×

SLE curves and natural parametrization. (English) Zbl 1288.60098

G. F. Lawler and S. Sheffield [Ann. Probab. 39, No. 5, 1896–1937 (2011; Zbl 1234.60087)] proposed a definition for a natural parametrization for the Schramm-Loewner evolution \(SLE_\kappa\) in terms of the Doob-Meyer decomposition of a path and conjectured that this definition is valid for all \(\kappa<8\), but could only establish the result for \(\kappa<4(7-\sqrt{33})\). In the paper under review it is shown that the definition is indeed valid for all \(\kappa<8\).

MSC:

60J67 Stochastic (Schramm-)Loewner evolution (SLE)
60D05 Geometric probability and stochastic geometry
60J60 Diffusion processes
30C20 Conformal mappings of special domains
28A80 Fractals

Citations:

Zbl 1234.60087
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Beffara, V. (2008). The dimension of the SLE curves. Ann. Probab. 36 1421-1452. · Zbl 1165.60007 · doi:10.1214/07-AOP364
[2] Cardy, J. (2005). SLE for theoretical physicists. Ann. Physics 318 81-118. · Zbl 1073.81068 · doi:10.1016/j.aop.2005.04.001
[3] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential : Theory of Martingales. B. North-Holland Mathematics Studies 72 . North-Holland, Amsterdam. · Zbl 0494.60002
[4] Gruzberg, I. A. and Kadanoff, L. P. (2004). The Loewner equation: Maps and shapes. J. Stat. Phys. 114 1183-1198. · Zbl 1072.81052 · doi:10.1023/B:JOSS.0000013973.40984.3b
[5] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus , 2nd ed. Graduate Texts in Mathematics 113 . Springer, New York. · Zbl 0734.60060
[6] Lawler, F. and Werness, B. (2013). Multi-point Green’s functions for SLE and an estimate of Beffara. Ann. Probab. 41 1513-1555. · Zbl 1277.60134
[7] Lawler, G. (2009). Schramm-Loewner evolution (SLE). In Statistical Mechanics (S. Sheffield and T. Spencer, eds.). IAS/Park City Mathematics Series 16 231-295. Amer. Math. Soc., Providence, RI. · Zbl 1180.82002
[8] Lawler, G. F. (1980). A self-avoiding random walk. Duke Math. J. 47 655-693. · Zbl 0445.60058 · doi:10.1215/S0012-7094-80-04741-9
[9] Lawler, G. F. (2005). Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114 . Amer. Math. Soc., Providence, RI. · Zbl 1074.60002
[10] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939-995. · Zbl 1126.82011 · doi:10.1214/aop/1079021469
[11] Lawler, G. F. and Sheffield, S. (2011). A natural parametrization for the Schramm-Loewner evolution. Ann. Probab. 39 1896-1937. · Zbl 1234.60087 · doi:10.1214/10-AOP560
[12] Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. of Math. (2) 161 883-924. · Zbl 1081.60069 · doi:10.4007/annals.2005.161.883
[13] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221-288. · Zbl 0968.60093 · doi:10.1007/BF02803524
[14] Schramm, O. and Sheffield, S. (2009). Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202 21-137. · Zbl 1210.60051 · doi:10.1007/s11511-009-0034-y
[15] Schramm, O. and Zhou, W. (2010). Boundary proximity of SLE. Probab. Theory Related Fields 146 435-450. · Zbl 1227.60101 · doi:10.1007/s00440-008-0195-1
[16] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239-244. · Zbl 0985.60090 · doi:10.1016/S0764-4442(01)01991-7
[17] Smirnov, S. (2010). Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2) 172 1435-1467. · Zbl 1200.82011 · doi:10.4007/annals.2010.172.1441
[18] Werner, W. (2004). Random planar curves and Schramm-Loewner evolutions. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1840 107-195. Springer, Berlin. · Zbl 1057.60078 · doi:10.1007/978-3-540-39982-7_2
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.