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

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 $\{$$k$-checkerboard, $j$-checkerboard$\}$ exhibit entirely different spectral behavior than the other anticommutator ensembles: while the spectrum of $\{$GOE, $k$-checkerboard$\}$ consists of 1 bulk regime of size $\Theta(N)$ and 1 blip regime of size $\Theta(N^{3/2})$, the spectrum of $\{$$k$-checkerboard, $j$-checkerboard$\}$ consists of 1 bulk regime of size $\Theta(N)$, 2 intermediary blip regimes of size $\Theta(N^{3/2})$, and 1 largest blip regime of size $\Theta(N^2)$. 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$\}$.