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July 2008 Fluctuation theory of connectivities for subcritical random cluster models
Massimo Campanino, Dmitry Ioffe, Yvan Velenik
Ann. Probab. 36(4): 1287-1321 (July 2008). DOI: 10.1214/07-AOP359


We develop a fluctuation theory of connectivities for subcritical random cluster models. The theory is based on a comprehensive nonperturbative probabilistic description of long connected clusters in terms of essentially one-dimensional chains of irreducible objects. Statistics of local observables, for example, displacement, over such chains obey classical limit laws, and our construction leads to an effective random walk representation of percolation clusters.

The results include a derivation of a sharp Ornstein–Zernike type asymptotic formula for two point functions, a proof of analyticity and strict convexity of inverse correlation length and a proof of an invariance principle for connected clusters under diffusive scaling.

In two dimensions duality considerations enable a reformulation of these results for supercritical nearest-neighbor random cluster measures, in particular, for nearest-neighbor Potts models in the phase transition regime. Accordingly, we prove that in two dimensions Potts equilibrium crystal shapes are always analytic and strictly convex and that the interfaces between different phases are always diffusive. Thus, no roughening transition is possible in the whole regime where our results apply.

Our results hold under an assumption of exponential decay of finite volume wired connectivities [assumption (1.2) below] in rectangular domains that is conjectured to hold in the whole subcritical regime; the latter is known to be true, in any dimensions, when q=1, q=2, and when q is sufficiently large. In two dimensions assumption (1.2) holds whenever there is an exponential decay of connectivities in the infinite volume measure. By duality, this includes all supercritical nearest-neighbor Potts models with positive surface tension between ordered phases.


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Massimo Campanino. Dmitry Ioffe. Yvan Velenik. "Fluctuation theory of connectivities for subcritical random cluster models." Ann. Probab. 36 (4) 1287 - 1321, July 2008.


Published: July 2008
First available in Project Euclid: 29 July 2008

zbMATH: 1160.60026
MathSciNet: MR2435850
Digital Object Identifier: 10.1214/07-AOP359

Primary: 60F15 , 60K15 , 60K17 , 60K35 , 82B20 , 82B41
Secondary: 37C30

Keywords: absence of roughening , Dependent percolation , Fortuin–Kasteleyn model , interfaces in Potts models , invariance principle , Local limit theorems , Ornstein–Zernike behavior for connectivities , random cluster model , renormalization , Ruelle operator , strict convexity of Wulff shape

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 4 • July 2008
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