Growth of periodic Grigorchuk groups

On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. In particular, for the first Grigorchuk group $G$ we show that its volume growth function $v_{G,S}(n)$ satisfies that $\lim_{n\to\infty}\log\log v_{G,S}(n)/\log n=\alpha_{0}$, where $\alpha_{0}=\frac{\log2}{\log\lambda_{0}}\approx0.7674$, $\lambda_{0}$ is the positive root of the polynomial $X^{3}-X^{2}-2X-4$.