Conjectured exact percolation thresholds of the Fortuin-Kasteleyn cluster for the ±J Ising spin glass model

The conjectured exact percolation thresholds of the Fortuin-Kasteleyn cluster for the +/- J Ising spin glass model are theoretically shown based on a conjecture. It is pointed out that the percolation transition of the Fortuin-Kasteleyn cluster for the spin glass model is related to a dynamical transition for the freezing of spins. The present results are obtained as locations of points on the so-called Nishimori line, which is a special line in the phase diagram. We obtain T-FK = 2/ln(z/z - 2) and p(FK) = z/2(z - 1) for the Bethe lattice, T-FK -> infinity and p(FK) -> 1/2 for the infinite-range model, T-FK = 2/ln 3 and p(FK) = 3/4 for the square lattice, T-FK similar to 3.9347 and p(FK) similar to 0.62441 for the simple cubic lattice, T-FK similar to 6.191 and p(FK) similar to 0.5801 for the 4-dimensional hypercubic lattice, and T-FK = 2/ln[1 + 2 sin(pi/18)/1 - 2 sin(pi/18)] and p(FK) = [1 + 2 sin(pi/18)]/2 for the triangular lattice, when J/k(B) = 1, where z is the coordination number, J is the strength of the exchange interaction between spins, k(B) is the Boltzmann constant, T-FK is the temperature at the percolation transition point, and p(FK) is the probability, that the interaction is ferromagnetic, at the percolation transition point. (c) 2012 Elsevier B.V. All rights reserved.