Abstract
Let θ(J) be the order parameter of a (ferromagnetic) Potts or randomcluster process with bond-variables J = (J e : e ∈ K). We discuss differential inequalities of the form
. Such inequalities may be established for all random-cluster processes that satisfy the FKG inequality, possibly in the presence of many-body interactions (subject to certain necessary and sufficient conditions on the sets of interactions). There are (at least) two principal consequences of this. First, for a process having ‘inverse-temperature’ β, the critical value ß c = ß c (J) is a strictly monotone function of J. Secondly, at any fixed point J lying on the critical surface of the process, the critical exponent of θ in the limit as J’ ↓ J is independent of the direction of approach of the limit. Such a conclusion should be valid for other critical exponents also; this amounts to a small amount of rigorous universality.
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Grimmett, G. (1993). Differential Inequalities for Potts and Random-Cluster Processes. In: Boccara, N., Goles, E., Martinez, S., Picco, P. (eds) Cellular Automata and Cooperative Systems. NATO ASI Series, vol 396. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1691-6_20
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DOI: https://doi.org/10.1007/978-94-011-1691-6_20
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