  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 TFK = 2 / ln [z / (z - 2)] and pFK = z / [2 (z -
1)] for the Bethe lattice, TFK -> infinity and pFK -> 1 / 2 for the
infinite-range model, TFK = 2 / ln 3 and pFK = 3 / 4 for the square lattice,
TFK ~ 3.9347 and pFK ~ 0.62441 for the simple cubic lattice, TFK ~ 6.191 and
pFK ~ 0.5801 for the 4-dimensional hypercubic lattice, and TFK = 2 / ln {[1 + 2
sin (pi / 18)] / [1 - 2 sin (pi / 18) ]} and pFK = [1 + 2 sin (pi / 18) ] / 2
for the triangular lattice, when J / kB = 1, where z is the coordination
number, J is the strength of the exchange interaction between spins, kB is the
Boltzmann constant, TFK is the temperature at the percolation transition point,
and pFK is the probability, that the interaction is ferromagnetic, at the
percolation transition point.
