Ballistic central-limit conjecture for the infinite Edwards model

Let $\mathbb{Q}^{\beta}$ be the probability measure of the one-dimensional Edwards model on polymers of infinite length, and let $X=(X_t)_{t\geq 0}$ be its canonical process. The parameters $a$ and $\sigma$ below are universal positive constants.

Infinite Edwards-model conjecture. Under $\mathbb{Q}^{\beta}$, the process $X$ is transient and

$$\mathbb{Q}^{\beta}\left(X_t\mathop{\longrightarrow}\limits_{t\to\infty}+\infty\right)=\mathbb{Q}^{\beta}\left(X_t\mathop{\longrightarrow}\limits_{t\to\infty}-\infty\right)=\frac12.$$

Moreover,

$$\frac{|X_t|}{t}\mathop{\longrightarrow}\limits_{t\to\infty}a\beta^{1/3}$$

a.s., and

$$\frac{|X_t|-a\beta^{1/3}t}{\sqrt{t}}$$

converges in law to a centered Gaussian variable of variance $\sigma^2$.

The paper constructs the infinite-length Edwards measure and gives an explicit expression for its finite-time density, but states that the behavior of the canonical process is not known. The conjecture predicts both ballistic motion and Gaussian fluctuations; the factor $\beta^{1/3}$ is dictated by Brownian scaling.