Self-avoiding walk's endpoint displacement

The endpoint of n-step self-avoiding walk in Z^d is predicted to have a typical distance from the origin of the order of n^{3/4} when d=2; this distance is numerically determined to be of the order of n^{0.59...} when d=3. In work with Hugo Duminil-Copin, and more recently also with Alexander Glazman and Ioan Manolescu, we have rigorous results excluding the extremes of fast and slow behaviour for endpoint displacement: ballisticity and localization near the origin. The talk will outline some of the main arguments used in establishing these two assertions.