### Abstract

The physics of k -core percolation pertains to those systems whose constituents require a minimum number of k connections to each other in order to participate in any clustering phenomenon. Examples of such a phenomenon range from orientational ordering in solid ortho-para H2 mixtures to the onset of rigidity in bar-joint networks to dynamical arrest in glass-forming liquids. Unlike ordinary (k=1) and biconnected (k=2) percolation, the mean field k≥3 -core percolation transition is both continuous and discontinuous, i.e., there is a jump in the order parameter accompanied with a diverging length scale. To determine whether or not this hybrid transition survives in finite dimensions, we present a 1/d expansion for k -core percolation on the d -dimensional hypercubic lattice. We show that to order 1/d3 the singularity in the order parameter and in the susceptibility occur at the same value of the occupation probability. This result suggests that the unusual hybrid nature of the mean field k -core transition survives in high dimensions.

Original language | English (US) |
---|---|

Article number | 046123 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 72 |

Issue number | 4 |

DOIs | |

State | Published - Oct 2005 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*72*(4), [046123]. https://doi.org/10.1103/PhysRevE.72.046123

**1/d expansion for k -core percolation.** / Harris, A. B.; Schwarz, Jennifer M.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 72, no. 4, 046123. https://doi.org/10.1103/PhysRevE.72.046123

}

TY - JOUR

T1 - 1/d expansion for k -core percolation

AU - Harris, A. B.

AU - Schwarz, Jennifer M

PY - 2005/10

Y1 - 2005/10

N2 - The physics of k -core percolation pertains to those systems whose constituents require a minimum number of k connections to each other in order to participate in any clustering phenomenon. Examples of such a phenomenon range from orientational ordering in solid ortho-para H2 mixtures to the onset of rigidity in bar-joint networks to dynamical arrest in glass-forming liquids. Unlike ordinary (k=1) and biconnected (k=2) percolation, the mean field k≥3 -core percolation transition is both continuous and discontinuous, i.e., there is a jump in the order parameter accompanied with a diverging length scale. To determine whether or not this hybrid transition survives in finite dimensions, we present a 1/d expansion for k -core percolation on the d -dimensional hypercubic lattice. We show that to order 1/d3 the singularity in the order parameter and in the susceptibility occur at the same value of the occupation probability. This result suggests that the unusual hybrid nature of the mean field k -core transition survives in high dimensions.

AB - The physics of k -core percolation pertains to those systems whose constituents require a minimum number of k connections to each other in order to participate in any clustering phenomenon. Examples of such a phenomenon range from orientational ordering in solid ortho-para H2 mixtures to the onset of rigidity in bar-joint networks to dynamical arrest in glass-forming liquids. Unlike ordinary (k=1) and biconnected (k=2) percolation, the mean field k≥3 -core percolation transition is both continuous and discontinuous, i.e., there is a jump in the order parameter accompanied with a diverging length scale. To determine whether or not this hybrid transition survives in finite dimensions, we present a 1/d expansion for k -core percolation on the d -dimensional hypercubic lattice. We show that to order 1/d3 the singularity in the order parameter and in the susceptibility occur at the same value of the occupation probability. This result suggests that the unusual hybrid nature of the mean field k -core transition survives in high dimensions.

UR - http://www.scopus.com/inward/record.url?scp=33244476457&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33244476457&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.72.046123

DO - 10.1103/PhysRevE.72.046123

M3 - Article

AN - SCOPUS:33244476457

VL - 72

JO - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

JF - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

SN - 1063-651X

IS - 4

M1 - 046123

ER -