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Asymptotic freeness through unitaries generated by polynomials of Wigner matrices

Félix Parraud, Kevin Schnelli

Abstract

We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on minimizing the assumption of smoothness on those functions while optimizing the error term with respect to $N$, the size of the matrices. As an application, we build on the idea that the long-time Heisenberg evolution associated to Wigner matrices generates asymptotic freeness as first shown in $[9]$. More precisely given $P$ a self-adjoint non-commutative polynomial and $Y^N$ a $d$-tuple of independent Wigner matrices, we prove that the quantum evolution associated to the operator $P(Y^N)$ yields asymptotic freeness for large times.

Asymptotic freeness through unitaries generated by polynomials of Wigner matrices

Abstract

We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on minimizing the assumption of smoothness on those functions while optimizing the error term with respect to , the size of the matrices. As an application, we build on the idea that the long-time Heisenberg evolution associated to Wigner matrices generates asymptotic freeness as first shown in . More precisely given a self-adjoint non-commutative polynomial and a -tuple of independent Wigner matrices, we prove that the quantum evolution associated to the operator yields asymptotic freeness for large times.
Paper Structure (10 sections, 12 theorems, 171 equations)

This paper contains 10 sections, 12 theorems, 171 equations.

Table of Contents

  1. Introduction and main results
  2. Framework and standard properties
  3. Definitions from Random Matrix Theory
  4. Definitions from Free Probability Theory
  5. Noncommutative polynomials and derivatives
  6. The Schwinger-Dyson equations and their generalization
  7. Reduction of the problem to the case of GUE matrices
  8. Concentration of the trace
  9. Concentration of the scalar product
  10. Proof of the main theorems

Key Result

Theorem 1.1

Let the following objects be given, Then with the convention $\left\Vert f_i\right\Vert_4 =1$ if $f_i = \mathop{\mathrm{id}}\nolimits_{\mathbb{R}}$, and otherwise, where $|\mu|$ is the variation of the measure $\mu$. Then we have the following result. For any $\varepsilon>0$, with high probability, where $\mathop{\mathrm{tr}}\nolimits_{N}$ is the normalized trace on $\mathbb{M}_N(\mathbb{C})$,

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 24 more