Random Walk on the Heisenberg group

November 25, 2014
Persi Diaconis | Stanford University

The Heisenberg group is the group of 3 x 3 uni-upper triangular matrices with entries in the reals,integers or mod n. This is in some sense the simplest non-commutative group. It is basic object of study in quantum physics, number theory,harmonic analysis and makes appearences in complexity theory. In joint work with Dan Bump, Angela Hicks, Laurent Miclo and Harold Widom we study simple random walk. For the integers mod n, we show that order n² steps are necessary and sufficient for convergence. For the integers, we get sharp rates for the behavior of the chance of returning to zero. These problems have been extensively studied. One novelty, This talk uses the apparently straight forward approach of Fourier analyis. This turns out to be surprisingly difficult, needing new techniques for bounding the spectrum of symmetric tri diagonal matrices with general entries. The good news is that these same matrices come up in a host of other problems (random walks on other groups,Harpers operator,Hofstaders butterfly,the 10 martini’s problem) and our results seem new and useful.

Speaker Details

Persi Diaconis is a Professor of Mathematics and Statistics at Stanford. He is famous for revolutionizing the field of Finite Markov Chains, showing their significant applications and their connections with diverse areas of Mathematics, in particular group representations and analysis. He is listed among the 20 most influential Scientists today at http://superscholar.org/features/20-most-influential-scientists-alive-today/

Among his awards (Partial list): •1981 – Awarded the Rollo Davidson Prize. •1982- Awarded a MacArthur Fellowship •1997 – Gibbs Lecturer, American Mathematical Society •2003 – Received an honorary D. Sci. degree from the University of Chicago. •2006 – Awarded the Van Wijngaarden Award. •2012 – Awarded the Levi L. Conant Prize. •2012 – Fellow of the American Mathematical Society[ •2013 – Received an Honorary Degree from the University of St Andrews. For more information see his web page at http://statweb.stanford.edu/~cgates/PERSI/ or the detailed biography at http://www-history.mcs.st-and.ac.uk/Biographies/Diaconis.html