A. Montanari,et al.  How to compute loop corrections to the Bethe approximation , 2005, cond-mat/0506769.

[18]  Mauro Sellitto,et al.  Facilitated spin models on Bethe lattice: Bootstrap percolation, mode-coupling transition and glassy dynamics , 2005 .

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

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

[21]  A. B. Harris.  Field-Theoretic Approach to Biconnectedness in Percolating Systems , 1983 .

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

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

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