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

TL;DR

This paper demonstrates that for even L, beta=L^2 random matrix ensembles are exactly solvable via the KP hierarchy, allowing explicit computation of physical observables through a hyperpfaffian tau-function and emergent quantized momentum.

Contribution

It establishes the KP hierarchy framework for beta=L^2 ensembles with even L, revealing a hyperpfaffian tau-function structure and a dimensional reduction mechanism.

Findings

Partition function is a hyperpfaffian tau-function.

Correlation functions are generated by differential operators.

Emergent quantized momentum enforces momentum conservation.

Abstract

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 $L\neq 1,2$. 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 ${LM\choose L}$ to $O(L^{2}M)$, enabling explicit computation of physical observables.