# zbMATH — the first resource for mathematics

SLEs as boundaries of clusters of Brownian loops. (English. Abridged French version) Zbl 1029.60085
Summary: We show that SLE curves can in fact be viewed as boundaries of certain clusters of Brownian loops (of the clusters in a Brownian loop soup). For small densities $$c$$ of loops, we show that the outer boundaries of the clusters created by the Brownian loop soup are SLE$$_{\kappa}$$-type curves where $$\kappa \in (8/3,4]$$ and $$c$$ related by the usual relation $$c=(3{\kappa}-8)(6-{\kappa})/2{\kappa}$$ (i.e., c corresponds to the central charge of the model). This gives (for any Riemann surface) a simple construction of a natural countable family of random disjoint SLE$$_{\kappa}$$ loops, that behaves “nicely” under perturbation of the surface and is related to various aspects of conformal field theory and representation theory.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
##### Keywords:
Schramm-Loewner evolution (SLE); SLE loops
Full Text:
##### References:
 [1] M. Bauer, D. Bernard, Conformal transformations and the SLE partition function martingales, Preprint, 2003 [2] V. Beffara, The dimension of the SLE curves, Preprint, 2002 · Zbl 1165.60007 [3] Belavin, A.A.; Polyakov, A.M.; Zamolodchikov, A.B., Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. phys. B, 241, 333-380, (1984) · Zbl 0661.17013 [4] Brydges, D.; Fröhlich, J.; Spencer, T., The random walk representation of classical spin systems and correlation inequalities, Comm. math. phys., 83, 123-150, (1982) [5] Cardy, J.L., Conformal invariance and surface critical behavior, Nucl. phys. B, 240, 514-532, (1984) [6] Chayes, J.T.; Chayes, L.; Durrett, R., Connectivity properties of Mandelbrot’s percolation process, Probab. theory related fields, 77, 307-324, (1988) · Zbl 0621.60110 [7] J. Dubédat, SLE(κ,ρ) martingales and duality, Preprint, 2003 [8] R. Friedrich, J. Kalkkinen, On conformal field theory and stochastic Löwner evolutions, Preprint, 2003 [9] R. Friedrich, W. Werner, Conformal restriction, highest-weight representations and SLE, Comm. Math. Phys. (2003), in press · Zbl 1030.60095 [10] Knizhnik, V.G.; Polyakov, A.M.; Zamolodchikov, A.B., Fractal structure of 2-D quantum gravity, Mod. phys. lett. A, 3, 819, (1988) [11] Lawler, G.F.; Schramm, O.; Werner, W., Values of Brownian intersection exponents I: half-plane exponents, Acta math., 187, 237-273, (2001) · Zbl 1005.60097 [12] Lawler, G.F.; Schramm, O.; Werner, W., Values of Brownian intersection exponents II: plane exponents, Acta math., 187, 275-308, (2001) · Zbl 0993.60083 [13] G.F. Lawler, O. Schramm, W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. (2001), in press · Zbl 1126.82011 [14] G.F. Lawler, O. Schramm, W. Werner, On the scaling limit of planar self-avoiding walks, in: M. Lapidus (Ed.), AMS Symp. Pure Math., Volume in honor of B.B. Mandelbrot, 2002, in press · Zbl 1069.60089 [15] Lawler, G.F.; Schramm, O.; Werner, W., Conformal restriction properties. the chordal case, J. amer. math. soc., 16, 915-955, (2003) · Zbl 1030.60096 [16] Lawler, G.F.; Werner, W., Universality for conformally invariant intersection exponents, J. eur. math. soc., 2, 291-328, (2000) · Zbl 1098.60081 [17] G.F. Lawler, W. Werner, The Brownian loop soup, Probab. Theory Related Fields, in press · Zbl 1049.60072 [18] Mandelbrot, B.B., The fractal geometry of nature, (1982), Freeman · Zbl 0504.28001 [19] Meester, R.; Roy, R., Continuum percolation, (1996), CUP · Zbl 0858.60092 [20] S. Rohde, O. Schramm, Basic properties of SLE, Ann. Math. (2001), in press · Zbl 1081.60069 [21] Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. math., 118, 221-288, (2000) · Zbl 0968.60093 [22] O. Schramm, S. Sheffield, (2003), in preparation [23] W. Werner, Random planar curves and Schramm-Loewner evolutions, in: 2002 St-Flour Summer School, Lecture Notes in Math., Springer, 2002, in press [24] W. Werner, Girsanov’s theorem for SLE(κ,ρ) processes, intersection exponents and hiding exponents, Ann. Fac. Sci. Toulouse, in press [25] W. Werner, Conformal restriction and related questions, Preprint, 2003 · Zbl 1189.60032 [26] W. Werner, (2003), in preparation
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.