Conservation operator processes from asymptotic representation theory and their CLT
Ryosuke Sato
TL;DR
The paper develops a general central limit theory for conservation operator processes on symmetric Fock spaces and applies it to the asymptotic representation theory of both classical and quantum unitary groups. By linking center elements of $U(\mathfrak{gl}_N)$ and its quantum analogue to operator-valued processes, it derives LLNs and CLTs as ranks grow, with Gaussian limits whose covariances are governed by Jacobi–Trudi–type symmetric functions evaluated at asymptotic parameters (e.g., $\omega$, $\omega_{\nu}$). The work unifies representations of $U(\infty)$, the symmetric group, and $U_q(\mathfrak{gl}_N)$ under a common quantum stochastic calculus framework, and it highlights independent increments and Wick-type moment calculus as central tools. These results offer a rigorous algebraic-probabilistic pathway to understand large-rank asymptotics in both classical and quantum settings, including connections to Plancherel-type characters and their stochastic realizations.
Abstract
In this paper, we examine applications of the theory of operator-valued processes to algebraic methods in probability theory. We show a central limit theorem for general conservation operator processes. Utilizing this, we analyze the asymptotic behavior of processes derived from unitary groups and quantum unitary groups as their ranks tend to infinity, thereby providing applications of asymptotic representation theory.