Proof of the Conjecture that the Planar Self-Avoiding Walk has Root Mean Square Displacement Exponent 3/4
Abstract
Description
This paper proves the long-standing open conjecture rooted in chemical physics (Flory (1949)) that the self-avoiding walk (SAW) in the square lattice has root mean square displacement exponent ν= 3/4. This value is an instance of the formula ν=1 on Z and ν= max(1/2, 1/4 + 1/d) in Z^d for dimensions d \geq 2, which will be proved in a subsequent paper. This expression differs from the one that Flory's arguments suggested. We consider (a) the point process of self-intersections defined via certain paths of the symmetric simple random walk in Z^2 and (b) a ``weakly self-avoiding cone process'' relative to this point process when in a certain "shape". We derive results on the asymptotic expected distance of the weakly SAW with parameter β>0 from its starting point, from which a number of distance exponents are immediately collectable for the SAW as well. Our method employs the Palm distribution of the point process of self-intersection points in a cone.
32 pages Change to first version: The lower bound for the normalized first two moments of the distance of the weakly SAW from its starting point is not uniform in βas βtends to infinity
32 pages Change to first version: The lower bound for the normalized first two moments of the distance of the weakly SAW from its starting point is not uniform in βas βtends to infinity