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

Speaker  James Lee

Affiliation  University of Washington

Host  Yuval Peres

Duration  01:09:06

Date recorded  15 November 2012

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).

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