# Abstract

Fluctuation theory of connectivities for subcritical random cluster models
M. Campanino, D. Ioffe and Y. Velenik
Ann. Probab.
36,
1287-1321
(2008).
We develop a fluctuation theory of connectivities for subcritical random cluster models. The theory is based on a comprehensive non-perturbative probabilistic description of long connected clusters in terms of essentially one-dimensional chains of irreducible objects. Statistics of local observables, e.g. 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-neighbour random cluster measures, in particular for nearest-neighbour $q$-states 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 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 our assumption holds whenever there is an exponential decay of connectivities in the infinite volume measure. By duality this includes all supercritical nearest-neighbour Potts models with positive surface tension between ordered phases.

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-neighbour random cluster measures, in particular for nearest-neighbour $q$-states 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 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 our assumption holds whenever there is an exponential decay of connectivities in the infinite volume measure. By duality this includes all supercritical nearest-neighbour Potts models with positive surface tension between ordered phases.

**Key words:**Random cluster model, Potts model, Ornstein-Zernike decay of correlations, Ruelle operator, renormalization, local limit theorems, invariance principle, equilibrium crystal shape.**Files:**PDF file, Published version, bibtex