# zbMATH — the first resource for mathematics

Conformal radii for conformal loop ensembles. (English) Zbl 1187.82044
From the abstract: The conformal loop ensembles $$\mathrm{CLE}_\kappa$$, defined for $$8/3\leq\kappa\leq8$$, are random collections of loops in a planar domain which are conjectured scaling limits of the $$O(n)$$ loop models. We calculate the distribution of the conformal radii of the nested loops surrounding a deterministic point. Our results agree with predictions made by Cardy and Ziff and by Kenyon and Wilson for the $$O(n)$$ model. We also compute the expectation dimension of the $$\mathrm{CLE}_\kappa$$ gasket, which consists of points not surrounded by any loop, to be $2 -\frac{ (8 - \kappa)(3\kappa - 8)}{32\kappa},$ which agrees with the fractal dimension given by Duplantier for the $$O(n)$$ model gasket.

##### MSC:
 82B31 Stochastic methods applied to problems in equilibrium statistical mechanics 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 60K99 Special processes 60J65 Brownian motion
##### Keywords:
conformal loop ensembles
Full Text:
##### References:
 [1] Borodin, A.N., Salminen, P.: Handbook of Brownian Motion–Facts and Formulae. Probability and its Applications. Basel: Birkhäuser Verlag, 2nd edition, 2002 · Zbl 1012.60003 [2] Cardy J. (2007) ADE and SLE. J. Phys. A 40(7): 1427–1438 · Zbl 1106.82007 [3] Camia F., Newman C.M. (2006) Two-dimensional critical percolation: the full scaling limit. Commun. Math. Phys. 268(1): 1–38 · Zbl 1117.60086 [4] Camia F., Newman C.M. (2007) Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Related Fields 139(3-4): 473–519 · Zbl 1126.82007 [5] Ciesielski Z., Taylor S.J. (1962) First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103(3): 434–450 · Zbl 0121.13003 [6] Cardy J., Ziff R.M. (2003) Exact results for the universal area distribution of clusters in percolation, Ising, and Potts models. J. Stat. Phys. 110(1-2): 1–33 · Zbl 1037.82020 [7] Dubédat, J.: 2005, Personal communication [8] Duplantier B. (1990) Exact fractal area of two-dimensional vesicles. Phys. Rev. Lett. 64(4): 493 [9] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. Vol. I., New York: McGraw-Hill Book Company, 1953, based, in part, on notes left by Harry Bateman · Zbl 0052.29502 [10] Fortuin C.M., Kasteleyn P.W. (1972) On the random-cluster model. I. Introduction and relation to other models. Physica 57: 536–564 [11] Grimmett, G.: The Random-Cluster Model, Volume 333 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag, (2006) · Zbl 1122.60087 [12] Kager W., Nienhuis B. (2004) A guide to stochastic Löwner evolution and its applications. J. Stat. Phys. 115(5-6): 1149–1229 · Zbl 1157.82327 [13] Kenyon, R.W., Wilson, D.B.: Conformal radii of loop models, 2004. Manuscript [14] Lawler, G.F., Schramm, O., Werner, W.: One-arm exponent for critical 2D percolation. Electron. J. Probab. 7: Paper No. 2, 13 pp. (2002) · Zbl 1015.60091 [15] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag, third edition, 1999 · Zbl 0917.60006 [16] Schramm O. (2001) A percolation formula. Electron. Comm. Probab. 6: 115–120 · Zbl 1008.60100 [17] Sheffield, S.: Exploration trees and conformal loop ensembles. http://arxiv.org/abs/math.PR/0609167 , 2006 Duke Math. J., to appear · Zbl 1170.60008 [18] Smirnov S. (2001) Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3): 239–244 · Zbl 0985.60090 [19] Sheffield, S., Werner, W.: Conformal loop ensembles: Construction via loop-soups, 2008, in preparation [20] Sheffield, S., Werner, W.: Conformal loop ensembles: The Markovian characterization, 2008, in preparation · Zbl 1271.60090
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.