Gumbel limiting-distribution conjecture for the FK coupling time

Let $T_L$ be the coupling time of the FK heat-bath process on $\mathbb{Z}_L^d$, with $d\geq2$, $q\geq1$, and $p\in(0,1)$. Assume that $p\neq p_{\mathrm{c}}$ whenever $q\geq q_*$. Let $\mu_T(L)=\mathbb{E}(T_L)$, $\sigma_T(L)=\sqrt{\operatorname{var}(T_L)}$, and let $G$ be the Gumbel distribution function.

Gumbel limiting-distribution conjecture.

$$\lim_{L\to\infty}\mathbb{P}\left[T_L\leq\mu_T(L)+x\sigma_T(L)\right]=G(x),\qquad x\in\mathbb{R}.$$

The conjecture extends the one-dimensional Gumbel limit to higher dimensions, both off criticality and at continuous critical points. The paper reports strong numerical evidence for both regimes.