Geodesic
The spectral measure of transition operator and harmonic functions, connected with the random walks on discrete groups
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VI, Tome 97 (1980), pp. 102-109 
V. A. Kaimanovich. The spectral measure of transition operator and harmonic functions, connected with the random walks on discrete groups. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VI, Tome 97 (1980), pp. 102-109. http://geodesic.mathdoc.fr/item/ZNSL_1980_97_a10/
@article{ZNSL_1980_97_a10,
     author = {V. A. Kaimanovich},
     title = {The spectral measure of transition operator and harmonic functions, connected with the random walks on discrete groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {102--109},
     year = {1980},
     volume = {97},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1980_97_a10/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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 $G$.