Conductivity of Coniglio-Klein clusters

We performed numerical simulations of the $q$-state Potts model to compute the reduced conductivity exponent $t/ \nu$ for the critical Coniglio-Klein clusters in two dimensions, for values of $q$ in the range $[1;4]$. At criticality, at least for $q<4$, the conductivity scales as $C(L) \sim L^{-\frac{t}{\nu}}$, where $t$ and $\nu$ are, respectively, the conductivity and correlation length exponents. For q=1, 2, 3, and 4, we followed two independent procedures to estimate $t / \nu$. First, we computed directly the conductivity at criticality and obtained $t / \nu$ from the size dependence. Second, using the relation between conductivity and transport properties, we obtained $t / \nu$ from the diffusion of a random walk on the backbone of the cluster. From both methods, we estimated $t / \nu$ to be $0.986 \pm 0.012$, $0.877 \pm 0.014$, $0.785 \pm 0.015$, and $0.658 \pm 0.030$, for q=1, 2, 3, and 4, respectively. We also evaluated $t /\nu$ for non integer values of $q$ and propose the following conjecture $40gt/ \nu=72+20g-3g^2$ for the dependence of the reduced conductivity exponent on $q$, in the range $0 \leq q \leq 4$, where $g$ is the Coulomb gas coupling.