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The Limiting Spectral Distribution of Various Matrix Ensembles Under the Anticommutator Operation

Glenn Bruda, Bruce Fang, Raul Marquez, Steven J. Miller, Beni Prapashtica, Vismay Sharan, Daeyoung Son, Saad Waheed, Janine Wang

Abstract

Inspired by the quantization of classical quantities and Rankin Selberg convolution, we study the anticommutator operation $\{\cdot, \cdot\}$, where $\{A,B\} = AB + BA$, applied to real symmetric random matrix ensembles including Gaussian orthogonal ensemble (GOE), the palindromic Toeplitz ensemble (PTE), the $k$-checkerboard ensemble, and the block $k$-circulant ensemble ($k$-BCE). Using combinatorial and topological techniques related to non-crossing and free matching properties of GOE and PTE, we obtain closed-form formulae for the moments of the limiting spectral distributions of $\{$GOE, GOE$\}$, $\{$PTE, PTE$\}$, $\{$GOE, PTE$\}$ and establish the corresponding limiting spectral distributions with generating functions and convolution. On the other hand, $\{$GOE, $k$-checkerboard$\}$ and $\{$$k$-checkerboard, $j$-checkerboard$\}$ exhibit entirely different spectral behavior than the other anticommutator ensembles: while the spectrum of $\{$GOE, $k$-checkerboard$\}$ consists of 1 bulk regime of size $Θ(N)$ and 1 blip regime of size $Θ(N^{3/2})$, the spectrum of $\{$$k$-checkerboard, $j$-checkerboard$\}$ consists of 1 bulk regime of size $Θ(N)$, 2 intermediary blip regimes of size $Θ(N^{3/2})$, and 1 largest blip regime of size $Θ(N^2)$. In both cases, with the appropriate weight function, we are able to isolate the largest regime for other regime(s) and analyze its moments and convergence results via combinatorics. We end with numerical computation of lower even moments of $\{$GOE, $k$-BCE$\}$ and $\{$$k$-BCE, $k$-BCE$\}$ based on genus expansion and discussion on the challenge with analyzing the intermediary blip regimes of $\{$$k$-checkerboard, $j$-checkerboard$\}$.

The Limiting Spectral Distribution of Various Matrix Ensembles Under the Anticommutator Operation

Abstract

Inspired by the quantization of classical quantities and Rankin Selberg convolution, we study the anticommutator operation , where , applied to real symmetric random matrix ensembles including Gaussian orthogonal ensemble (GOE), the palindromic Toeplitz ensemble (PTE), the -checkerboard ensemble, and the block -circulant ensemble (-BCE). Using combinatorial and topological techniques related to non-crossing and free matching properties of GOE and PTE, we obtain closed-form formulae for the moments of the limiting spectral distributions of GOE, GOE, PTE, PTE, GOE, PTE and establish the corresponding limiting spectral distributions with generating functions and convolution. On the other hand, GOE, -checkerboard and -checkerboard, -checkerboard exhibit entirely different spectral behavior than the other anticommutator ensembles: while the spectrum of GOE, -checkerboard consists of 1 bulk regime of size and 1 blip regime of size , the spectrum of -checkerboard, -checkerboard consists of 1 bulk regime of size , 2 intermediary blip regimes of size , and 1 largest blip regime of size . In both cases, with the appropriate weight function, we are able to isolate the largest regime for other regime(s) and analyze its moments and convergence results via combinatorics. We end with numerical computation of lower even moments of GOE, -BCE and -BCE, -BCE based on genus expansion and discussion on the challenge with analyzing the intermediary blip regimes of -checkerboard, -checkerboard.

Paper Structure

This paper contains 20 sections, 47 theorems, 158 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Results
  5. Limiting Expected Moments of Some Anticommutator Ensembles
  6. Limiting Expected Moments of the Anticommutator of GOE and GOE
  7. Limiting Expected Moments of the Anticommutator of PTE and PTE
  8. Limiting Expected Moments of the Anticommutator of GOE and PTE
  9. Limiting Expected Moments of the Anticommutator of GOE and BCE and of the Anticommutator of BCE and BCE
  10. Limiting Expected Moments of Blips of Checkerboard Ensemble
  11. Preliminaries
  12. Limiting Expected Moments of Blips of the Anticommutator of Checkerboard and Checkerboard
  13. Moments of anticommutators
  14. Moments of Anticommutators
  15. Bulk Moments of the Anticommutator of GOE and Checkerboard
  16. ...and 5 more sections

Key Result

Proposition 1.8

Let $A_N$ be an $N\times N$ real symmetric matrix, then where $\textup{Tr}(\cdot)$ denotes the trace of a matrix and $a_{ij}$'s are the entries of $A_N$ indexed by $ij$. Similarly, if $A_N$ is a random matrix drawn from an $N\times N$ real symmetric matrix ensemble, then where by $\mathbb{E}\left[\textup{Tr}(A^m_N)\right]$ we mean averaging over the $N\times N$ random matrix ensemble with each m

Figures (12)

  • Figure 1: Comparison between the normalized empirical spectral density for one hundred 1000$\times$1000 matrices from $\{\textup{PTE, PTE}\}$ and the theoretical spectral density.
  • Figure 2: Comparison between normalized empirical spectral density for one hundred 1000$\times$1000 matrices from $\{\textup{GOE, GOE}\}$ and the theoretical spectral density.
  • Figure 3: Normalized empirical spectral density for one hundred 1000$\times$1000 matrices from $\{\textup{GOE, PTE}\}$.
  • Figure 4: Normalized empirical spectral density for one hundred 1500$\times$1500 matrices from $\{\textup{GOE, $2$-BCE}\}$.
  • Figure 5: Normalized empirical spectral density for one hundred 1500$\times$1500 matrices from $\{\textup{$2$-BCE, $2$-BCE}\}$.
  • ...and 7 more figures

Theorems & Definitions (112)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Definition 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 102 more