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Random matrices associated to Young diagrams

Fabio Deelan Cunden, Marilena Ligabò, Tommaso Monni

Abstract

We consider the singular values of certain Young diagram shaped random matrices. For block-shaped random matrices, the empirical distribution of the squares of the singular eigenvalues converges almost surely to a distribution whose moments are a generalisation of the Catalan numbers. The limiting distribution is the density of a product of rescaled independent Beta random variables and its Stieltjes-Cauchy transform has a hypergeometric representation. In special cases we recover the Marchenko-Pastur and Dykema-Haagerup measures of square and triangular random matrices, respectively. We find a further factorisation of the moments in terms of two complex-valued random variables that generalises the factorisation of the Marcenko-Pastur law as product of independent uniform and arcsine random variables.

Random matrices associated to Young diagrams

Abstract

We consider the singular values of certain Young diagram shaped random matrices. For block-shaped random matrices, the empirical distribution of the squares of the singular eigenvalues converges almost surely to a distribution whose moments are a generalisation of the Catalan numbers. The limiting distribution is the density of a product of rescaled independent Beta random variables and its Stieltjes-Cauchy transform has a hypergeometric representation. In special cases we recover the Marchenko-Pastur and Dykema-Haagerup measures of square and triangular random matrices, respectively. We find a further factorisation of the moments in terms of two complex-valued random variables that generalises the factorisation of the Marcenko-Pastur law as product of independent uniform and arcsine random variables.
Paper Structure (9 sections, 6 theorems, 61 equations, 1 figure)

This paper contains 9 sections, 6 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Partitions and Young diagrams
  3. $\lambda$-shaped random matrices
  4. Full matrices
  5. Triangular matrices
  6. Balanced shapes
  7. Block-shaped random matrices
  8. Proofs
  9. Outlook

Key Result

Theorem 1

Let $\lambda^{(N)}=N \raisebox{-1pt}{!}\begin{picture}(1,1)(0,0) \put(0,0){\line(1,0){0.33}} \put(0.33,0){\line(0,1){1}} \put(0.,0.33){\line(1,0){0.66}} \put(0.66,0.33){\line(0,1){0.66}} \put(0,0.66){\line(1,0){1}} \put(1,0.66){\line(0,1){0.33}} \put(0,0){\line(0,1){1}} \put(0,1){\line(1,0){1}} \end

Figures (1)

  • Figure 1: Plot of the densities $F'_{\langle r\rangle}(x)$ for several values of $r$.

Theorems & Definitions (18)

  • Example 1
  • Theorem 1
  • Definition 1
  • Proposition 1: Gu, Prodinger and Wagner Gu10
  • Remark 1
  • Proposition 2
  • Remark 2
  • Corollary 1
  • Remark 3
  • proof : Proof of Theorem \ref{['thm:main']}
  • ...and 8 more