Markov Type and the Multi-scale Geometry of Metric Spaces – How Well Can Martingales Aim?


November 15, 2012


James Lee


University of Washington


The behavior of random walks on metric spaces can sometimes be understood by embedding such a walk into a nicer space (e.g. a Hilbert space) where the geometry is more readily approachable. This beautiful theme has seen a number of geometric and probabilistic applications. We offer a new twist on this study by showing that one can employ mappings that are significantly weaker than bi-Lipschitz. This is used to answer questions of Naor, Peres, Schramm, and Sheffield (2004) by proving that planar graph metrics and doubling metrics have Markov type 2. The main new technical idea is that martingales are significantly worse at aiming than one might at first expect. (Joint work with Jian Ding and Yuval Peres).


James Lee

James Lee is an Associate Professor at the Department of Computer Science and Engineering, University of Washington. He received a PhD in CS from Berkeley, After a postdoc in Avi Wigderson’s group at the Institute for Advanced Study in Princeton he joined UW.

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