Mathematics – Probability
Scientific paper
2010-01-10
Mathematics
Probability
56 pages, 7 figures. Comm. Math. Phys., to appear. Several figures and a glossary of notation have been added, and numerous ty
Scientific paper
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 Fortuin-Kasteleyn planar random cluster model on the presence of an open dual circuit Gamma_0 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(Gamma_0), this being the maximum distance from a point in the circuit Gamma_0 to the boundary of the circuit's convex hull; and in a longitudinal sense by what we term maximum facet length, MFL(Gamma_0), namely, the length of the longest line segment of which the boundary of the convex hull is formed. The principal conclusion of the series of papers is the following uniform control on local deviation: that there are positive constants c and C such that the conditional probability that the normalised quantity n^{-1/3}\big(\log n \big)^{-2/3} MLR(Gamma_0) lies in the interval [c,C] tends to 1 in the high n-limit; and that the same statement holds for n^{-2/3}\big(\log n \big)^{-1/3} MFL(Gamma_0). 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 a range of radially defined stochastic interface models, including several in the Kardar-Parisi-Zhang universality class. This paper is devoted to proving the upper bounds in these assertions, and includes a heuristic overview of the surgical technique used in the three papers.
No associations
LandOfFree
Phase separation in random cluster models I: uniform upper bounds on local deviation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Phase separation in random cluster models I: uniform upper bounds on local deviation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Phase separation in random cluster models I: uniform upper bounds on local deviation will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-128540