Random walk on unipotent matrix groups

2015/12/20 by Diaconis, Persi, Hough, Bob · 2 citations
#20B25 #22E25 #60B15 #60E10 #60F05 #60F25 #60G42 #60J10 #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR)

paper · doi:10.48550/arxiv.1512.06304

Abstract

We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving measure. As a second illustration, the method is used to study walks on the n× n uni-upper triangular group with entries taken modulo p. The method allows sharp answers to the behavior of individual coordinates: coordinates immediately above the diagonal require order p² steps for randomness, coordinates on the second diagonal require order p steps; coordinates on the kth diagonal require order p^(2)/(k) steps.

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• Cutoff for random walks on dihedral groups
• On Rapid mixing for random walks on nilmanifolds

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