This paper describes a second order perturbation analysis of the BK property in the space of Hermitean determinantal probability measures around the subspace of product measures, showing that the second order Taylor approximation of the BK inequality holds for increasing events.
"Perturbation analysis of the van den Berg Kesten inequality for determinantal probability measures." Electron. J. Probab. 18 1 - 20, 2013. https://doi.org/10.1214/EJP.v18-2339