Some geometric critical exponents for percolation and the random-cluster model
Abstract
We introduce several infinite families of new 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 new 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 d_min in two dimensions: d_min = (g+2)(g+18)/(32g) where g is the Coulomb-gas coupling, related to the cluster fugacity q via q = 2 + 2 cos(g\pi/2) with 2 \le g \le 4.
Cite
@article{arxiv.0904.3448,
title = {Some geometric critical exponents for percolation and the random-cluster model},
author = {Youjin Deng and Wei Zhang and Timothy M. Garoni and Alan D. Sokal and Andrea Sportiello},
journal= {arXiv preprint arXiv:0904.3448},
year = {2010}
}
Comments
LaTeX2e/Revtex4. Version 2 is completely rewritten to make the exposition more reader-friendly; it consists of a 4-page main paper (including 3 figures) and a 2-page EPAPS appendix (given as a single Postscript file). To appear in Phys Rev E