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 $d_{\text{min}}$ in two dimensions: $d_{\text{min}}\stackrel{?}{=}\left(g+2\right)\left(g+18\right)/\left(32g\right)$, where $g$ is the Coulomb-gas coupling, related to the cluster fugacity $q$ via $q=2+2\text{cos}\left(g\pi /2\right)$ with $2\le g\le 4$.

• Received 22 April 2009

DOI:https://doi.org/10.1103/PhysRevE.81.020102