Spacing distribution in the 2D Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at non-integer $β$

Abstract

A random matrix representation is proposed for the two-dimensional (2D) Coulomb gas at inverse temperature β. For 2 × 2 matrices with Gaussian distribution this yields a surmise for the nearest neighbour spacing distribution of complex eigenvalues in radial distance. It reproduces the 2D Poisson distribution at β = 0 and approximates the complex Ginibre ensemble at β = 2. The surmise is used to fit data from open quantum spin chains and ecology. The spacing distributions of complex symmetric and complex quaternion self-dual ensembles are fitted by non-integer values β = 1.4 and β = 2.6, respectively. They have been suggested as the only two symmetry classes with 2D bulk statistics different from the Ginibre ensemble.

Territorial behaviour of buzzards versus random matrix spacing distributions

“…The spacing distributions we use in the main text are local quantities which a priori seem to be less sensitive to the geometry. In this appendix we illustrate the deviation between the numerically determined NN spacing distribution from [3] and the surmise [26] given in (2.4) together with (2.5), for small values of β. The deviation quantified in terms of the standard deviation and the Kolmogorov-Smirnov distance can be found in [26,Table 1.…”

“…In order to improve the approximation of the surmise (2.4) in 2D to larger values of β (and with exact results for N > 2 being unavailable), an effective β eff was introduced in [26], by fitting a third order polynomial in β to the numerically determined spacing of the 2D Coulomb gas in the range of β ∈ [0, 3]:…”

“…2 right for a comparison at β = 2, where the surmise clearly fails. In [26] a more detailed comparison is presented, including higher β-values and their Kolmogorov-Smirnov distances. Obviously, the fit using an effective β eff spoils the heuristically expected proportionality at very small argument ∼ s 1+β eff .…”

“…sur,β eff (s) from the surmise [26], and the numerically determined NN and NNN spacing from [3] as functions of β, in order to quantify the repulsion within one species, and between different species.…”

“…In 2D, it has been proposed [3] to directly use the 2D Coulomb gas at intermediate values of β, and to determine the NN and NNN spacing distributions numerically, that can then be compared with data. These findings are reviewed in Section 2.2, in particular a corresponding approximate surmise for the 2D NN spacing distribution valid for small values of β [26], cf. Appendix B.…”