We study the moments of the distance traveled by a walk in the plane with unit steps in random directions. While this historically interesting random walk is well understood from a modern probabilistic point of view, our own interest is in determining explicit closed forms for the moment functions and their arithmetic values at integers when only a small number of steps is taken. As a consequen...

We consider random walks in a random environment of the type p0 + γξz, where p0 denotes the transition probabilities of a stationary random walk on Z, to nearest neighbors, and ξz is an i.i.d. random perturbation. We give an explicit expansion, for small γ, of the asymptotic speed of the random walk under the annealed law, up to order 2. As an application, we construct, in dimension d ≥ 2, a wa...

For the random Dirichlet series ∞ ∑ n=0 Xn(ω) e−sλn (s = σ + it ∈ C, 0 = λ0 < λn ↑ ∞), whose coefficients are uniformly nondegenerate independent random variables, we provide some explicit conditions for the line of convergence to be its natural boundary a.s. Running Title Natural Boundary of Random Dirichlet Series

We prove that in various natural models of a random quotient of a group, depending on a density parameter, for each hyperbolic group there is some critical density under which a random quotient is still hyperbolic with high probability, whereas above this critical value a random quotient is very probably trivial. We give explicit characterizations of these critical densities for the various mod...