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+2 cos(gπ/2) with 2≤g≤4.

Some geometric critical exponents for percolation and the random-cluster model / Y. Deng, W. Zhang, T.M. Garoni, A.D. Sokal, A. Sportiello. - In: PHYSICAL REVIEW E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS. - ISSN 1539-3755. - 81:2(2010), pp. 020102.020102.1-020102.020102.4. [10.1103/PhysRevE.81.020102]

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

A. Sportiello
Ultimo
2010

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+2 cos(gπ/2) with 2≤g≤4.
Settore FIS/02 - Fisica Teorica, Modelli e Metodi Matematici
2010
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/156226
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