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A Palm hierarchy for determinantal point processes with the confluent hypergeometric kernel, the decomposing measures in the problem of harmonic analysis on the infinite-dimensional unitary group

Alexander I. Bufetov

TL;DR

The paper proves a precise recursion for Hua--Pickrell decomposing measures by showing that the parameter shift $s\to s+1$ corresponds to taking the reduced Palm measure at infinity of the associated determinantal point process. Central to the argument is the confluent hypergeometric kernel $K^{(s)}$ that represents the $s$-family, together with a demonstration that the Palm operation commutes with the scaling limit from finite orthogonal polynomial ensembles to the infinite-particle determinantal process. The authors develop a general framework for integrable projection kernels, including delicate handling when $\Pi(p,p)=0$, and they establish the Palm-continuity and convergence needed to pass to the limit. This yields not only the main recursion but also a robust methodology for deriving similar $s\to s+1$ relations in related infinite-dimensional harmonic analysis problems and their finite-dimensional approximations. The results connect harmonic analysis on $U(\infty)$, determinantal processes, and orthogonal polynomial ensembles, providing a unified view of how parameter shifts correspond to conditioning operations at infinity.

Abstract

The main result of this note is that the shift of the parameter by 1 in the parameter space of decomposing measures in the problem of harmonic analysis on the infinite-dimensional unitary group corresponds to the taking of the reduced Palm measure at infinity for our decomposing measures. The proof proceeds by finite-dimensional approximation of our measures by orthogonal polynomial ensembles. The key remark is that the taking the reduced Palm measure commutes with the scaling limit transition from finite to infinite particle systems.

A Palm hierarchy for determinantal point processes with the confluent hypergeometric kernel, the decomposing measures in the problem of harmonic analysis on the infinite-dimensional unitary group

TL;DR

The paper proves a precise recursion for Hua--Pickrell decomposing measures by showing that the parameter shift corresponds to taking the reduced Palm measure at infinity of the associated determinantal point process. Central to the argument is the confluent hypergeometric kernel that represents the -family, together with a demonstration that the Palm operation commutes with the scaling limit from finite orthogonal polynomial ensembles to the infinite-particle determinantal process. The authors develop a general framework for integrable projection kernels, including delicate handling when , and they establish the Palm-continuity and convergence needed to pass to the limit. This yields not only the main recursion but also a robust methodology for deriving similar relations in related infinite-dimensional harmonic analysis problems and their finite-dimensional approximations. The results connect harmonic analysis on , determinantal processes, and orthogonal polynomial ensembles, providing a unified view of how parameter shifts correspond to conditioning operations at infinity.

Abstract

The main result of this note is that the shift of the parameter by 1 in the parameter space of decomposing measures in the problem of harmonic analysis on the infinite-dimensional unitary group corresponds to the taking of the reduced Palm measure at infinity for our decomposing measures. The proof proceeds by finite-dimensional approximation of our measures by orthogonal polynomial ensembles. The key remark is that the taking the reduced Palm measure commutes with the scaling limit transition from finite to infinite particle systems.
Paper Structure (13 sections, 14 theorems, 94 equations)

This paper contains 13 sections, 14 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Hua---Pickrell measures
  4. Spaces of configurations
  5. Weak topology on space of probability measures on the space of configurations
  6. Locally trace-class operators
  7. Determinantal point processes
  8. Palm measures of determinantal point processes
  9. Admissible weights
  10. Change of variables in kernel of determinantal processes
  11. Hua---Pickrell measures and determinantal point processes
  12. The proof of Theorem \ref{['thm:main']}
  13. Concluding remarks: convergence of circular orthogonal polynomial ensembles

Key Result

Proposition 1

A Borel probability measure $\mathbb{P}\in \mathfrak{M}_{\mathrm{fin}}({\mathrm {Conf}}(E))$ is defined by joint distributions $(\#_{\varphi_1}, \dots, \#_{\varphi_l})$ of all finite collections of continuous functions $\varphi_1, \dots, \varphi_l\colon E\to\mathbb{R}$ with pairwise disjoint support

Theorems & Definitions (25)

  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 1
  • Remark
  • Proposition 4
  • Proposition 5
  • Remark
  • ...and 15 more