Off-critical mean conjecture for the FK heat-bath coupling

Let $T_L$ be the coupling time and $W_L$ the coupon-collector time for the FK heat-bath coupling on the torus $\mathbb{Z}_L^d$, where $d\geq2$, $q\geq1$, and $p\in(0,1)$ with $p\neq p_{\mathrm{c}}$. Define $\mu_T(L)=\mathbb{E}(T_L)$ and $\mu_W(L)=\mathbb{E}(W_L)$.

Off-critical mean conjecture. There exists $C(p,q,d)\geq1$ such that, as $L\to\infty$,

$$\mu_T(L)\sim C(p,q,d)\,\mu_W(L).$$

The conjecture formalizes the expectation that off-critical coupling is governed by coupon-collector behavior; the paper reports numerical evidence and notes consequences for the mixing-time scale in two dimensions.