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We consider random walks on a non-elementary hyperbolic group endowed with aword distance. To a probability measure on the group are associated twonumerical quantities, the rate of escape and the entropy. On the set ofadmissible probability measures whose support is contained in a given finiteset, we show that both quantities depend in an analytic way on the probabilitymeasure. Our spectral techniques also give a new proof of the central limittheorem, and imply that the corresponding variance is analytic.

Analyticity of the entropy and the escape rate of random walks in hyperbolic groups