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 d(min) in two dimensions: d(min)=[over ?](g+2)(g+18)/(32 g) , where g is the Coulomb-gas coupling, related to the cluster fugacity q via q=2+2 cos(gpi/2) with 2< or =g< or =4 .