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Matrix harmonic analysis at high temperature via the Dirichlet process

Jiyuan Zhang

TL;DR

The work analyzes harmonic analysis of large random matrices in the high-temperature limit, showing that rank-one deformations of spherical-type functions converge to transforms of a Markov-Krein pair ρ^(c) linked to the limiting spectral measure ρ via the Dirichlet-process Markov-Krein correspondence. It provides contour-integral representations and convergence results for both the multivariate Bessel function and the Heckman-Opdam hypergeometric function, establishing that in the classical regime the limit is the Fourier transform of ρ, while in the high-temperature regime it is the Fourier transform (and Mellin transform) of ρ^(c). The Dirichlet process underpins the ρ↔ρ^(c) correspondence, enabling robust handling of unbounded tails and random empirical measures, with precise assumptions on convergence and tail behavior. Overall, the paper unifies random-matrix asymptotics with nonparametric Bayesian tools to characterize spectral-limit transforms across regimes, providing new insights and technical tools (contour formulas, convergence criteria) for MK-based limits.

Abstract

We investigate harmonic analysis of random matrices of large size with their Dyson indices going simultaneous to zero, that is in the high temperature limit. In this regime, we show that the multivariate Bessel function/Heckman-Opdam hypergeometric function of the empirical spectral distribution converges to the Fourier/Mellin transform of a measure, which and the limiting empirical distribution are intimately related by the Markov-Krein correspondence. The uniqueness, existence and other properties of the Markov-Krein correspondence can be studied using the theory of the Dirichlet process.

Matrix harmonic analysis at high temperature via the Dirichlet process

TL;DR

The work analyzes harmonic analysis of large random matrices in the high-temperature limit, showing that rank-one deformations of spherical-type functions converge to transforms of a Markov-Krein pair ρ^(c) linked to the limiting spectral measure ρ via the Dirichlet-process Markov-Krein correspondence. It provides contour-integral representations and convergence results for both the multivariate Bessel function and the Heckman-Opdam hypergeometric function, establishing that in the classical regime the limit is the Fourier transform of ρ, while in the high-temperature regime it is the Fourier transform (and Mellin transform) of ρ^(c). The Dirichlet process underpins the ρ↔ρ^(c) correspondence, enabling robust handling of unbounded tails and random empirical measures, with precise assumptions on convergence and tail behavior. Overall, the paper unifies random-matrix asymptotics with nonparametric Bayesian tools to characterize spectral-limit transforms across regimes, providing new insights and technical tools (contour formulas, convergence criteria) for MK-based limits.

Abstract

We investigate harmonic analysis of random matrices of large size with their Dyson indices going simultaneous to zero, that is in the high temperature limit. In this regime, we show that the multivariate Bessel function/Heckman-Opdam hypergeometric function of the empirical spectral distribution converges to the Fourier/Mellin transform of a measure, which and the limiting empirical distribution are intimately related by the Markov-Krein correspondence. The uniqueness, existence and other properties of the Markov-Krein correspondence can be studied using the theory of the Dirichlet process.

Paper Structure

This paper contains 27 sections, 30 theorems, 229 equations, 2 figures.

Table of Contents

  1. Introduction and Main results
  2. Overview
  3. Preliminaries and main results
  4. Probability measures and convergence assumptions
  5. Symmetric functions
  6. Main results
  7. Strategy of the proofs
  8. Organisation of the paper
  9. Dirichlet process and the Markov-Krein correspondence
  10. Dirichlet process and its random mean
  11. Convergence of measures
  12. Further properties of the Markov-Krein correspondence
  13. Fourier transform of TEXT
  14. Mellin transform of rho(c)
  15. Proof of the main theorems
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $\{\rho_N\}$ be a sequence of discrete probability measures given in discrete, and $\rho$ be its limiting measure under the convergence specified by Assumption a1. We also assume that $\rho\in\mathcal{V}$, and that for some $u\in\mathbb C\setminus\{0\}$ one has Then the following statements hold.

Figures (2)

  • Figure 1: A standard choice for the Hankel loop $\mathcal{C}$.
  • Figure 2: Deformation of the contour from $-i\mathcal{C}$ to $\mathbb R-iM$.

Theorems & Definitions (60)

  • Remark 1.1: Premise of Assumption \ref{['a1']}
  • Remark 1.2
  • Theorem 1.1: The Fourier transform
  • Theorem 1.2: The Fourier transform; the random case
  • Remark 1.3
  • Theorem 1.3: The Mellin transform
  • Remark 1.4: Technical conditions and generalisations
  • Theorem 2.1: LR04 Existence of the random mean
  • Remark 2.1
  • Theorem 2.2: LR04 Existence and Uniqueness of the Markov-Krein correspondence
  • ...and 50 more