Infinite paths with bounded or recurrent partial sums

Abstract

We consider problems of the following type. Assign independently to each vertex of the square lattice the value +1, with probability p, or - 1, with probability 1 - p. We ask whether an innite path ? exists, with the property that the partial sums of the 1s along ? are uniformly bounded, and whether there exists an innite path ? with the property that the partial sums along ? are equal to zero innitely often. The answers to these question depend on the type of path one allows, the value of p and the uniform bound specied. We show that phase transitions occur for these phenomena. Moreover, we make a surprising connection between the problem of nding a path to innity (not necessarily self-avoiding, but visiting each vertex at most nitely many times) with a given bound on the partial sums, and the classical Boolean model with squares around the points of a Poisson process in the plane. For the recurrence problem, we also show that the probability of nding such a path is monotone in p, for p 1 2 .