×

zbMATH — the first resource for mathematics

Conformal loop ensembles: the Markovian characterization and the loop-soup construction. (English) Zbl 1271.60090
We have at hand an algebra for curves endowed with certain selected properties. A chordal Schramm-Loewner evolution (SLE) is a random non-self-traversing curve in a simply connected domain, joining two prescribed boundary points of the domain; and a conformal loop ensemble (CLE) is a random collection of loops which combine conformal invariance with a natural restriction property suggested by the fact that the discrete analog of this property trivially holds for some discrete models. These CLE ensembles are constructed by two different ways: one is based on SLE and the other one is based on loop-soup.The main purpose of the paper is to show that, to some extent, there is a complete equivalence between SLE and CLE. All these results, which refer to Brownian motion, fractals, percolation, Poisson processes, find their use in the application of conformal field theory to physics.

MSC:
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] V. Beffara, ”The dimension of the SLE curves,” Ann. Probab., vol. 36, iss. 4, pp. 1421-1452, 2008. · Zbl 1165.60007
[2] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, ”Infinite conformal symmetry in two-dimensional quantum field theory,” Nuclear Phys. B, vol. 241, iss. 2, pp. 333-380, 1984. · Zbl 0661.17013
[3] J. van den Berg, ”A note on disjoint-occurrence inequalities for marked Poisson point processes,” J. Appl. Probab., vol. 33, iss. 2, pp. 420-426, 1996. · Zbl 0860.60013
[4] K. Burdzy and G. F. Lawler, ”Nonintersection exponents for Brownian paths. II. Estimates and applications to a random fractal,” Ann. Probab., vol. 18, iss. 3, pp. 981-1009, 1990. · Zbl 0719.60085
[5] F. Camia and C. M. Newman, ”Two-dimensional critical percolation: the full scaling limit,” Comm. Math. Phys., vol. 268, iss. 1, pp. 1-38, 2006. · Zbl 1117.60086
[6] J. T. Chayes, L. Chayes, and R. Durrett, ”Connectivity properties of Mandelbrot’s percolation process,” Probab. Theory Related Fields, vol. 77, iss. 3, pp. 307-324, 1988. · Zbl 0621.60110
[7] D. Chelkak and S. Smirnov, ”Conformal invariance of the Ising model at criticality,” Invent. Math., vol. 189, pp. 515-580, 2012. · Zbl 1257.82020
[8] B. Doyon, Conformal loop ensembles and the stress-energy tensor. I. Fundamental notions of CLE, 2009.
[9] B. Doyon, Conformal loop ensembles and the stress-energy tensor. II. Construction of the stress-energy tensor, 2009. · Zbl 1263.81253
[10] J. Dubédat, ”SLE and the free field: partition functions and couplings,” J. Amer. Math. Soc., vol. 22, iss. 4, pp. 995-1054, 2009. · Zbl 1204.60079
[11] B. Duplantier and S. Sheffield, ”Liouville quantum gravity and KPZ,” Invent. Math., vol. 185, iss. 2, pp. 333-393, 2011. · Zbl 1226.81241
[12] A. Kemppainen, On Random Planar Curves and Their Scaling Limits, 2009.
[13] W. Kager and B. Nienhuis, ”A guide to stochastic Löwner evolution and its applications,” J. Statist. Phys., vol. 115, iss. 5-6, pp. 1149-1229, 2004. · Zbl 1157.82327
[14] S. Lalley, G. Lawler, and H. Narayanan, ”Geometric interpretation of half-plane capacity,” Electron. Commun. Probab., vol. 14, pp. 566-571, 2009. · Zbl 1191.60094
[15] G. F. Lawler, Conformally Invariant Processes in the Plane, Providence, RI: Amer. Math. Soc., 2005, vol. 114. · Zbl 1074.60002
[16] G. F. Lawler, ”Schramm-Loewner evolution,” in Statistical Mechanics, Sheffield, S. and Spencer, T., Eds., Providence, RI: Amer. Math. Soc., 2009, pp. 231-295. · Zbl 1180.82002
[17] G. F. Lawler, O. Schramm, and W. Werner, ”The dimension of the planar Brownian frontier is \(4/3\),” Math. Res. Lett., vol. 8, iss. 4, pp. 401-411, 2001. · Zbl 1114.60316
[18] G. F. Lawler, O. Schramm, and W. Werner, ”Values of Brownian intersection exponents. I. Half-plane exponents,” Acta Math., vol. 187, iss. 2, pp. 237-273, 2001. · Zbl 1005.60097
[19] G. F. Lawler, O. Schramm, and W. Werner, ”Values of Brownian intersection exponents. II. Plane exponents,” Acta Math., vol. 187, iss. 2, pp. 275-308, 2001. · Zbl 0993.60083
[20] G. F. Lawler, O. Schramm, and W. Werner, ”Conformal invariance of planar loop-erased random walks and uniform spanning trees,” Ann. Probab., vol. 32, iss. 1B, pp. 939-995, 2004. · Zbl 1126.82011
[21] G. F. Lawler, O. Schramm, and W. Werner, ”Conformal restriction: the chordal case,” J. Amer. Math. Soc., vol. 16, iss. 4, pp. 917-955, 2003. · Zbl 1030.60096
[22] G. F. Lawler and J. A. Trujillo Ferreras, ”Random walk loop soup,” Trans. Amer. Math. Soc., vol. 359, iss. 2, pp. 767-787, 2007. · Zbl 1120.60037
