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)|
|Journal||Physical Review E - Statistical, Nonlinear, and Soft Matter Physics|
|State||Published - Feb 10 2010|
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics