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Convergence rate of the diagonal-valued Cauchy-transform for permutation invariant random matrices

Alexis Imbert

Abstract

Let $A$ be a permutation invariant random matrix and $B$ another random matrix. We give a quantitative bound on the difference between the diagonal of the resolvent of $A+B$ and the diagonal of the resolvent of the free sum with amalgamation over the diagonal of $A$ and $B$. Moreover, we improve the rate of convergence whenever the matrices $A$ and $B$ are sparse and bounded in operator norm. Doing so, we explicitly construct the free sum over the diagonal of $A$ and $B$ as an adjacency operator of a weighted locally finite graph.

Convergence rate of the diagonal-valued Cauchy-transform for permutation invariant random matrices

Abstract

Let be a permutation invariant random matrix and another random matrix. We give a quantitative bound on the difference between the diagonal of the resolvent of and the diagonal of the resolvent of the free sum with amalgamation over the diagonal of and . Moreover, we improve the rate of convergence whenever the matrices and are sparse and bounded in operator norm. Doing so, we explicitly construct the free sum over the diagonal of and as an adjacency operator of a weighted locally finite graph.
Paper Structure (20 sections, 13 theorems, 97 equations, 2 figures)

This paper contains 20 sections, 13 theorems, 97 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Statement of the main Theorem
  4. Notions of diagonal-valued free probability
  5. Main theorems and sketch of the proof
  6. Explicit construction of the amalgamated free sum
  7. Traffic notions and assumptions
  8. Traffic notions
  9. Assumption on the injective trace
  10. Diluted matrices with bounded entries.
  11. Erdos-Renyi matrices.
  12. Diluted Wigner matrices with moment conditions.
  13. Entry wise product.
  14. Assumption on the Frobenius norm
  15. Proof of the main theorems
  16. ...and 5 more sections

Key Result

Theorem 2.3

Let $\mu$ be a positive unital $\mathcal{B}$-bimodular map that satisfies the first point above. It satisfies the second point if and only if there exists a $\mathcal{B}$-valued non-commutative probability space $(\mathcal{A},\mathcal{B}, E)$ and a self-adjoint element $a\in\mathcal{A}$ such that $\

Figures (2)

  • Figure 1: Construction of $A\boxplus_{\Delta}B$ as an iterating process. The blue (resp. red) edges are labeled $a$ (resp. $b$).
  • Figure 2: Examples of the modification of the $\mathcal{GCC}$ when adding an edge. The top picture is a detail of the $\mathcal{GCC}$ with a blue dashed line representing the edge we add at step $l$ in the graph. The bottom picture is the local change of the $\mathcal{GCC}$. Colored squares are vertex from colored components ($a$ is blue, $b$ is red) and gray circles are connectors.

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Proposition 2.2 of popa_non-commutative_2013
  • Remark 2.1
  • Definition 2.4
  • Theorem 2.5: Theorem 1.2 of au_large_2021
  • Theorem 2.6
  • Theorem 2.7
  • Remark 3.1
  • Proposition 3.1
  • ...and 28 more