Loop-erased Walks Intersect Infinitely Often in Four Dimensions

In this short note we show that the paths two independent loop-erased random walks in four dimensions intersect infinitely often. We actually prove the stronger result that the cut-points of the two walks intersect infinitely often. Let S(t) be a transient Markov chain with integer time t on a countable state space. Associated to S, is the loop-erased process Ŝ obtained by erasing loops in chronological order defined as follows. Let s0 = sup{t : S(t) = S(0)}, and for n > 0, sn = sup{t : S(t) = S(sn−1 + 1)}. Then the loop-erased process Ŝ(n) is defined by