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

TL;DR

This work delivers a detailed large-deviation analysis for the number of real eigenvalues in the elliptic real Ginibre ensemble, quantifying the probability that almost all eigenvalues are real. It derives explicit asymptotic expansions for $p_{n,n-2l}$ in two distinct non-Hermitian regimes: strong non-Hermiticity with $\tau$ fixed and weak non-Hermiticity with $\tau=1-\alpha^2/n$, including full-order expansions for $l=1$. The authors employ two complementary methodologies: a potential-theoretic approach with Szegő-type limits in the strong regime, and skew-orthogonal polynomial techniques with Hermite asymptotics in the weak regime, complemented by an integrable Pfaffian structure for $p_{n,m}$. Collectively, the results provide a precise electrostatic/variational picture of how the real-eigenvalue count deviates from its typical scale and yield explicit rate functions and subleading terms, with broader implications for real-eigenvalue statistics in non-Hermitian random matrices.

Abstract

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.

Figures (4)

• Figure 1: The plot shows a schematic illustration of the graph $m \mapsto p_{n,m}$, along with markings of the left and right tail regimes.
• Figure 2: The plots (A) and (B) illustrate $\tau \mapsto \log p_{n,n-2l} - (a_1 n^2 + a_2 n + a_3 \log n)$ for $n = 10$ (red, dotted), $30$ (blue, dot-dashed), and $100$ (purple, dashed). From these plots, it can be observed that the values remain bounded as $n$ increases. Additionally, plot (A) compares this with $\tau \mapsto \log ( \frac{(3-\tau)^{1/2} (1+\tau)^{3/2}}{8\sqrt{\pi} (1-\tau)^{3/2}} )$ (black, solid), as given in \ref{['eq. LDP sh l=1 v2']}. The plots (C) and (D) display $\alpha \mapsto \log p_{n,n-2l} - (b_1 n + b_2 \log n + b_3)$ for $n = 10$ (red, dotted), $30$ (blue, dot-dashed), and $100$ (purple, dashed), along with $\alpha \mapsto 0$ (black, solid). These plots demonstrate that the values converge to $0$ as $n$ increases. We used \ref{['eq. pnm zonal formula']} for the numerical evaluation at each data point.
• Figure 3: The plots show the approximated potential $Q_n^{(r)}(z)$ on $z\in\mathbb{H}$ with $r=0$, $n=100$ for $\tau = 0$ in plot (A) and $\tau = 1-1/n$ in plot (B).
• Figure 4: Illustration of the regions $\rm I, II^{\pm}, III$ and ${\rm I}_n, {\rm II}_n^\pm, {\rm III}_n$ in the upper half plane.

Theorems & Definitions (30)

• Theorem 1.1: The probability of having a finite number $l$ of complex eigenvalue pairs
• Theorem 1.2: The probability of having one complex conjugate pair of eigenvalues, i.e. $l=1$
• Lemma 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['Thm. LDP rate function l=1']}
• Lemma 4.1
• proof