The two-dimensional moving-polymer scaling conjecture

Let $J>0$ be the intrinsic length of a moving polymer taking values in $\mathbf{R}^2$. Because the path $x\mapsto\mathbf{u}(t,x)$ behaves like a two-dimensional Brownian motion, its local time is replaced by either Varadhan's renormalized self-intersection local time or a mollified local time, and $R(T,J)$ denotes the effective radius defined using one of these alternatives. The two-dimensional moving-polymer scaling conjecture. With high probability,

$$R(T,J)\approx J^{5/4}.$$

This conjecture is based on Flory's physical reasoning. The corresponding one-dimensional model has a different scaling result, while in two dimensions the local time must be renormalized or mollified; the dependence of $R(T,J)$ on $\beta$ is not specified.