Conjectured exact percolation thresholds of the Fortuin–Kasteleyn cluster for the 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 F K = 2 / ln ( z / z − 2 ) and p F K = z / 2 ( z − 1 ) for the Bethe lattice, T F K → ∞ and p F K → 1 / 2 for the infinite-range model, T F K = 2 / ln 3 and p F K = 3 / 4 for the square lattice, T F K ∼ 3.9347 and p F K ∼ 0.62441 for the simple cubic lattice, T F K ∼ 6.191 and p F K ∼ 0.5801 for the 4-dimensional hypercubic lattice, and T F K = 2 / ln [ 1 + 2 sin ( π / 18 ) / 1 − 2 sin ( π / 18 ) ] and p F K = [ 1 + 2 sin ( π / 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 F K is the temperature at the percolation transition point, and p F K is the probability, that the interaction is ferromagnetic, at the percolation transition point.