Spacing distribution in the two-dimensional Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at nonintegerβ

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 we analytically compute the nearest-neighbor spacing distribution of complex eigenvalues in radial distance. Because it does not provide such a good approximation as the Wigner surmise in 1D, we introduce an effective β eff (β ) in our analytic formula that describes the spacing obtained numerically from the 2D Coulomb gas well for small values of β. It reproduces the 2D Poisson distribution at β = 0 exactly, that is valid for a large particle number. The surmise is used to fit data in two examples, from open quantum spin chains and ecology. The spacing distributions of complex symmetric and complex quaternion self-dual ensembles of non-Hermitian random matrices, that are only known numerically, are very well fitted by noninteger values β = 1.4 and β = 2.6 from a 2D Coulomb gas, respectively. These two ensembles have been suggested as the only two symmetry classes, where the 2D bulk statistics is different from the Ginibre ensemble.

Interactions between different birds of prey as a random point process

“…In 2D, it has been proposed [4] to directly use the 2D Coulomb gas at intermediate values of β, and to determine the NN and NNN spacing distributions numerically, which can then be compared with data. This is reviewed in section 2.2, in particular a corresponding approximate surmise for the 2D NN spacing distribution valid for small values of β [27]; appendix B. These NN and NNN spacings were then used for fits as a function of β, for the spacing of eigenvalues of the Liouville operator of dissipative, boundary-driven systems in 2D [4], and in the yearly spacing distributions of nests of the common buzzard [10].…”

“…For the local correlations among points at a distance of order 1/ √ N , very little is known analytically for fixed β > 0, apart from the Poisson statistics at β = 0 (that extends to β ∼ 1/N ; see [39]) and the integrable case β = 2, when the point process (2.3) becomes determinantal; see [28]. Inspired by the Wigner surmise for the 1D Dyson gas, and its generalisation to general β based on a 2 × 2 β-ensemble [40], a surmise (sur) for the NN spacing distribution was derived from complex normal 2 × 2 random matrices in [27] 5…”

“…This characteristic is opposite to the Wigner surmise in 1D, which becomes more accurate for increasing values of β, rather than for β → 0. 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 [27], 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]:…”

“…See appendix B for plots in this range, and figure 2 right for a comparison at β = 2, where the surmise clearly fails. In [27] 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 a very small argument ∼ s 1+β eff .…”

“…sur,β eff (s) from the surmise [27], and the numerically determined NN and NNN spacing from [4] as functions of β, in steps of 0.1, and we refer to [4] for details of the numerics. In particular, around 200 000 spacings have been used there to generate these numerical distributions.…”