Phase Separation in Random Cluster Models III: Circuit Regularity - Hammond, A (2011)
We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn random cluster model on the presence of an open circuit Γ 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 site v∈Γ, the circuit Γ 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 Hammond (arXiv:1001.1527, 2009; arXiv:1001.1528, 2009). © 2010 Springer Science+Business Media, LLC.