Bond percolation in frustrated systems (Q1577737)

As is well known, the presence of phase transition in classical ferromagnetic (Ising, Potts) systems at low temperatures is equivalent to percolation in the related random cluster model. For non-ferromagnetic systems, like the famous Edwards-Anderson spin glass model on \({\mathbb Z}^d\), the presence of negative couplings introduces long range correlations between bonds in the related random cluster model. Such dependence becomes decisive in the zero temperature limit since it forbids any bond configuration that contains frustrated circuits (a circuit is called frustrated if the product of all its coupling constants is negative). It thus becomes natural and important to study percolation in random cluster models subject to ``non-frustration'' property. The simplest situation of this type occurs as follows: to any bond \(b\) of \({\mathbb Z}^d\) assign a value \(J_b\in\{-1,+1\}\) and consider the usual percolation model at density \(p\) of occupied bonds conditioned to avoid full occupation of frustrated circuits. Using a variant of Peierls' argument, the authors show that if \(p>8/9\), then uniformly in all configurations \(\{J_b\}\) the percolation of occupied bonds in such a system takes place with probability one. A similar result is obtained for the (conditioned on ``non-frustration'') FK random cluster measure on~\({\mathbb Z}^d\). To stress the fact that the percolation/non-percolation transition in systems with frustration is not necessarily sharp, the authors show that for a particular graph and a particular choice of the values assumed by \(J\), it is possible to identify four regions of the interval \((0,1)\) such that if \(p\) is in these regions, percolation occurs for none, almost none, almost all or all of the \(J\)'s, respectively. Nontrivial estimates for the lengths of these intervals are given for the triangular lattice and \(J\in\{-t,-1,+1,+t\}\) with~\(t>0\) .