Central limit theorem for the determinantal point process with the confluent hypergeometric kernel

The theorem on the diagonalization of an $n$-by $n$ Hermitian matrix can be formulated as follows: "Ergodic measures on Hermitian matrices with respect to the action of the unitary group by conjugation are parametrized by $n$-point subsets of the real line (the spectra of the matrices)."
As shown by G. Olshanski and A. Vershik, this formulation remains valid even when $n$ is infinite. The generalization of the spectrum in this case is a countable subset of the real line. This result can be informally interpreted as a method for diagonalizing semi-infinite matrices.
Just as any unitarily invariant measure on finite matrices induces a measure on $n$-point subsets of the real line by taking the spectrum, a measure on semi-infinite matrices induces a random countable subset. An interesting example of measures on infinite matrices are the Hua–Pickrell measures; the induced measure on subsets is called the point process with the confluent hypergeometric kernel.
A. Borodin and G. Olshansky proved that this process is determinantal: it is connected to a certain Hilbert space of holomorphic functions. The talk will focus on describing this space and its connection to the central limit theorem – specifically, the convergence of the logarithm of the “characteristic polynomial” of a random semi-infinite matrix (in the sense described above) to a Gaussian distribution under the contraction of the random subset.