**General Problem:** Suppose $X_1,\ldots,X_n \sim \mathbb{P}_X^{\otimes n}$ is a finite sequence of i.i.d. (real- or integer-valued) random variables. Suppose $A\subseteq \mathbb{R}^n$ is a set of "admissible configurations".

*Are there efficient methods of sampling from the restriction of $\mathbb{P}_X^{\otimes n}$ onto $A$?*

**More specific context:** I understand that above formulation of the problem is hopelessly general, therefore I would like to give more context on my actual setting. In my case, I am considering a model with $\mathbb{P}_X = \mathrm{Poi}(\lambda)$ and where $A$ is given as the solution set of a finite numbers of equations (representing parity constraints on the Poisson variables). More specifically, I am considering the *random current model* (cf. [this survey article by Duminil-Copin][1]).

Therefore a (perhaps more approachable) version of the problem would be as follows: Consider also a finite set of linear functions $F_1,\ldots,F_N \colon \mathbb{R}^n \to \mathbb{R}$ and an "admissible value set" $V_\mathrm{adm}$. Then let $A:= \{\mathbf{x} \in \mathbb{R}^n\mid F_i(\mathbf{x}) \in V_\mathrm{adm}, \, i=1,\ldots,N\}$.

*Is it in this context possible to efficiently sample from the restriction of $\mathbb{P}_X^{\otimes n}$ onto $A$?*

I would be glad about input either on the general of more specific formulation of the problem. Also hints to general literature on such problems would be very helpful.



**EDIT**: As requested, I will try to make the random current setting more explicit: One considers a simple graph $G = (\mathcal{V},\mathcal{E})$. For a subset $S\subseteq \mathcal{V}$, we define the "admissible set" $A_S :=\{ \mathbf{n} = \{\mathbf{n}_e\}_{e\in \mathcal{E}}\in \mathbb{N}_0^\mathcal{E}\mid \mathrm{deg}_\mathbf{n}(v) \text{ odd for $v$ in $S$ and even otherwise}\}$, where $\mathrm{deg}_\mathbf{n}(v) = \sum_{vw \in \mathcal{E}}\mathbf{n}_{vw}$.


  [1]: https://arxiv.org/abs/1607.06933