Infinite paths with bounded or recurrent partial sums

We consider problems of the following type. Assign independently to each ve rtex of the square lattice the value tl, with probability p, or -1, with pr obability 1-p. We ask whether an infinite path rr exists, with the property that the partial sums of the +/- 1s along pi are uniformly bounded. and wh ether there exists an infinite path pi ' with the property that the partial sums along pi ' are equal to zero infinitely often. The answers to these q uestion depend on the type of path one allows, the value of p and the unifo rm bound specified. We show that phase transitions occur for these phenomen a. Moreover. we make a surprising connection between the problem of finding a path to infinity (not necessarily self-avoiding, but visiting each verte x at most finitely many times) with a given bound on the partial sums, and the classical Boolean model with squares around the points of a Poisson pro cess in the plane. For the recurrence problem, we also show that the probab ility of finding such a path is monotone in p. for p greater than or equal to 1/2.