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

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.

Statistical Mechanics (cond-mat.stat-mech); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Quantum Physics (quant-ph)

arXiv:2609.00164 [cond-mat.stat-mech]

(or arXiv:2609.00164v1 [cond-mat.stat-mech] for this version)

https://doi.org/10.48550/arXiv.2609.00164 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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