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Probability and Geometry on Groups  Lecture notes for a graduate course
, 2015
"... These notes have grown (and are still growing) out of two graduate courses I gave at the University of Toronto. The main goal is to give a selfcontained introduction to several interrelated topics of current research interests: the connections between 1) coarse geometric properties of Cayley grap ..."
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These notes have grown (and are still growing) out of two graduate courses I gave at the University of Toronto. The main goal is to give a selfcontained introduction to several interrelated topics of current research interests: the connections between 1) coarse geometric properties of Cayley graphs of infinite groups; 2) the algebraic properties of these groups; and 3) the behaviour of probabilistic processes (most importantly, random walks, harmonic functions, and percolation) on these Cayley graphs. I try to be as little abstract as possible, emphasizing examples rather than presenting theorems in their most general forms. I also try to provide guidance to recent research literature. In particular, there are presently over 150 exercises and many open problems that might be accessible to PhD students. It is also hoped that researchers working either in probability or in geometric group theory will find these notes useful to enter the other field.
ISSN: 1083589X ELECTRONIC COMMUNICATIONS in PROBABILITY
, 2014
"... NonLiouville groups with return probability exponent at most 1/2 Michał Kotowski * Bálint Virág† We construct a finitely generated group G without the Liouville property such that the return probability of a random walk satisfies p2n(e, e) & e−n 1/2+o(1). This shows that the constant 1/2 in a ..."
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NonLiouville groups with return probability exponent at most 1/2 Michał Kotowski * Bálint Virág† We construct a finitely generated group G without the Liouville property such that the return probability of a random walk satisfies p2n(e, e) & e−n 1/2+o(1). This shows that the constant 1/2 in a recent theorem by SaloffCoste and Zheng, saying that return probability exponent less than 1/2 implies the Liouville property, cannot be improved. Our construction is based on permutational wreath products over treelike Schreier graphs and the analysis of large deviations of inverted orbits on such graphs.
NonLiouville groups with return probability exponent at most 1/2
, 2014
"... We construct a finitely generated group G without the Liouville property such that the return probability of a random walk satisfies p2n(e, e) & e−n γ+o(1) for γ = 1/2. This shows that the constant 1/2 in a recent theorem by Gournay, saying that return probability exponent less than 1/2 implies ..."
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We construct a finitely generated group G without the Liouville property such that the return probability of a random walk satisfies p2n(e, e) & e−n γ+o(1) for γ = 1/2. This shows that the constant 1/2 in a recent theorem by Gournay, saying that return probability exponent less than 1/2 implies the Liouville property, cannot be improved. Our construction is based on permutational wreath products over treelike Schreier graphs and the analysis of large deviations of inverted orbits on such graphs. 1