On the displacement exponent of the self-avoiding walk in three and higher dimensions.
Preprint (2003)

This paper proves that the self-avoiding walk (SAW) in Z^d has its lower and upper displacement exponents between 1/4 + 1/d and (d+1)/2d for d >1 and at exactly 1 for d=1. The problem on the value of the displacement exponent of the SAW, in three dimensions especially, dates back to work (P.J. Flory, 1949) on linear polymer chains in chemical physics and is surrounded by cascades of open conjectures on the SAW, mostly for d=2. While little is mathematically known about the SAW for d=2 or d=3, as early as in the 1980ies, Monte Carlo simulations produced confidence intervals for the displacement exponent and other critical exponents. Our results extend the ones for d=2, presented in earlier work of Hueter, and are based on the method which involves the weakly self-avoiding cone process in a certain shape of the point process of self-intersections introduced there.