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Operator-valued Khintchine inequality for $ε$-free semicircles

Benoît Collins, Akihiro Miyagawa

TL;DR

This paper addresses bounding the operator norm of the sum $\sum_{i=1}^d a_i \otimes s_i$ where $s_i$ are $\epsilon$-free semicircles, connecting the bounds to the adjacency spectrum of $\epsilon$. A concrete Fock-space representation on $\mathcal F_{\epsilon}$ with $s_i = l_i + l_i^*$ and the relation $l_i^*l_j - \epsilon_{ij} l_j l_i^* = \delta_{ij} I$ is used to derive sharp bounds. The main contributions are a Khintchine-type upper bound with constant $2\sqrt{\lambda_1+1}$, a refined bound involving $\lambda_2$ for connected regular graphs, and a lower bound in terms of the clique number $\omega(\epsilon)$; the results extend Haagerup–Pisier’s operator-valued Khintchine inequality to the $\epsilon$-free setting. Additionally, the paper discusses extensions to $\epsilon$-free Haar unitaries and situates the work within graph-dependent random-matrix models, highlighting new optimality regimes for multipartite and XY-model graphs.

Abstract

We exhibit several bounds for operator norms of the sum of $ε$-free semicircular random variables introduced in the paper of Speicher and Wysoczański. In particular, using the first and second largest eigenvalues of the adjacency matrix $ε$, we show analogs of the operator-valued Khintchine-type inequality obtained by Haagerup and Pisier.

Operator-valued Khintchine inequality for $ε$-free semicircles

TL;DR

This paper addresses bounding the operator norm of the sum where are -free semicircles, connecting the bounds to the adjacency spectrum of . A concrete Fock-space representation on with and the relation is used to derive sharp bounds. The main contributions are a Khintchine-type upper bound with constant , a refined bound involving for connected regular graphs, and a lower bound in terms of the clique number ; the results extend Haagerup–Pisier’s operator-valued Khintchine inequality to the -free setting. Additionally, the paper discusses extensions to -free Haar unitaries and situates the work within graph-dependent random-matrix models, highlighting new optimality regimes for multipartite and XY-model graphs.

Abstract

We exhibit several bounds for operator norms of the sum of -free semicircular random variables introduced in the paper of Speicher and Wysoczański. In particular, using the first and second largest eigenvalues of the adjacency matrix , we show analogs of the operator-valued Khintchine-type inequality obtained by Haagerup and Pisier.

Paper Structure

This paper contains 5 sections, 7 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\epsilon$-independence
  4. Representation of $\epsilon$-free semicircles
  5. Main results

Key Result

Theorem 1.1

Let $s_1,\dots,s_d$ be a standard $\epsilon$-free semicircle. Then, for any $a_1,\ldots,a_d \in B(H)$, we have where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $\epsilon$. Moreover, when $\epsilon$ is a connected regular graph with a degree less than $d-1$, we also have where $\lambda_2$ is the second largest eigenvalue of the adjacency matrix $\epsilon$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 5 more