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.

💡 Research Summary

The paper investigates three universal edge statistics that arise in non‑Hermitian random matrix theory (RMT), focusing on the three simplest symmetry classes: class A (complex Ginibre ensemble), class AI† (complex symmetric matrices) and class AII† (complex self‑dual matrices). These classes represent Gaussian ensembles and can be interpreted as two‑dimensional Coulomb gases with inverse temperatures β = 2, β ≈ 1 and β ≈ 2.6 respectively.

In the first part the authors analytically study the complex spacing ratio ρ(N)(z), defined as the ratio of the distance to the nearest neighbour (NN) and the next‑to‑nearest neighbour (NNN) eigenvalues. By conditioning one eigenvalue at the origin (a typical bulk point) they derive a conditional point‑process representation that eliminates the troublesome 1/N terms present in earlier uncontrolled approximations. Using Andréief’s integration formula they show that, for any rotationally invariant weight ω(|z|), the conditional spacing‑ratio distribution can be expressed as an average over a five‑diagonal matrix, greatly simplifying numerical evaluation.

They then extend the analysis to the elliptic Ginibre–Girkó ensemble (eGinUE), which interpolates between the Ginibre ensemble (τ = 0) and the Gaussian unitary ensemble (GUE) as τ → 1. An N = 3 “surmise” for the spacing ratio is derived that smoothly connects the Ginibre result to the exact GUE result, demonstrating that the same β = 2 Coulomb‑gas description holds at the spectral edge.

The second part presents extensive numerical simulations for all three classes, both in the bulk and at the spectral edge. For classes AI† and AII†, where no determinantal or Pfaffian formulas are known, the authors compare the spacing‑ratio statistics and its moments with those of a 2‑D Poisson process and with the predictions of a 2‑D Coulomb gas at effective β values. They find that the degree of eigenvalue repulsion at the edge can be understood by varying β, confirming that AI† behaves like a β ≈ 1 gas and AII† like a β ≈ 2.6 gas.

A careful unfolding procedure is introduced to account for the varying mean density near the edge. After unfolding, the nearest‑neighbour (NN) and next‑to‑nearest‑neighbour (NNN) spacing distributions are measured. In all three symmetry classes, the small‑s behavior of the NN spacing follows the universal cubic repulsion P(s) ∼ s³, both in the bulk and at the edge, a result also derived analytically via a small‑argument expansion of the Fredholm determinant at the edge (Appendix A).

The authors observe that the complex spacing ratio alone does not fully capture the local edge statistics; additional information from NN and NNN spacing distributions is required for a complete description. Nevertheless, the combined analysis provides a coherent picture: the edge statistics of non‑Hermitian Gaussian ensembles fall into three universal classes, each corresponding to a distinct effective inverse temperature of a 2‑D Coulomb gas. This work deepens the understanding of non‑Hermitian RMT, offers practical tools for analyzing spectral data in open quantum systems, and opens avenues for further exploration of edge universality in more general non‑Hermitian ensembles.