The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911228663693312 |
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| author | Akemann, Gernot Byun, Sung-Soo Lee, Yong-Woo |
| author_facet | Akemann, Gernot Byun, Sung-Soo Lee, Yong-Woo |
| contents | We investigate real eigenvalues of real elliptic Ginibre matrices of size $n$, indexed by the parameter of asymmetry $τ\in [0,1]$. In both the strongly and weakly non-Hermitian regimes, where $τ\in [0,1)$ is fixed or $1-τ=O(1/n)$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_18310 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble Akemann, Gernot Byun, Sung-Soo Lee, Yong-Woo Probability Statistical Mechanics Mathematical Physics We investigate real eigenvalues of real elliptic Ginibre matrices of size $n$, indexed by the parameter of asymmetry $τ\in [0,1]$. In both the strongly and weakly non-Hermitian regimes, where $τ\in [0,1)$ is fixed or $1-τ=O(1/n)$, 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. |
| title | The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble |
| topic | Probability Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2503.18310 |