Limiting behavior of determinantal point processes associated with weighted Bergman kernels

Let Ω be a bounded pseudoconvex domain in Cⁿ and φ be strictly plurisubharmonic function on Ω. For each k, we consider determinantal point process Λₖ with kernel Kkφ, where Kkφ is the reproducing kernel of weighted Bergman space H(kφ) with weight e-kφ. We show that the limit of scaled cumulant generating funcion for Λₖ converges as k→∞ to a certain limit, which can be explicitly expressed in terms of φ and a test function u. Note that we need to restrict the shape of the test functions u to be φ-admissible.