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Beta Ensembles in the Freezing Regime and Finite Free Convolutions

Fumihiko Nakano, Khanh Duy Trinh, Ziteng Wang

Abstract

In the freezing regime where the system size N is fixed and the inverse temperature beta tends to infinity, the eigenvalues of Gaussian beta ensembles converge to zeros of the Nth Hermite polynomial. That law of large numbers has been proved by analyzing the joint density or reading off the random matrix model. This paper studies its dynamical version of this phenomenon. We show that in the freezing regime the eigenvalue processes called beta Dyson Brownian motions converge to deterministic limiting processes which can be written as the finite free convolution of the initial data and the zeros of Hermite polynomials. This result is a counterpart of those in the random matrix regime where N tends to infinity with fixed beta, as well as to the high temperature regime where N tends to infinity while beta N remains bounded. We also establish Gaussian fluctuations around the limit and deal with the Laguerre case.

Beta Ensembles in the Freezing Regime and Finite Free Convolutions

Abstract

In the freezing regime where the system size N is fixed and the inverse temperature beta tends to infinity, the eigenvalues of Gaussian beta ensembles converge to zeros of the Nth Hermite polynomial. That law of large numbers has been proved by analyzing the joint density or reading off the random matrix model. This paper studies its dynamical version of this phenomenon. We show that in the freezing regime the eigenvalue processes called beta Dyson Brownian motions converge to deterministic limiting processes which can be written as the finite free convolution of the initial data and the zeros of Hermite polynomials. This result is a counterpart of those in the random matrix regime where N tends to infinity with fixed beta, as well as to the high temperature regime where N tends to infinity while beta N remains bounded. We also establish Gaussian fluctuations around the limit and deal with the Laguerre case.
Paper Structure (12 sections, 23 theorems, 224 equations)

This paper contains 12 sections, 23 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Finite free convolution
  3. Beta Dyson's Brownian motions in the freezing regime
  4. Limit theorems for the empirical measure processes
  5. Law of large numbers
  6. Central limit theorems
  7. Beta Laguerre ensembles and beta Laguerre processes
  8. Law of large numbers for beta Laguerre ensembles
  9. Law of large numbers for beta Laguerre processes
  10. Central limit theorems for the empirical distributions
  11. Orthogonal polynomials and their duals
  12. Finite free convolutions and the Fourier transform

Key Result

Theorem 1.1

Let $\lambda_1^{(\beta)}(t) \le \cdots \le \lambda_N^{(\beta)}(t)$ be the unique strong solution to the system of SDEs eq:DBM. (Here we have added the superscript ${}^{(\beta)}$ to indicate the dependence on $\beta$.) Then as $\beta \to \infty$, uniformly for $t \in [0, T]$, where $\boxplus_N$ denotes the finite free convolution (see Definition defn:FFC).

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 36 more