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On the continuity of SLE(k) curves in k and their behavior at the tip

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On the continuity of SLE curves in k and their behavior at the tip

The Schramm-Loewner evolution with parameter k > 0, SLE , is a family of random fractal curves constructed using the Loewner differential equation driven by a standard Brownian motion times the square-root of k. These curves arise as scaling limits of cluster interfaces in certain planar critical lattice models. A natural question that has been asked is whether the (parameterized) SLE curves almost surely change continuously if the Brownian motion sample is kept fixed while k is varied.

In the talk I will present recent work giving a positive answer to this question, at least for an interval of k. I will also give some background on SLE and the Loewner equation, and describe the basic tools we use for the proof, in particular certain bounds characterizing the behavior of a growing SLE curve close to its tip.

The talk is based on joint work with Steffen Rohde and Carto Wong, and Greg Lawler.

This talk is part of the Probability series.

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