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.)


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.