2010/01/31 by Alan Hammond
Mathematics · Physics and Astronomy · #Artificial intelligence #Cluster (spacecraft) #Combinatorics #Computer science #Control (management) #Control theory (sociology) #Logarithm #Mathematical analysis #Mathematics #Open-circuit voltage #Phase (matter) #Physics #Quantum mechanics #Random Matrices and Applications #Set (abstract data type) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topology (electrical circuits) #Voltage #math-ph #math.MP #math.PR
paper · pdf · doi:10.1007/s10955-010-0081-y
published as J. Statist. Phys. 2011, 142 (2), 229-276 · 50 pages, 7 figures. A few typos have been corrected
openalex publication_date 2010/12/15 · arxiv created 2011/06/12 · arxiv updated 2015/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn planar random cluster model on the presence of an open circuit Gamma0 encircling the origin and enclosing an area of at least (or exactly) n2. 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 Gamma0, the circuit Gamma0 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.