[23] G. F. Lawler and W. Werner, ”The Brownian loop soup,” Probab. Theory Related Fields, vol. 128, iss. 4, pp. 565-588, 2004. · Zbl 1049.60072
[24] J. Le Gall and G. Miermont, ”Scaling limits of random planar maps with large faces,” Ann. Probab., vol. 39, iss. 1, pp. 1-69, 2011. · Zbl 1204.05088
[25] B. B. Mandelbrot, The Fractal Geometry of Nature, San Francisco, CA: W. H. Freeman and Co., 1982. · Zbl 0504.28001
[26] R. Meester and R. Roy, Continuum Percolation, Cambridge: Cambridge Univ. Press, 1996, vol. 119. · Zbl 0866.60088
[27] J. Miller, ”Fluctuations for the Ginzburg-Landau \(\nabla\phi\) interface model on a bounded domain,” Comm. Math. Phys., vol. 308, iss. 3, pp. 591-639, 2011. · Zbl 1237.82030
[28] J. Miller, Universality for SLE(4). · Zbl 1429.60066
[29] . Nacu and W. Werner, ”Random soups, carpets and fractal dimensions,” J. Lond. Math. Soc. (2), vol. 83, iss. 3, pp. 789-809, 2011. · Zbl 1223.28012
[30] B. Nienhuis, ”Exact critical point and critical exponents of \({ O}(n)\) models in two dimensions,” Phys. Rev. Lett., vol. 49, iss. 15, pp. 1062-1065, 1982.
[31] C. Pommerenke, Boundary Behaviour of Conformal Maps, New York: Springer-Verlag, 1992, vol. 299. · Zbl 0762.30001
[32] J. Pitman and M. Yor, ”A decomposition of Bessel bridges,” Z. Wahrsch. Verw. Gebiete, vol. 59, iss. 4, pp. 425-457, 1982. · Zbl 0484.60062
[33] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, New York: Springer-Verlag, 1991, vol. 293. · Zbl 0731.60002
[34] S. Rohde and O. Schramm, ”Basic properties of SLE,” Ann. of Math., vol. 161, iss. 2, pp. 883-924, 2005. · Zbl 1081.60069
[35] O. Schramm, ”Scaling limits of loop-erased random walks and uniform spanning trees,” Israel J. Math., vol. 118, pp. 221-288, 2000. · Zbl 0968.60093
[36] O. Schramm, ”A percolation formula,” Electron. Comm. Probab., vol. 6, pp. 115-120, 2001. · Zbl 1008.60100
[37] O. Schramm and S. Sheffield, ”Harmonic explorer and its convergence to \({ SLE}_4\),” Ann. Probab., vol. 33, iss. 6, pp. 2127-2148, 2005. · Zbl 1095.60007
[38] O. Schramm and S. Sheffield, ”Contour lines of the two-dimensional discrete Gaussian free field,” Acta Math., vol. 202, iss. 1, pp. 21-137, 2009. · Zbl 1210.60051
[39] O. Schramm and S. Sheffield, A contour line of the continuum Gaussian Free Field, 2010. · Zbl 1331.60090
[40] O. Schramm, S. Sheffield, and D. B. Wilson, ”Conformal radii for conformal loop ensembles,” Comm. Math. Phys., vol. 288, iss. 1, pp. 43-53, 2009. · Zbl 1187.82044
[41] O. Schramm and S. Smirnov, ”On the scaling limits of planar percolation,” Ann. Probab., vol. 39, iss. 5, pp. 1768-1814, 2011. · Zbl 1231.60116
[42] O. Schramm and D. B. Wilson, ”SLE coordinate changes,” New York J. Math., vol. 11, pp. 659-669, 2005. · Zbl 1094.82007
[43] S. Sheffield, ”Exploration trees and conformal loop ensembles,” Duke Math. J., vol. 147, iss. 1, pp. 79-129, 2009. · Zbl 1170.60008
[44] S. Sheffield and N. Sun, Strong path convergence from Loewner driving convergence, 2010. · Zbl 1255.60148
[45] S. Smirnov, ”Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits,” C. R. Acad. Sci. Paris Sér. I Math., vol. 333, iss. 3, pp. 239-244, 2001. · Zbl 0985.60090
[46] S. Smirnov, ”Towards conformal invariance of 2D lattice models,” in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1421-1451. · Zbl 1112.82014
[47] S. Smirnov, ”Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model,” Ann. of Math., vol. 172, iss. 2, pp. 1435-1467, 2010. · Zbl 1200.82011
[48] N. Sun, ”Conformally invariant scaling limits in planar critical percolation,” Probab. Surv., vol. 8, pp. 155-209, 2011. · Zbl 1245.60096
[49] W. Werner, ”Random planar curves and Schramm-Loewner evolutions,” in Lectures on Probability Theory and Statistics, New York: Springer-Verlag, 2004, vol. 1840, pp. 107-195. · Zbl 1057.60078
[50] W. Werner, ”SLEs as boundaries of clusters of Brownian loops,” C. R. Math. Acad. Sci. Paris, vol. 337, iss. 7, pp. 481-486, 2003. · Zbl 1029.60085
[51] W. Werner, ”Conformal restriction and related questions,” Probab. Surv., vol. 2, pp. 145-190, 2005. · Zbl 1189.60032
[52] W. Werner, ”Some recent aspects of random conformally invariant systems,” in Mathematical Statistical Physics, Elsevier B. V., Amsterdam, 2006, pp. 57-99. · Zbl 1370.60142
[53] W. Werner, ”The conformally invariant measure on self-avoiding loops,” J. Amer. Math. Soc., vol. 21, iss. 1, pp. 137-169, 2008. · Zbl 1130.60016
[54] W. Werner, ”Conformal restriction properties,” in International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 741-762. · Zbl 1142.60066
[55] D. Zhan, ”Reversibility of chordal SLE,” Ann. Probab., vol. 36, iss. 4, pp. 1472-1494, 2008. · Zbl 1157.60051
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.