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Strong convergence of tensor products of independent G.U.E. matrices

Serban Belinschi, Mireille Capitaine

Abstract

Given tuples of properly normalized independent $N\times N$ G.U.E. matrices $(X_N^{(1)},\dots,X_N^{(r_1)})$ and $(Y_N^{(1)},\dots,Y_N^{(r_2)})$, we show that the tuple $(X_N^{(1)}\otimes I_N,\dots,X_N^{(r_1)}\otimes I_N,I_N\otimes Y_N^{(1)},\dots,I_N\otimes Y_N^{(r_2)})$ of $N^2\times N^2$ random matrices converges strongly as $N$ tends to infinity. It was shown by Ben Hayes that this result implies that the Peterson-Thom conjecture is true.

Strong convergence of tensor products of independent G.U.E. matrices

Abstract

Given tuples of properly normalized independent G.U.E. matrices and , we show that the tuple of random matrices converges strongly as tends to infinity. It was shown by Ben Hayes that this result implies that the Peterson-Thom conjecture is true.
Paper Structure (14 sections, 10 theorems, 209 equations)

This paper contains 14 sections, 10 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. From G.U.E. to unitary matrices
  3. Haagerup, Thorbjørnsen and Schultz's approach
  4. Noncommutative analysis
  5. Noncommutative polynomials
  6. Noncommutative functions
  7. Relations between the difference-differential operator and the free difference quotient
  8. Domains for $(z-\mathcal{S})^{-1}$
  9. Differentiation and evaluation
  10. Asymptotic expansion of polynomials in Cayley transforms of GUE's
  11. A fundamental result of Parraud
  12. Parraud's formulae for polynomials in Cayley transforms
  13. Parraud's formulae on rational functions
  14. Expansion of the Cauchy transform

Key Result

Theorem 1

Almost surely, for any noncommutative polynomial $P$ in $r_1+r_2$ variables,

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Remark 5
  • Remark 6
  • Example 7
  • Proposition 8
  • Proposition 9
  • Corollary 10
  • ...and 6 more