Phase separation in random cluster models II: the droplet at equilibrium, and local deviation lower bounds

We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn planar random cluster model on the presence of an open circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. 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. We prove that that there exists a constant c > 0 such that the conditional probability that the normalised quantity n^{-1/3}\big(\log n \big)^{-2/3} MLR(Gamma_0) exceeds 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). To obtain these bounds, we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a Markov chain that resamples sections of the circuit boundary. We analyse the chain at equilibrium to prove the local roughness lower bounds. Alongside complementary upper bounds provided in arXiv:1001.1527, the fluctuations MLR(Gamma_0) and MFL(Gamma_0) are determined up to a constant factor.Comment: 54 pages, 9 figures. Ann. Probab., to appear. A few typos have been correcte

Sharp phase transition in the random stirring model on trees

We establish that the phase transition for infinite cycles in the random stirring model on an infinite regular tree of high degree is sharp. That is, we prove that there exists d_0 such that, for any d \geq d_0, the set of parameter values at which the random stirring model on the rooted regular tree with offspring degree d almost surely contains an infinite cycle consists of a semi-infinite interval. The critical point at the left-hand end of this interval is at least 1/d + 1/(2d^2) and at most 1/d + 2/(d^2). This version is a major revision, with a much shorter proof. Principal among the changes are a reworking of the argument in Section 4 of the old version, which was proposed by a referee, and the use of a simpler means of handling a boundary case, which eliminates the previous Section 6.Comment: 20 pages, three figures. A short explanation of Proposition 3.2 has been added. Probab. Theory and Related Fields, to appea

Phase separation in random cluster models III: circuit regularity

We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn planar random cluster model on the presence of an open circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. In this paper, we prove that the resulting circuit is highly regular: we define a notion of a regeneration site in such a way that, for any such element v of Gamma_0, the circuit Gamma_0 cuts through the radial line segment through v only at v. We show that, provided that the conditioned circuit is centred at the origin in a natural sense, the set of regeneration sites reaches into all parts of the circuit, with maximal distance from one such site to the next being at most logarithmic in n with high probability. The result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations, and, as such, it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in arXiv:1001.1527 and arXiv:1001.1528.Comment: 50 pages, 7 figures. A few typos have been correcte

Infinite cycles in the random stirring model on trees

We prove that, in the random stirring model of parameter T on an infinite rooted tree each of whose vertices has at least two offspring, infinite cycles exist almost surely, provided that T is sufficiently high. In the appendices, the bound on degree above which the result holds is improved slightly.Comment: 23 pages, two figure

Phase separation in random cluster models I: uniform upper bounds on local deviation

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.Comment: 56 pages, 7 figures. Comm. Math. Phys., to appear. Several figures and a glossary of notation have been added, and numerous typos have been correcte