The resolvent for simple random walks on the free product of two discrete groups (Q1085869)

The author studies symmetric random walks on the free product \(G=G_ 1*G_ 2\) of two discrete groups; such a random walk is given by a probability \(p=a_ 1p_ 1+a_ 2p_ 2\), where \(p_ i\) are symmetric probabilities on \(G_ i\), \(p_ i(e)=0\) and \(a_ 1+a_ 2=1\). The author gives a method using Stieltjes transforms to calculate the resolvent \((=power\) series in \(z^{-1}\) with coefficients \(p^{*n}(e))\) of p from the resolvents of \(p_ i\). In some particular cases he gets from this local limit theorems. Similar results using another method were obtained by \textit{W. Woess}, Random walks on free products of discrete groups. Boll. Unione Mat. Ital. (1987).