On the probability that self-avoiding walk ends at a given point

We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on Zd for d.2. We show that the probability that a walk of length n ends at a point x tends to 0 as n tends to infinity, uniformly in x. Also, when x is fixed, with .x.=1, this probability decreases faster than n.1/4+. for any .>0. This provides a bound on the probability that a self-avoiding walk is a polygon.