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Sharp phase transition for the random-cluster and Potts models via decision trees. (English) Zbl 1482.82009

This paper introduces a strong new tool, the (generalised) OSSS inequality for decision trees, which originated in computer science, into the study of statistical mechanical lattice models with a Fortuin-Kasteleyn random-cluster representation. Inequalities for Boolean functions have been made useful in statistical mechanics before, but the present one clearly improves on earlier existing proofs and results. As a consequence, various existing results are generalised, provided with shorter proofs, or both. In particular, the high-temperature phase (subcritical percolation phase) displays exponential decay down to the critical point; there holds a mean-field bound on some critical exponents, and for graphs which satisfy a duality relation, there is a relation between their percolation threshold. The strength of these results is in their generality, for planar graphs and for percolation or Ising models most of these results were known, but the authors here can in one go handle general finite-range ferromagnets, including Potts spins, on general graphs (and in general dimensions). Since the paper appeared, the results have already been extended for further models, e.g., Gaussian fields and loop \(O(n)\) models. It promises to be a classic paper in my opinion.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B43 Percolation
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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