The metastable random-walk conjecture for the dynamical sine-Gordon model

Let the dynamical sine-Gordon system be given by the stochastic evolution in~, and let the mean transition time be the quantity in~ or~, according to whether $\gamma\beta>1$ or $\gamma\beta<1$. Consider the process obtained by rescaling time by this mean transition time.

Metastable random-walk conjecture. As $\epsilon\to0$, the rescaled process approaches a symmetric simple random walk on

$$\{k2\pi/\beta\colon k\in\mathbb Z\},$$

with exponentially distributed times between jumps of mean $1$.

This describes the expected effective Markov-chain behaviour of the metastable sine-Gordon system: after rescaling by the long transition time, successive jumps between neighbouring wells should become symmetric and memoryless. The statement is presented as a natural expectation; its proof is beyond the scope of the article.