Laplacian growth, sandpiles and scaling limits

Laplacian growth, sandpiles and scaling limits

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Yuval Peres , Microsoft Research & UC Berkeley
Fine Hall 214

How can repeating simple local operations lead to an intricate large scale structure? This phenomenon arises in several growth models originating in Physics: Internal diffusion limited aggregation (IDLA) and the Abelian sandpile. The first of these is closely related to free boundary problems for the Laplacian and an algebraic operation introduced by Diaconis and Fulton known as ``smash sum’’. These connections allow a precise description of large scale geometry, using a least action principle. The abelian sandpile, discovered independently by Statistical Physicists and Combinatorialists is harder to analyze, yet has recently yielded many of its secrets in works of Pegden, Smart and Levine. I will also discuss the rotor-router model, where (with random initial conditions) the range is conjectured to grow like t^{2/3} at time t; recently, with L. Florescu, we showed this holds as a lower bound.  This talk is based on joint works with Lionel Levine.