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In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusA Class of Random Walks on the Hypercube

In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius: A Class of Random Walks on the... [We introduce a general class of time inhomogeneous random walks on the N-hypercube. These random walks are reversible with an N-product Bernoulli stationary distribution and have a property of local change of coordinates in a transition. Several types of representations for the transition probabilities are found. The paper studies cut-off for the mixing time. We observe that for a sub-class of these processes with long range (i.e. non-local) there exists a critical value of the range that allows an almost-perfect mixing in at most two steps. That is, the total variation distance between the two steps transition and stationary distributions decreases to zero as the dimension of the hypercube N increases. Notice that a well-known result (Theorem 1 in [6]) shows that there does not exist a random walk on Abelian groups (such as the hypercube) which mixes perfectly in exactly two steps.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusA Class of Random Walks on the Hypercube

Part of the Progress in Probability Book Series (volume 77)
Editors: Vares, Maria Eulália; Fernández, Roberto; Fontes, Luiz Renato; Newman, Charles M.

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References (14)

Publisher
Springer International Publishing
Copyright
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
ISBN
978-3-030-60753-1
Pages
265 –298
DOI
10.1007/978-3-030-60754-8_13
Publisher site
See Chapter on Publisher Site

Abstract

[We introduce a general class of time inhomogeneous random walks on the N-hypercube. These random walks are reversible with an N-product Bernoulli stationary distribution and have a property of local change of coordinates in a transition. Several types of representations for the transition probabilities are found. The paper studies cut-off for the mixing time. We observe that for a sub-class of these processes with long range (i.e. non-local) there exists a critical value of the range that allows an almost-perfect mixing in at most two steps. That is, the total variation distance between the two steps transition and stationary distributions decreases to zero as the dimension of the hypercube N increases. Notice that a well-known result (Theorem 1 in [6]) shows that there does not exist a random walk on Abelian groups (such as the hypercube) which mixes perfectly in exactly two steps.]

Published: Nov 4, 2020

Keywords: Hypercube; Random walk; Markov chain; Mixing; 60J10

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