### Abstract

We introduce several infinite families of critical exponents for the random-cluster model and present scaling arguments relating them to the k -arm exponents. We then present Monte Carlo simulations confirming these predictions. These exponents provide a convenient way to determine k -arm exponents from Monte Carlo simulations. An understanding of these exponents also leads to a radically improved implementation of the Sweeny Monte Carlo algorithm. In addition, our Monte Carlo data allow us to conjecture an exact expression for the shortest-path fractal dimension dmin in two dimensions: dmin =? (g+2) (g+18) / (32g), where g is the Coulomb-gas coupling, related to the cluster fugacity q via q=2+2cos (gπ/2) with 2≤g≤4.

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

Article number | 020102 |

Journal | Physical Review E |

Volume | 81 |

Issue number | 2 |

DOIs | |

State | Published - Feb 10 2010 |

### Fingerprint

### ASJC Scopus subject areas

- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Statistics and Probability

### Cite this

*Physical Review E*,

*81*(2), [020102]. https://doi.org/10.1103/PhysRevE.81.020102

**Some geometric critical exponents for percolation and the random-cluster model.** / Deng, Youjin; Zhang, Wei; Garoni, Timothy M.; Sokal, Alan D.; Sportiello, Andrea.

Research output: Contribution to journal › Article

*Physical Review E*, vol. 81, no. 2, 020102. https://doi.org/10.1103/PhysRevE.81.020102

}

TY - JOUR

T1 - Some geometric critical exponents for percolation and the random-cluster model

AU - Deng, Youjin

AU - Zhang, Wei

AU - Garoni, Timothy M.

AU - Sokal, Alan D.

AU - Sportiello, Andrea

PY - 2010/2/10

Y1 - 2010/2/10

N2 - We introduce several infinite families of critical exponents for the random-cluster model and present scaling arguments relating them to the k -arm exponents. We then present Monte Carlo simulations confirming these predictions. These exponents provide a convenient way to determine k -arm exponents from Monte Carlo simulations. An understanding of these exponents also leads to a radically improved implementation of the Sweeny Monte Carlo algorithm. In addition, our Monte Carlo data allow us to conjecture an exact expression for the shortest-path fractal dimension dmin in two dimensions: dmin =? (g+2) (g+18) / (32g), where g is the Coulomb-gas coupling, related to the cluster fugacity q via q=2+2cos (gπ/2) with 2≤g≤4.

AB - We introduce several infinite families of critical exponents for the random-cluster model and present scaling arguments relating them to the k -arm exponents. We then present Monte Carlo simulations confirming these predictions. These exponents provide a convenient way to determine k -arm exponents from Monte Carlo simulations. An understanding of these exponents also leads to a radically improved implementation of the Sweeny Monte Carlo algorithm. In addition, our Monte Carlo data allow us to conjecture an exact expression for the shortest-path fractal dimension dmin in two dimensions: dmin =? (g+2) (g+18) / (32g), where g is the Coulomb-gas coupling, related to the cluster fugacity q via q=2+2cos (gπ/2) with 2≤g≤4.

UR - http://www.scopus.com/inward/record.url?scp=77249148424&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77249148424&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.81.020102

DO - 10.1103/PhysRevE.81.020102

M3 - Article

AN - SCOPUS:77249148424

VL - 81

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 - 2

M1 - 020102

ER -