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

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.