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Loop-erased random walk on planar graphs

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This talk focuses on loop-erased random walk, or LERW . LERW is a random self-avoiding curve obtained by erasing the loop in the trajectory of a random walk in chronological order. Lawler, Schramm, and Werner proved that LERW on the Euclidean plane converges to SLE as the mesh goes to 0. SLE , or Schramm-Loewner Evolution, is a fascinating random process discovered by Oded Schramm in 1999. SLE arises as the scaling limit of many models in mathematical physics. It has many wonderful properties, perhaps the most important is “conformal invariance”. Lawler, Schramm, and Werner’s proof for the scaling limit of LERW essentially uses the symmetry of the lattice structure. The question arises whether a similar result holds even under perturbed lattices; for example, if only a small portion of edges are removed from the original lattice. We extend Lawler, Schramm and Werner’s result: For any planar Markov chain (that is a Markov chain embedded into the complex plane so that edges do not cross one another), if the scaling limit of the Markov chain is planar Brownian motion, then the scaling limit of the loop erasure of the Markov chain is SLE . One main example, is loop-erased random walk on the super-critical percolation cluster; that is, the infinite component after super-critical percolation on Z^2. Berger and Biskup showed that the random walk on the super-critical percolation cluster converges to Brownian motion. Thus, our result implies that the loop-erased random walk on the super-critical percolation cluster converges to SLE . Joint work with Amir Yehudayoff.

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