Conjectured Exact Percolation Thresholds of the Fortuin-Kasteleyn Cluster for the +-J Ising Spin Glass Model

TL;DR

This paper conjectures exact percolation thresholds for the Fortuin-Kasteleyn cluster in the +-J Ising spin glass model, relating them to phase transitions on various lattice structures along the Nishimori line.

Contribution

It provides conjectured exact formulas for percolation thresholds in the spin glass model on different lattices, connecting percolation and spin freezing transitions.

Findings

Thresholds are derived for Bethe, infinite-range, square, cubic, hypercubic, and triangular lattices.

Results relate percolation transition points to lattice structure and interaction probabilities.

Formulas are expressed in terms of lattice coordination number and interaction parameters.

Abstract

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.