Red-bond exponents of the critical and the tricritical Ising model in three dimensions -: art. no. 056132

Using the Wolff and geometric cluster algorithms and finite-size scaling analysis, we investigate the critical Ising and the tricritical Blume-Capel models with nearest-neighbor interactions on the simple-cubic lattice. The sampling procedure involves the decomposition of the Ising configuration into geometric clusters, each of which consists of a set of nearest-neighboring spins of the same sign connected with bond probability p. These clusters include the well-known Kasteleyn-Fortuin clusters as a special case for p = 1 - exp(-2K), where K is the Ising spin-spin coupling. Along the critical line K=K-c, the size distribution of geometric clusters is investigated as a function of p. We observe that, unlike in the case of two-dimensional tricriticality, the percolation threshold in both models lies at p(c)=1 - exp(-2K(c)). Further, we determine the corresponding red-bond exponents as y(r)=0.757(2) and 0.501(5) for the critical Ising and the tricritical Blume-Capel models, respectively. On this basis, we conjecture y(r) = 1/2 for the latter model.