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arXiv · 0904.3448

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

Youjin Deng·Wei Zhang·Timothy M. Garoni·Alan D. Sokal·Andrea Sportiello
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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.

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Youjin Deng, Wei Zhang, Timothy M. Garoni, Alan D. Sokal, Andrea Sportiello. 2009-04-22. Some geometric critical exponents for percolation and the random-cluster model. https://doi.org/10.1103/physreve.81.020102

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