2010/01/10 by Alan Hammond, Hammond, Alan
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.1001.1528
54 pages, 9 figures. Ann. Probab., to appear. A few typos have been corrected
openalex publication_date 2010/01/10 · arxiv created 2011/06/12 · arxiv updated 2011/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Gamma0), this being the maximum distance from a point in the circuit Gamma0 to the boundary of the circuit's convex hull; and in a longitudinal sense by what we term maximum facet length, MFL(Gamma0), 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(log n )-2/3 MLR(Gamma0) exceeds c tends to 1 in the high n-limit; and that the same statement holds for n-2/3(log n )-1/3 MFL(Gamma0). 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(Gamma0) and MFL(Gamma0) are determined up to a constant factor.