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 with probability p or with probability p We ask whether an innite path exists with the property that the partial sums of the s 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 selfavoiding 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