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

Abstract

We investigate real eigenvalues of real elliptic Ginibre matrices of size $n$, indexed by the parameter of asymmetry $\tau \in [0,1]$. In both the strongly and weakly non-Hermitian regimes, where $\tau \in [0,1)$ is fixed or $1-\tau =O(1/n)$, respectively, we derive the asymptotic expansion of the probability $p_{n,n-2l}$ that all but a finite number $2l$ of eigenvalues are real. In particular, we show that the expansion is of the form \begin{align*} \log p_{n, n-2l} = \begin{cases} a_1 n^2 +a_2 n + a_3 \log n +O(1) &\text{at strong non-Hermiticity}, \\ b_1 n +b_2 \log n + b_3 +o(1) &\text{at weak non-Hermiticity}, \end{cases} \end{align*} 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 $n-2l$ real eigenvalues, together with the strong Szegő limit theorems. In the latter case, we utilise the skew-orthogonal polynomial formalism and the asymptotic behaviour of the Hermite polynomials.

30 pages, 4 figures. Journal accepted version. Several remarks are added compared to the previous version

References in corpus (25)

• Almost-Hermitian Random Matrices: Crossover from Wigner-Dyson to Ginibre eigenvalue statistics
• Asymptotic expansion of beta matrix models in the one-cut regime
• Eigenvalue statistics of the real Ginibre ensemble
• Directed Quantum Chaos
• Almost-Hermitian Random Matrices: Eigenvalue Density in the Complex Plane
• Random matrices: Universality of local spectral statistics of non-Hermitian matrices
• Nonlinear Analogue of the May-Wigner Instability Transition
• Fluctuations of linear eigenvalue statistics of $\beta$ matrix models in the multi-cut regime
• Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble
• Integrable Structure of Ginibre's Ensemble of Real Random Matrices and a Pfaffian Integration Theorem
• Orthogonal and symplectic matrix models: universality and other properties
• Topology trivialization transition in random non-gradient autonomous ODE's on a sphere
• A method to calculate correlation functions for $\beta =1$ random matrices of odd size
• Probability of all eigenvalues real for products of standard Gaussian matrices
• What is the probability that a large random matrix has no real eigenvalues?
• The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices
• Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble
• A Note on the Pfaffian Integration Theorem
• Finite size corrections for real eigenvalues of the elliptic Ginibre matrices
• On the number of real eigenvalues of a product of truncated orthogonal random matrices
• The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$
• Moments of Characteristic Polynomials of Non-Symmetric Random Matrices
• Zeros of conditional Gaussian analytic functions, random sub-unitary matrices and q-series
• Fluctuations in Various Regimes of Non-Hermiticity and a Holographic Principle
• Real Eigenvalues of Asymmetric Wishart Matrices: Expected Number, Global Density and Integrable Structure