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Multi-point Green’s functions for SLE and an estimate of Beffara. (English) Zbl 1277.60134
Summary: We define and prove of the existence of the multi-point Green function for Schramm-Loewner evolution (SLE) – a normalized limit of the probability that an SLE\(_{\kappa}\)-curve passes near a pair of marked points in the interior of a domain. When \(\kappa<8\), this probability is nontrivial, and an expression can be written in terms two-sided radial SLE. One of the main components of our proof is a refinement of a bound first provided by V. Beffara [Ann. Probab. 36, No. 4, 1421–1452 (2008; Zbl 1165.60007)]. This work contains a proof of this bound independent from the original.

MSC:
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
82B27 Critical phenomena in equilibrium statistical mechanics
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[1] Alberts, T. and Kozdron, M. J. (2008). Intersection probabilities for a chordal SLE path and a semicircle. Electron. Commun. Probab. 13 448-460. · Zbl 1187.82034
[2] Bass, R. F. (1995). Probabilistic Techniques in Analysis . Springer, New York. · Zbl 0817.60001
[3] Beffara, V. (2008). The dimension of the SLE curves. Ann. Probab. 36 1421-1452. · Zbl 1165.60007
[4] 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
[5] Lawler, G. (2009). Schramm-Loewner evolution (SLE). In Statistical Mechanics. IAS/Park City Mathematics Series 16 231-295. Amer. Math. Soc., Providence, RI. · Zbl 1180.82002
[6] Lawler, G. (2010). Fractal and multifractal properties of SLE.
[7] Lawler, G. (2011). Continuity of radial and two-sided radial SLE at the terminal point. Available at . 1104.1620
[8] Lawler, G. and Zhou, W. (2010). SLE curves and natural parametrization. Available at . 1006.4936 · Zbl 1288.60098
[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
[11] Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. of Math. (2) 161 883-924. · Zbl 1081.60069
[12] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221-288. · Zbl 0968.60093
[13] Schramm, O. and Sheffield, S. (2009). Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202 21-137. · Zbl 1210.60051
[14] 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
[15] 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
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