Exact joint eigenvalue densities of non-Hermitian random matrices are Calogero scattering states

• URL: http://arxiv.org/abs/2609.00164v1
• Date: Mon, 31 Aug 2026 18:00:43 GMT
• Title: Exact joint eigenvalue densities of non-Hermitian random matrices are Calogero scattering states
• Abstract summary: Solving exact joint eigenvalue densities is central to random matrix theory.<n>Up to a Vandermonde factor, they are scattering-state wave functions of the Calogero model.<n>In contrast to many previously known joint densities, the densities cannot be written as a gas of eigenvalues with pairwise interactions.
• Score: 19.53154279337076
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Determining exact joint eigenvalue densities is central to random matrix theory. We solve this long-standing problem for non-Hermitian matrices with transposition symmetry (complex symmetric and complex self-dual) at arbitrary matrix size. Up to a Vandermonde factor, they are scattering-state wave functions of the Calogero model, a line of particles interacting through an inverse-square potential, with the coupling strength set by the symmetry. In contrast to many previously known joint densities, the densities cannot be written as a gas of eigenvalues with pairwise interactions. We further compute the complex level spacing distributions and two-point spectral correlation functions, which carry power-law tails, absent in a Coulomb gas. Our results shed light on the interplay among random matrices, integrability, and symmetry.

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