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Mathematics Colloquium

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Date:  Tuesday, January 28, 2014

Title:  Loop-erased random walk

Abstract:  A number of models of random fractals arise as limits of discrete models with interactions. I will give a survey of one model, the loop-erased random walk (LERW), which arises from erasing loops of simple random walks. This process is also known as the Laplacian random walk in that the distribution of its growth is tied to the solution of the Dirichlet problem with value zero on the path. The most interesting dimensions for this process are two and three. I will explain why and then discuss some of the behavior in these dimensions. I will emphasize two dimensions where the scaling limit is a ``conformally invariant'' random fractal with fractal dimension 5/4. The Schramm-Loewner evolution (SLE) was developed to understand this process. If time allows, I will describe some recent work with M. Rezaei on the Minkowski content of the path and with C. BeneÅ¡ and F. Viklund on the convergence of the discrete Green's function of the loop-erased walk to the SLE Green's function. This talk is intended for a general mathematical audience. It is not expected that the audience knows the terms in this abstract.


Speaker:  Greg Lawler
Institution:  University of Chicago

 

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