Mathematics – Probability
Scientific paper
2010-01-10
J. Statist. Phys. 2011, 142 (2), 229-276
Mathematics
Probability
50 pages, 7 figures. A few typos have been corrected
Scientific paper
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.
No associations
LandOfFree
Phase separation in random cluster models III: circuit regularity 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 III: circuit regularity, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Phase separation in random cluster models III: circuit regularity will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-128549