Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups

Abstract

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. We show that deviation inequalities have several consequences including Central Limit Theorems, 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. In a second part of the paper, we show that the (exponential) deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups. These include non-elementary (relatively) hyperbolic groups, Mapping Class Groups, many groups acting on CAT(0) spaces and small cancellation groups.

v2: several new results, including a Central Limit Theorem for random walks on acylindrically hyperbolic groups; v3: final version, to appear in Duke Mathematical Journal

References in corpus (4)

• Asymptotic entropy and green speed for random walks on countable groups
• Central limit theorems for mapping class groups and $\text{Out}(F_N)$
• Analyticity of the entropy and the escape rate of random walks in hyperbolic groups
• Central limit theorem for linear groups

Cited by in corpus (13)

• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• Exponential bounds for random walks on hyperbolic spaces without moment conditions
• Central limit theorems for mapping class groups and $\text{Out}(F_N)$
• Counting loxodromics for hyperbolic actions
• Central limit theorem and geodesic tracking on hyperbolic spaces and Teichmüller spaces
• Sublinearly Morse Boundary II: Proper geodesic spaces
• Gromov's random monsters do not act non-elementarily on hyperbolic spaces
• An embedding of the Morse boundary in the Martin boundary
• Markov chains on hyperbolic-like groups and quasi-isometries
• Random walks and contracting elements I: Deviation inequality and Limit laws
• Large deviation principles for non-elementary random walks on hyperbolic spaces
• A central limit theorem for the degree of a random product of Cremona transformations
• Large deviations for irreducible random walks on relatively hyperbolic groups