×

zbMATH — the first resource for mathematics

Exploration trees and conformal loop ensembles. (English) Zbl 1170.60008
Two-dimensional statistical physics models often involve random collections of disjoint, non-intersecting loops in a planar lattice. The paper departs from a conjecture: when suitable boundary conditions are set in a random loop model on a simply connected planar domain, then as the grid size gets finer, the law of a random path connecting a pair of boundary points converges to the law of the chordal Schramm-Loewner evolution. The primary purpose of the paper is to investigate a family of candidate loop collections that have a scaling limit, called conformal loop ensembles. A secondary purpose is to formulate a series of conjectures and open questions for these ensembles, like e.g. about a continuum analog of the Fortuin-Kasteleyn cluster expansion for the Potts model, or scaling limits of the q-state Potts models. The focus is on hexagonal lattice graphs and so-called O(N) loop models. Bessel, skew Levy and Schramm-Loewner processes are considered. The radia Schramm-Loewner evolutions are found to be most natural candidates for the limiting laws of the exploration tree.

MSC:
60D05 Geometric probability and stochastic geometry
82B27 Critical phenomena in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
05C80 Random graphs (graph-theoretic aspects)
60G52 Stable stochastic processes
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] M. Aizenman and A. Burchard, Hölder regularity and dimension bounds for random curves , Duke Math. J. 99 (1999), 419–453. · Zbl 0944.60022
[2] F. Camia and C. M. Newman, The full scaling limit of two-dimensional critical percolation , preprint,\arxivmath/0504036v1[math.PR] · Zbl 1117.60086
[3] J. Cardy and R. M. Ziff, Exact results for the universal area distribution of clusters in percolation; Ising and Potts models, preprint,\arxivcond-mat/0205404v2[cond-mat.dis-nn] · Zbl 1037.82020
[4] M. Decamps, M. Goovaerts, and W. Schoutens, Asymmetric skew Bessel processes and their applications to finance , J. Comput. Appl. Math. 186 (2006), 130–147. · Zbl 1087.91022
[5] B. Duplantier, Exact fractal area of two-dimensional vesicles , Phys. Rev. Lett. 64 (1990), 493. · Zbl 1297.01037
[6] W. Kager and B. Nienhuis, A guide to stochastic Löwner evolution and its applications , J. Statist. Phys. 115 (2004), 1149–1229. · Zbl 1157.82327
[7] G. F. Lawler, Conformally Invariant Processes in the Plane , Math. Surveys Monogr. 114 , Amer. Math. Soc., Providence, 2005. · Zbl 1074.60002
[8] G. F. Lawler, O. Schramm, and W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees , Ann. Probab. 32 (2004), 939–995. · Zbl 1126.82011
[9] J. Pitman, Partition structures derived from Brownian motion and stable subordinators , Bernoulli 3 (1997), 79–96. · Zbl 0882.60081
[10] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion , 3rd ed., Grundlehren Math. Wiss. 293 , Springer, Berlin, 1999. · Zbl 0917.60006
[11] S. Rohde and O. Schramm, Basic properties of SLE , Ann. of Math. (2) 161 (2005), 883–924. · Zbl 1081.60069
[12] O. Schramm, S. Sheffield, and D. B. Wilson, Conformal radii for conformal loop ensembles , preprint,\arxivmath/0611687v4[math.PR] · Zbl 1187.82044
[13] O. Schramm and D. B. Wilson, SLE coordinate changes , New York J. Math. 11 (2005), 659–669. · Zbl 1094.82007
[14] S. Sheffield and W. Werner, Loop soup clusters and simple CLEs , in preparation.
[15] S. Smirnov, Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits , C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 239–244. · Zbl 0985.60090
[16] S. Watanabe, “Generalized arc-sine laws for one-dimensional diffusion processes and random walks” in Stochastic Analysis (Ithaca, N.Y., 1993) , Proc. Sympos. Pure Math. 57 , Amer. Math. Soc., Providence, 1995, 157–172. · Zbl 0824.60080
[17] W. Werner, SLE s as boundaries of clusters of Brownian loops, C. R. Math. Acad. Sci. Paris 337 (2003), 481–486. · Zbl 1029.60085
[18] -, “Random planar curves and Schramm-Loewner evolutions” in Lectures on Probability Theory and Statistics , Lecture Notes in Math. 1840 , Springer, Berlin, 2004, 107–195. · Zbl 1057.60078
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.