Exact Solvability via the KP Hierarchy for $\beta =L^2$ Random Matrix Ensembles

Random matrix ensembles with Dyson index $\beta =L^2$ describe systems of $M$ charge-$L$ particles interacting logarithmically in the presence of an external potential, yet exact formulas for their physical observables have remained elusive for $L\mathrm{\ne }1,2$. We show that, for $L$ even, $\beta =L^2$ ensembles are governed by the KP hierarchy at finite particle number–paralleling the KP solvability of classical $\beta =1,2,4$ ensembles. The partition function is a hyperpfaffian $\tau$-function satisfying the Hirota bilinear identity, and correlation functions are generated by finite-order differential operators acting on this $\tau$-function. The key mechanism is an emergent quantized momentum that stratifies the system into discrete sectors, enforcing momentum conservation as a selection rule. This produces a dramatic dimensional reduction from $\left(\genfrac{}{}{0px}{}{LM}{L}\right)$ to $O(L^{2}M)$, enabling explicit computation of physical observables.