We obtain explicit formulae, in the form of regularized multiplicative functionals related to certain Blaschke products, of the Radon–Nikodym derivatives between reduced Palm measures of all orders for determinantal point processes associated with a large class of weighted Bergman spaces on the disk. Our method also applies to determinantal point processes associated with weighted Fock spaces.
The Thoma cone is an infinite-dimensional locally compact space, which is closely related to the space of extremal characters of the infinite symmetric group. In another context, the Thoma cone appears as the set of parameters for totally positive, upper triangular Toeplitz matrices of infinite size.
The purpose of the paper is to construct a family of continuous time Markov processes on the Thoma cone, depending on two continuous parameters. Our construction largely exploits specific properties of the Thoma cone related to its representation-theoretic origin, although we do not use representations directly. On the other hand, we were inspired by analogies with random matrix theory coming from models of Markov dynamics related to orthogonal polynomial ensembles.
We show that our processes possess a number of nice properties, namely: (1) every process X is a Feller process; (2) the infinitesimal generator of X, its spectrum, and the eigenfunctions admit an explicit description; (3) in the equilibrium regime, the finite-dimensional distributions of X can be interpreted as (the laws of) infinite-particle systems with determinantal correlations; (4) the corresponding time-dependent correlation kernel admits an explicit expression, and its structure is similar to that of time-dependent correlation kernels appearing in random matrix theory.