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+ε for any ε>0. This provides a bound on the probability that a self-avoiding walk is a polygon.