Abstract
Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero.
The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be Presented.
Mathematics Subject Classification (2000). Primary 60K35; Secondary 82B20, 82B43, 30C35.
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Acknowledgements
Greg Lawler, Wendelin Werner and Steffen Rohde have collaborated with me during the early stages of the development of SLE. Without them the subject would not be what it is today. I wish to thank Itai Benjamini, Gil Kalai, Richard Kenyon, Scott Sheffield, Jeff Steif and David Wilson for numerous inspiring conversations. Thanks are also due to Yuval Peres for useful advice, especially concerning Problem 7.1.
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Schramm, O. (2011). Conformally invariant scaling limits: an overview and a collection of problems. In: Benjamini, I., Häggström, O. (eds) Selected Works of Oded Schramm. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9675-6_34
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