arXiv subjects

Noah Aygün

Publications and source records attributed to Noah Aygün.

3 recordsLinked to original sources

arXiv2026

Three non-Hermitian random matrix universality classes of complex edge statistics: Spacing ratios and distributions

The conjectured three generic local bulk statistics amongst all non-Hermitian random matrix symmetry classes have recently been extended to three generic local edge statistics. We study analytically and numerically complex spacing ratios and nearest-neighbour (NN) spacing distributions that characterise such local statistics. We choose the three simplest representatives of these universality classes, given by the Gaussian ensembles of complex Ginibre, complex symmetric and complex self-dual matrices, denoted by class A, AI$^†$ and AII$^†$. In the first part, we analytically study the complex spacing ratio in class A, at finite matrix size $N$. Introducing a conditional point process, we simplify existing expressions and show why an uncontrolled approximation introduced earlier converges well in the large-$N$ limit in the bulk. When specifying to the elliptic Ginibre ensemble, we present a parameter-dependent $N=3$ surmise for the complex spacing ratio, interpolating to that of the Gaussian unitary ensemble (GUE), where such a surmise is very accurate. In the second numerical part, we compare complex spacing ratios, its moments, and NN spacing distributions for all three ensembles with that of uncorrelated points, the two-dimensional (2D) Poisson process, both in the bulk and at the edge. The varying degree of repulsion within these different edge universality classes can be well understood in terms of an effective 2D Coulomb gas description, at different values of inverse temperature $β$. We find indications that the complex spacing ratio does not fully unfold the local statistics at the edge. Finally we verify that for small argument, in all three symmetry classes the NN spacing distributions in the bulk and at the edge are consistent with the universal cubic repulsion.

math-ph↗

arXiv2025

Complex symmetric, self-dual, and Ginibre random matrices: Analytical results for three classes of bulk and edge statistics

Recently, a conjecture about the local bulk statistics of complex eigenvalues has been made based on numerics. It claims that there are only three universality classes, which have all been observed in open chaotic quantum systems. Motivated by these new insights, we compute and compare the expectation values of $k$ pairs of complex conjugate characteristic polynomials in three ensembles of Gaussian non-Hermitian random matrices representative for the three classes: the well-known complex Ginibre ensemble, complex symmetric and complex self-dual matrices. In the Cartan classification scheme of non-Hermitian random matrices they are labelled as class A, AI$^†$ and AII$^†$, respectively. Using the technique of Grassmann variables, we derive explicit expressions for a single pair of expected characteristic polynomials for finite as well as infinite matrix dimension. For the latter we consider the global limit as well as zoom into the edge and the bulk of the spectrum, providing new analytical results for classes AI$^†$ and AII$^†$. For general $k$, we derive the effective Lagrangians corresponding to the non-linear $σ$-models in the respective physical systems. Interestingly, they agree for all three ensembles, while the corresponding Goldstone manifolds, over which one has to perform the remaining integrations, are different and equal the three classical compact groups in the bulk. In particular, our analytical results show that these three ensembles have indeed different local bulk and edge spectral statistics, corroborating the conjecture further.

math-ph↗

arXiv2023

Generalised unitary group integrals of Ingham-Siegel and Fisher-Hartwig type

We generalise well-known integrals of Ingham-Siegel and Fisher-Hartwig type over the unitary group $U(N)$ with respect to Haar measure, for finite $N$ and including fixed external matrices. When depending only on the eigenvalues of the unitary matrix, they can be related to a Toeplitz determinant with jump singularities. After introducing fixed deterministic matrices as external sources, the integrals can no longer be solved using Andréiéf's integration formula. Resorting to the character expansion as put forward by Balantekin, we derive explicit determinantal formulae containing Kummer's confluent and Gauss' hypergeometric function. They depend only on the eigenvalues of the deterministic matrices and are analytic in the parameters of the jump singularities. Furthermore, unitary two-matrix integrals of the same type are proposed and solved in the same manner. When making part of the deterministic matrices random and integrating over them, we obtain similar formulae in terms of Pfaffian determinants. This is reminiscent to a unitary group integral found recently by Kanazawa and Kieburg.

math-ph↗