Percolation of Fortuin-Kasteleyn clusters for the random-bond Ising model

We apply generalisations of the Swendson-Wang and Wolff cluster algorithms, which are based on the construction of Fortuin-Kasteleyn clusters, to the three-dimensional $\pm 1$ random-bond Ising model. The behaviour of the model is determined by the temperature $T$ and the concentration $p$ of negative (anti-ferromagnetic) bonds. The ground state is ferromagnetic for $0 \le p<p_c$, and a spin glass for $p_c < p \le 0.5$ where $p_c \simeq 0.222$. We investigate the percolation transition of the Fortuin-Kasteleyn clusters as function of temperature. Except for $p=0$ the Fortuin-Kasteleyn percolation transition occurs at a higher temperature than the magnetic ordering temperature. This was known before for $p=1/2$ but here we provide evidence for a difference in transition temperatures even for $p$ arbitrarily small. Furthermore, for all values of $p>0$, our data suggest that the percolation transition is universal, irrespective of whether the ground state exhibits ferromagnetic or spin-glass order, and is in the universality class of standard percolation. This shows that correlations in the bond occupancy of the Fortuin-Kasteleyn clusters are irrelevant, except for $p=0$ where the clusters are tied to Ising correlations so the percolation transition is in the Ising universality class.

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