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FUNCTIONS OF RANDOM WALKS ON HYPERPLANE ARRANGEMENTS
 

Summary: FUNCTIONS OF RANDOM WALKS ON HYPERPLANE
ARRANGEMENTS
CHRISTOS A. ATHANASIADIS AND PERSI DIACONIS
Abstract. Many seemingly disparate Markov chains are unified when viewed as random
walks on the set of chambers of a hyperplane arrangement. These include the Tsetlin
library of theoretical computer science and various shuffling schemes. If only selected
features of the chains are of interest, then the mixing times may change. We study the
behavior of hyperplane walks, viewed on a subarrangement of a hyperplane arrangement.
These include many new examples, for instance a random walk on the set of acyclic
orientations of a graph. All such walks can be treated in a uniform fashion, yielding
diagonalizable matrices with known eigenvalues, stationary distribution and good rates of
convergence to stationarity.
1. Introduction
Many seemingly disparate Markov chains may be successfully studied by viewing them
as random walks on the set of chambers of a hyperplane arrangement [9]. These include
the Tsetlin library of theoretical computer science, a variety of walks on the hypercube and
various shuffling schemes [13]. If only selected features of such a Markov chain are of interest
(for instance, only a few sites on the hypercube or the relative ordering of the top few cards),
then the mixing time may change. Following a suggestion of Uyemura Reyes [46], we
study the behavior of hyperplane walks, viewed on subarrangements of a given hyperplane

  

Source: Athanasiadis, Christos - Department of Mathematics, University of Athens

 

Collections: Mathematics