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

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.