The spectral measure of transition operator and harmonic functions, connected with the random walks on discrete groups

The random walk on a countable group $G$ determined by probability measure $\mu$ is under consideration. We obtain an estimation of the spectral measure of the random walk transition operator by the Folner's sets growth for amenable groups. This estimation allows to find a lower bound for the probability of returning to the unit of $G$ on the $n$-th step. These estimations are given for the groups $G_k=\mathbb Z^k\times\mathbb Z_2(\mathbb Z^k)$. In the second part of the paper we obtain a lower bound for the entropy $h(G,\mu)$ by the variation of nontrivial bounded $\mu$-harmonic function on $G$.