We prove existence of asymptotic entropy of random walks on regular languages over a finite alphabet and we give formulas for it. Furthermore, we show that the entropy varies real-analytically in terms of probability measures of constant support, whi
ch describe the random walk. This setting applies, in particular, to random walks on virtually free groups.
We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities d
epend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.
