Abstract
This paper examines how close the chordal SLE κ curve gets to the real line asymptotically far away from its starting point. In particular, when κ ϵ (0, 4), it is shown that if β > β κ := 1/(8/κ − 2), then the intersection of the SLE κ curve with the graph of the function y = x/(log x)β, x > e, is a.s. bounded, while it is a.s. unbounded if β = β κ . The critical SLE4 curve a.s. intersects the graph of \(y=x^{{-({\rm log\,log\,x})}^{\alpha}}, x > e^e\), x > e e, in an unbounded set if α ≤ 1, but not if α > 1. Under a very mild regularity assumption on the function y(x), we give a necessary and sufficient integrability condition for the intersection of the SLE κ path with the graph of y to be unbounded. When the intersection is bounded a.s., we provide an estimate for the probability that the SLE κ path hits the graph of y. We also prove that the Hausdorff dimension of the intersection set of the SLE κ curve and the real axis is 2 − 8/κ when 4 < κ < 8.
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References
Alberts, T., Sheffield, S.: Hausdorff dimension of the SLE curve intersected with the real line. arXiv:0711.4070v1 (2007)
Beffara V.: Hausdorff dimensions for SLE(6). Ann. Probab. 32, 2606–2629 (2004)
Beffara, V.: The dimension of the SLE curves. Ann. Prob. (2007, to appear)
Berg R. van den, Járai A.A.: The lowest crossing in two-dimensional critical percolation. Ann. Probab. 31, 1241–1253 (2003)
Camia, F., Newman, C.M.: The full scaling limit of two-dimensional critical percolation. arXiv:math. Pr/0504036 (2005)
Cardy J.: SLE for theoretical physicists. Ann. Phys. 318, 81–118 (2005)
Gruzberg I.A., Kadanoff L.P.: The Loewner equation: maps and shapes. J. Statist. Phys. 114, 1183–1198 (2004)
Lawler, G.F.: Conformally Invariant Processes in the Plane. American Mathematical Society, Providence, RI (2005)
Lawler G., Schramm O., Werner W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32, 939–995 (2004)
Rohde S., Schramm O.: Basic properties of SLE. Ann. Math. 161, 883–924 (2005)
Schramm O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000)
Schramm, O., Sheffield, S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. arXiv:math/0605337 (2008, to appear)
Schramm O., Wilson D.B.: SLE coordinate changes. New York J. Math. 11, 659–669 (2005)
Smirnov S.: Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333, 239–244 (2001)
Smirnov, S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. arXiv:0708.0039v1 (2007)
Werner, W.: Random planar curves and Schramm–Loewner evolutions, Lectures on probability theory and statistics, Lecture Notes in Math., 1840, pp. 107–195. Springer, Berlin. arXiv:math.PR/0303354 (2004)
Zhan, D.: Reversibility of chordal SLE. arXiv:0705.1852 (2007)
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W. Zhou has been supported in part by grants R-155-000-076-112 and R-155-000-083-112 at the National University of Singapore.
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Schramm, O., Zhou, W. Boundary proximity of SLE. Probab. Theory Relat. Fields 146, 435–450 (2010). https://doi.org/10.1007/s00440-008-0195-1
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DOI: https://doi.org/10.1007/s00440-008-0195-1