The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble

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Abstract
We investigate real eigenvalues of real elliptic Ginibre matrices of size n , indexed by the parameter of asymmetry τ ∈ [ 0 , 1 ] . In both the strongly and weakly non-Hermitian regimes, where τ ∈ [ 0 , 1 ) is fixed or 1 − τ = O ( 1 / n ) , respectively, we derive the asymptotic expansion of the probability p n , n − 2 l that all but a finite number 2 l of eigenvalues are real. In particular, we show that the expansion is of the form log p n , n − 2 l = { a 1 n 2 + a 2 n + a 3 log n + O ( 1 ) at   strongnon-Hermiticity , b 1 n + b 2 log n + b 3 + o ( 1 ) at   weaknon-Hermiticity , and we determine all coefficients explicitly. Furthermore, in the special case where l = 1, we derive the full-order expansions. For the proofs, we employ distinct methods for the strongly and weakly non-Hermitian regimes. In the former case, we utilise potential-theoretic techniques to analyse the free energy of elliptic Ginibre matrices conditioned to have n − 2 l real eigenvalues, together with the strong Szegö limit theorems. In the latter case, we utilise the skew-orthogonal polynomial formalism and the asymptotic behaviour of the Hermite polynomials.
Funding Information
  • National Research Foundation of Korea (RS-2023-00301976)
  • Deutsche Forschungsgemeinschaft (NRF-2016K2A9A2A13003815)
  • Seoul National University (New Faculty Startup Fund)

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