The bunkbed problem and the random cluster model

Abstract

The well known bunkbed conjecture about percolation on finite graphs is now resolved; Gladkov, Pak and Zimin, building upon work of Hollom, have constructed a counterexample. We revisit this conjecture and study it in the broader context of the class of random cluster measures. We show that the major partial (positive) results on the bunkbed conjecture can also be proved for all random cluster measures, including the results for complete graphs, complete bipartite graphs, and the case when $p \uparrow 1$. The arboreal gas measure for forests is another limit of the random cluster measure for which we conjecture the inequality to be true and provide proofs in special cases. We identify a setting where the conjecture does hold, that of ``almost spanning tree measures''. A further analysis leads to intriguing correlation inequalities that complement Rayleigh's inequalities for spanning tree measures.