Consider a random process defined on $2^{\mathcal{X}}$, i.e. all subsets of a set $\mathcal{X}$.

It's well known that this process is determinantal if one can find a positive semidefinite matrix $K$, s.t. if $Y$ is a random sample from this process, for any $A \in 2^{\mathcal{X}}$, $\Pr[A \subseteq Y] = det(K_A) $, where $K_A$ is the principal submatrix of $K$ indexed by the elements of $A$.

There is a myriad of results proving that certain processes are determinantal, canonical examples of which are self-avoiding random walks, uniformly random spanning trees, dimer covers in bipartite planar graphs, etc.

My question is if there are any such results of approximate flavor, i.e. proving that for a certain random process $p$ on $2^{\mathcal{X}}$, there is a determinantal point process $q$, s.t. $||p-q||_{TV}$ is bounded by $\epsilon$. ($\epsilon$ could be a constant or a function depending on $|\mathcal X|$, I am not imposing any conditions on it.)