Monotone coupling from the past for finite ferromagnetic Ising weights

Input a finite nonempty set $V$, a set $E\subseteq [V{]}^2$ of unordered pairs of distinct sites, finite dimensionless couplings $K_e\ge 0$, finite dimensionless fields $h_i\in \mathbb{R}$, and a two-sided iid stream $(I_s,U_s{)}_{s\in \mathbb{Z}}$ in which $I_s$ is uniform on $V$, $U_s$ is uniform on $[0,1]$, and all these variables are mutually independent. On $\mathrm{\Omega }=\{-1,+1{\}}^V$, define $\sigma \le \tau$ by ${\sigma }_i\le {\tau }_i$ for every $i\in V$, using $-1<+1$; its least and greatest elements are $-$ and $+$. For the finite-spin heat-bath update, use the total sampling order $+1\prec -1$, which is distinct from $\le$. Define the strictly positive finite weight $w(\sigma )=\exp \text{\hspace{0.17em}⁣}({\sum }_{\{i,j\}\in E}{K}_{\{i,j\}}{\sigma }_i{\sigma }_j+{\sum }_{i\in V}{h}_i{\sigma }_i)\mathrm{.}$ At time $s$, let $F_s:\mathrm{\Omega }\to \mathrm{\Omega }$ be the one-site finite-spin heat-bath update at site $I_s$ with uniform value $U_s$, leaving every other spin unchanged; explicitly, site $i$ is set to $+1$ exactly when $U_s<\frac{\exp (H_i(\eta ))}{\exp (H_i(\eta ))+\exp (-H_i(\eta ))},H_i(\eta )=h_i+{\sum }_{\{i,j\}\in E}{K}_{\{i,j\}}{\eta }_j,$ where $\eta$ is the exterior configuration at $i$, and otherwise is set to $-1$. For $T=1,2,4,\dots$, initialize lower and upper configurations at time $-T$ to $-$ and $+$, respectively, and apply in chronological order the same stored maps $F_{-T+1},\dots ,F_0$ to both. If their time-zero outputs agree, return that common configuration. Otherwise double $T$ and replay all maps in the enlarged interval, reusing the already stored maps at every overlapping time. This is a random procedure; no finite running-time bound is part of its definition.

Finite-spin random-scan heat-bath update

The order-preserving update is part of the specified coupling. Replacing a previously stored seed when a horizon is doubled changes the procedure. Termination and exactness are proved separately.