N ov 2 00 4 The Effect of Finite Memory Cutoff on Loop

Let Sn be a simple random walk (SRW) defined on Z . We construct a stochastic process from Sn by erasing loops of length at most N , where α ∈ (0,∞) and N is the scaling parameter that will be taken to infinity in determining the limiting distribution. We call this process the N loop erased walk (N LEW). Under some assumptions we will prove that for 0 < α < 1 1+2ζ , the limiting distribution is Gaussian. Here ζ is the intersection exponent of random walks in Z. For α > 2 the limiting distribution is equal to the limiting distribution of the loop erased walk.