Deviation inequalities for random walks

By Pierre Mathieu, Alessandro Sisto. 2014

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We study random walks on groups with the feature that, roughly speaking, successive positions of the walk tend to be "aligned". We formalize and quantify this property by means of the notion of deviation inequalities. On one hand, we show that the (exponential) deviation inequality holds for measures with exponential tail on what we call acylindrically hyperbolic groups with hierarchy paths. These include non-elementary (relatively) hyperbolic groups, Mapping Class Groups, groups acting on CAT(0) cube complexes and small cancellation groups. On the other hand, we show that deviation inequalities have several consequences including the local Lipschitz continuity of the rate of escape and entropy, as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a6f737fc972d15feff5a29ddb4b959929a1376fd