The question about existence of groups of intermediate growth (super-polynomial but sub-exponential) was raised by Milnor in the 60s. First examples of such groups were constructed by Grigorchuk in the early 1980s. We will discuss some probabilistic ideas in studying such groups, and explain near optimal volume growth lower estimates of Grigorchuk groups coming from random walks with nontrivial Poisson-Furstenberg boundary on these groups. Joint with Anna Erschler.