Percolation and criticality of systems with competing interactions on Bethe lattices: limitations and potential strengths of cluster schemes

The random clusters introduced by Fortuin and Kasteleyn (FK) and analyzed by Coniglio and Klein (CK) for Ising and related models have led first Swendsen and Wang and then Wolff to formulate remarkably efficient Markov chain Monte Carlo sampling schemes that weaken the critical slowing down. In frustrated models, however, no standard way to produce a comparable gain at small frustration -- let alone efficiently sample the large frustration regime -- has yet been identified. In order to understand why formulating appropriate cluster criteria for frustrated models has thus far been elusive, we here study minimal short-range attractive and long-range repulsive as well as spin-glass models on Bethe lattices. Using a generalization of the CK approach and the cavity-field method, the appropriateness and limitations of the FK--CK type clusters are identified. We find that a standard, constructive cluster scheme is then inoperable, and that the frustration range over which generalized FK--CK clusters are even definable is finite. These results demonstrate the futility of seeking constructive cluster schemes for frustrated systems but leaves open the possibility that alternate approaches could be devised.