Metastable winding-number random-walk conjecture for the periodic XY chain

Let $X^{(N)}$ be the periodic XY chain, let $\tau_{0\pm1}$ denote the first transition from winding number $0$ to one of the neighboring winding-number phases, and let $\mathbb{P}^{(N)}_{\mu|0}$ be the dynamics started from equilibrium conditioned on winding number $0$. Assume

$$\liminf \frac{J}{\sigma^2\log N}>1,$$

and define

$$\tau^{(N)}=\frac{1}{2}\inf\left\{t:\mathbb{P}^{(N)}_{\mu|0}\left(\tau_{0\pm1}>t\right)<e^{-1}\right\}.$$

Metastable random-walk conjecture. (i) The metastable time satisfies $\tau^{(N)}=C(J_N,\sigma_N)e^{J_N/\sigma_N^2-\log N}$, where $C(J,\sigma)$ is polynomial in $J$ and $\sigma$. (ii) For $S>0$, define $n_s^{(N)}=\mathcal{W}\left(X^{(N)}(s/\tau^{(N)})\right)$ for $s\in[0,S]$. As $N\to\infty$, the process $(n_s)_{0\leq s\leq S}$ converges to a continuous-time simple random walk with exponential waiting times of rate $1$.

This conjecture describes the loss of memory between metastable winding-number phases in the regime $J/\sigma^2$ above the logarithmic threshold. It predicts both the transition-time scale and a limiting effective random walk, but the supplied source does not state a resolution.