The Flory scaling conjecture for the barycentric excluded-volume random walk

Let $(W_n,G_n)$ be the barycentric excluded-volume random walk described in the paper, with $W_n\in\mathbb R^2$ its position, and let $\|\cdot\|$ denote Euclidean norm. Let $\Theta$ be a uniform random unit vector. Flory scaling conjecture. Almost surely,

$$\lim_{n\to\infty}\frac{\log\|W_n\|}{\log n}=\frac34, \qquad \lim_{n\to\infty}\frac{W_n}{\|W_n\|}=\Theta.$$

The conjecture predicts anomalous, superdiffusive behaviour with scale exponent $3/4$, motivated by the Flory exponent for planar self-avoiding walks. The paper reports heuristic and numerical evidence, but rigorous analysis of this model remains open.