Formula for the Mean Square Displacement Exponent of the Self-Avoiding Walk in 3, 4 and All Dimensions

Abstract

This paper proves the formula \nu(d) =1 for d=1 and \nu(d) = max(1/4 +1/d, 1/2) for d > 1 for the root mean square displacement exponent \nu(d) of the self-avoiding walk (SAW) in Z^d, and thus, resolves some major long-standing open conjectures rooted in chemical physics (Flory, 1949). The values \nu(2) =3/4 and \nu(4) = 1/2 coincide with those that were believed on the basis of heuristic and "numerical evidence". Perhaps surprisingly, there was no precise conjecture in dimension 3. Yet as early as in the 1980ies, Monte Carlo simulations produced a couple of confidence intervals for the exponent \nu(3). This work is a follow-up to Hueter (2001), which proves the result for d=2 and lays out the fundamental building blocks for the analysis in all dimensions. We consider (a) the point process of self-intersections defined via certain paths of length n of the symmetric simple random walk in Z^d and (b) a ``weakly self-avoiding cone process'' relative to this point process in a certain "shape". The asymptotic expected distance of the process in (b) can be calculated rather precisely as n tends large and, if the point process has circular shape, can be shown to asymptotically equal (up to error terms) the one of the weakly SAW with parameter \beta >0. From these results, a number of distance exponents are immediately collectable for the SAW as well. Our approach invokes the Palm distribution of the point process of self-intersections in a cone.