consider the random walk among n places with n(n - 1)/2 transports. we attach an exponential random variable xij to each transport between places pi and pj and take these random variables mutually independent. if transports are possible or impossible independently with probability p and 1-p, respectively, then we give a lower bound for the distribution function of the smallest path at point log...

Suppose that G and H are finite, connected graphs, G regular, X is a lazy random walk on G and Z is a reversible ergodic Markov chain on H. The generalized lamplighter chain X associated with X and Z is the random walk on the wreath product H o G, the graph whose vertices consist of pairs (f, x) where f = (fv)v∈V (G) is a labeling of the vertices of G by elements of H and x is a vertex in G. In...

Main theorems of the article concern the problem of M. Atiyah on possible values of l2-Betti numbers. It is shown that all non-negative real numbers are l2-Betti numbers, and that “many” (for example all non-negative algebraic) real numbers are l2-Betti numbers of aspherical manifolds with respect to a free cocompact action. Also an explicit example is constructed which leads to an aspherical m...

Let T1, . . . , Td be homogeneous trees with degrees q1 + 1, . . . , qd + 1 ≥ 3, respectively. For each tree, let h : Tj → Z be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of T1, . . . , Td is the graphDL(q1, . . . , qd) consisting of all d-tuples x1 · · ·xd ∈ T1×· · ·×Td with h(x1) + · · ·+ h(xd) = 0, equipped wi...