Suppose we are given the free product $V$ of a finite family of finite or countable sets. We consider a transient random walk on the free product arising naturally from a convex combination of random walks on the free factors. We prove the existence of the asymptotic entropy and present three different, equivalent formulas, which are derived by three different techniques. In particular, we will show that the entropy is the rate of escape with respect to the Greenian metric. Moreover, we link asymptotic entropy with the rate of escape and volume growth resulting in two inequalities.
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机译：假设我们得到了有限集或可数集的有限族的自由积$ V $。我们考虑对自由因子的随机游走的凸组合自然产生的对自由产品的瞬态随机游走。我们证明了渐近熵的存在，并给出了三种不同的等效公式，它们是通过三种不同的技术得出的。特别是，我们将证明熵是相对于格林度量的逃逸率。此外，我们将渐近熵与逃逸率和体积增长联系起来，从而导致两个不等式。
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