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 [Formula: see text], where [Formula: see text], applied to real symmetric random matrix ensembles including Gaussian orthogonal ensemble (GOE), the palindromic Toeplitz ensemble (PTE), the [Formula: see text]-checkerboard ensemble and the block [Formula: see text]-circulant ensemble ([Formula: see text]-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 [Formula: see text], [Formula: see text], [Formula: see text] and establish the corresponding limiting spectral distributions with generating functions and convolution. On the other hand, [Formula: see text] and [Formula: see text] exhibit entirely different spectral behavior than the other anticommutator ensembles: while the spectrum of [Formula: see text] consists of 1 bulk regime of size [Formula: see text] and 1 blip regime of size [Formula: see text], the spectrum of [Formula: see text] consists of 1 bulk regime of size [Formula: see text], 2 intermediary blip regimes of size [Formula: see text] and 1 largest blip regime of size [Formula: see text]. In both cases, with the appropriate weight function, we can 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 [Formula: see text] and [Formula: see text] based on genus expansion and discussion on the challenge with analyzing the intermediary blip regimes of [Formula: see text].