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Abstract We show that the distribution of bulk spacings between pairs of adjacent eigenvalue real parts of a random matrix drawn from the complex elliptic Ginibre ensemble is asymptotically given by a generalization of the Gaudin-Mehta distribution, in the limit of weak non-Hermiticity. The same generalization is expressed in terms of an integro-differential Painlevé function and it is shown that the generalized Gaudin-Mehta distribution describes the crossover, with increasing degree of non-Hermiticity, from Gaudin-Mehta nearest-neighbor bulk statistics in the Gaussian Unitary Ensemble to Poisson gap statistics for eigenvalue real parts in the bulk of the Complex Ginibre Ensemble.
Bothner et al. (Fri,) studied this question.