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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 andquantify this property by means of the notion of deviation inequalities. Weshow that deviation inequalities have several consequences including CentralLimit Theorems, the local Lipschitz continuity of the rate of escape andentropy, as well as linear upper and lower bounds on the variance of thedistance of the position of the walk from its initial point. In a second partof the paper, we show that the (exponential) deviation inequality holds formeasures with exponential tail on acylindrically hyperbolic groups. Theseinclude non-elementary (relatively) hyperbolic groups, Mapping Class Groups,many groups acting on CAT(0) spaces and small cancellation groups.

Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups