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Phase Separation in Random Cluster Models I: Uniform Upper Bounds on Local Deviation

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Abstract

This is the first in a series of three papers that addresses the behaviour of the droplet that results, in the percolating phase, from conditioning the planar Fortuin-Kasteleyn random cluster model on the presence of an open dual circuit Γ0 encircling the origin and enclosing an area of at least (or exactly) n 2. (By the Fortuin-Kasteleyn representation, the model is a close relative of the droplet formed by conditioning the Potts model on an excess of spins of a given type.) We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Γ0), this being the maximum distance from a point in the circuit Γ0 to the boundary ∂ of the circuit’s convex hull; and in a longitudinal sense by what we term maximum facet length, MLF(Γ0), namely, the length of the longest line segment of which the polygon ∂ is formed. The principal conclusion of the series of papers is the following uniform control on local deviation: that there are constants 0 < c < C < ∞ such that the conditional probability that the normalized quantity n −1/3(log n )−2/3MLR lies in the interval [c, C] tends to 1 in the high n-limit; and that the same statement holds for n −2/3 (log n )−1/3 MLF. In this way, we confirm the anticipated n 1/3 scaling of maximum local roughness, and provide a sharp logarithmic power-law correction. This local deviation behaviour occurs by means of locally Gaussian effects constrained globally by curvature, and we believe that it arises in many radially defined stochastic interface models, including growth models belonging to the Kardar-Parisi-Zhang universality class.

The present paper is devoted to proving the upper bounds in these assertions. In fact, we derive bounds valid in the moderate deviations’ regime. The second paper (Hammond in Ann Probab, arXiv:1001.1528, 142(2):229–276, 2011) provides the lower bounds. Crucial to our approach are surgical techniques that renew the conditioned circuit on the scale at which the local deviation manifests itself. A successful analysis of the surgeries depends on the circuit possessing a renewal structure, with only local backtracking occurring from its overall progress in a direction specified by the macroscopic Wulff profile. The third paper (Hammond in J Stat Phys 142(2):229–276, 2010) presents the required tool on regeneration structure of the conditioned circuit.

The present paper includes a heuristic presentation of the surgical technique that is also used in Hammond (J Stat Phys 142(2):229–276, 142(2):229–276, 2010), and a discussion of the significance of local deviation and of problems raised by our approach.

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Authors and Affiliations

  1. Department of Statistics, University of Oxford, 1 South Parts Road, Oxford, UK

    Alan Hammond

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  1. Alan Hammond
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Correspondence to Alan Hammond.

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Communicated by H. Spohn

Supported in part by U.S. NSF grant OISE-07-30136 and U.K. EPSRC grant EP/1004378/1. This work was undertaken during visits to the Theory Group at Microsoft Research in Redmond, WA, and to Ecole Normale Superieure in Paris.

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Hammond, A. Phase Separation in Random Cluster Models I: Uniform Upper Bounds on Local Deviation. Commun. Math. Phys. 310, 455–509 (2012). https://doi.org/10.1007/s00220-011-1370-2

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