We show that the asymptotic entropy of a random walk on a nonelementary hyperbolic group, with symmetric and bounded increments, is differentiable and we identify its derivative as a correlation. We also prove similar results for the rate of escape.
"Differentiating the entropy of random walks on hyperbolic groups." Ann. Probab. 43 (1) 166 - 187, February 2015. https://doi.org/10.1214/13-AOP901