Dynamical relationship between CAR algebras and determinantal point processes: point processes at finite temperature and stochastically positive KMS systems
Ryosuke Sato
TL;DR
This work extends the static link between determinantal point processes and CAR algebras to a dynamical setting by employing stochastically positive KMS states and their quasi-free realizations. The authors construct stochastic processes on configuration spaces that are stationary with respect to finite-temperature DPPs, deriving space-time determinantal correlations and analyzing zero-temperature limits via strong resolvent convergence of the underlying self-adjoint operators. The framework unifies abstract operator-algebraic constructions with concrete examples from orthogonal polynomial ensembles and hypergeometric difference operators, and it demonstrates convergence and Markov properties across finite-temperature and zero-temperature regimes. These results provide an algebraic, representation-theoretic lens for studying interacting particle systems and their temporal evolution, with potential extensions to Pfaffian processes and continuum settings.
Abstract
The aim of this paper is threefold. Firstly, we develop the author's previous work on the dynamical relationship between determinantal point processes and CAR algebras. Secondly, we present a novel application of the theory of stochastic processes associated with KMS states for CAR algebras and their quasi-free states. Lastly, we propose a unified theory of algebraic constructions and analysis of stationary processes on point configuration spaces with respect to determinantal point processes. As a byproduct, we establish an algebraic derivation of a determinantal formula for space-time correlations of stochastic processes, and we analyze several limiting behaviors of these processes.