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arXiv · 2001.10987

A new approach to SLE phase transition

Dmitry Beliaev·Terry J. Lyons·Vlad Margarint
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Abstract

It is well know that $SLE_\kappa$ curves exhibit a phase transition at $\kappa=4$. For $\kappa\le 4$ they are simple curves with probability one, for $\kappa>4$ they are not. The standard proof is based on the analysis of the Bessel SDE of dimension $d=1+4/\kappa$. We propose a different approach which is based on the analysis of the Bessel SDE with $d=1-4/\kappa$. This not only gives a new perspective, but also allows to describe the formation of the SLE `bubbles' for $\kappa>4$.

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Dmitry Beliaev, Terry J. Lyons, Vlad Margarint. 2020-01-29. A new approach to SLE phase transition. https://arxiv.org/abs/2001.10987

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