Supercritical self-avoiding walks are space-filling
  We consider random self-avoiding walks between two points on the boundary of a finite subdomain of Z^d (the probability of a self-avoiding trajectory gamma is proportional to mu^{-length(gamma)}). We show that the random trajectory becomes space-filling in the scaling limit when the parameter mu is supercritical. 
Hugo Duminil-Copin, Gady Kozma and Ariel Yadin
math.PR math-ph math.CO math.MP