Consider the standard Ising model on $\mathbb{Z}^2$ for some inverse temperature $\beta\leq \beta_c$ (such that the corresponding inifinite volume measure $\mu$ is unique). For a finite subset $A\subset \mathbb{Z}^2$, is there a method to sample directly from the exact marginal distribution $\mu_{|A}$? An attempt to do this is this recent paper, it however seems to be only an approximation to the marginal.
It is clear to me that this is possible when considering the conditional measure. Introducing a boundary condition $\psi$ on $\mathbb{Z}^2 \setminus A$ and sampling from $$ \mu_\psi = \mu(x \mid x=\psi \text{ on } \mathbb{Z}^2\setminus A)$$ can be done using the Markov chain technique 'coupling from the past'. For $\beta < \beta_c$, one has exponential decay of correlations; therefore, one can enclose $A$ in a much larger set $B$, sample from $\mu_\psi$ for some boundary condition $\psi$ on $\mathbb{Z}^2 \setminus B$ and then restrict the sample to $A$ to obtain a reasonably good approximation, but I would like to know whether there is a more elegant method.