Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices

TL;DR

This work analyzes the distribution of the number of real eigenvalues in the elliptic real Ginibre ensemble, focusing on moderate-to-large deviations in two asymmetry regimes: strong (τ fixed) and weak (τ_n → 1). By exploiting the Pfaffian/determinant structure of the generating function, the authors derive explicit asymptotics for the generating function and its cumulants, then employ a Gregory-type ansatz and a local large-deviation principle to obtain rate functions φ_s and φ_w that describe the decay of p_{n,m} in the intermediate and large deviation regimes. The results connect the Gaussian fluctuation regime to extreme large deviations, reveal a universal form in the strong regime, and establish a rigorous framework that may extend to related ensembles and to real-root counting problems in random polynomials. The findings provide new insights into the real eigenvalue statistics of non-Hermitian random matrices and open avenues for localized interval counts and joint real/complex eigenvalue statistics in integrable models.

Abstract

We study the statistics of the number of real eigenvalues in the elliptic deformation of the real Ginibre ensemble. As the matrix dimension grows, the law of large numbers and the central limit theorem for the number of real eigenvalues are well understood, but the probabilities of rare events remain largely unexplored. Large deviation type results have been obtained only in extreme cases, when either a vanishingly small proportion of eigenvalues are real or almost all eigenvalues are real. Here, in both the strong and weak asymmetry regimes, we derive the probabilities of rare events in the moderate-to-large deviation regime, thereby providing a natural connection between the previously known regime of Gaussian fluctuations and the large deviation regime. Our results are new even for the classical real Ginibre ensemble.

Figures (4)

• Figure 1: Eigenvalues of $100$ realisations of the eGinOE with $n=20$ for a given $\tau$.
• Figure 2: A schematic sketch of $m\mapsto p_{n,m}$. The graph on the left (blue) illustrates the case with strong asymmetry, while the graph on the right (orange) illustrates the case with weak asymmetry. The regimes studied in this paper are depicted on the $x$-axis as shaded regions.
• Figure 3: Figures (A)--(C) show the numerical plots of $m \mapsto p_{n,m}$ in the strong asymmetry regime for $n = 8, 32, 128$, respectively. Figures (D)--(F) display the corresponding plots in the weak asymmetry regime. Figures (G) and (H) present the graphs of $m \mapsto -\log p_{n,m}$ for both regimes (dotted line), together with their comparison to $m \mapsto \mathbb{E}\mathcal{N}_n\,\phi_{\mathrm{s}}(m/\mathbb{E}\mathcal{N}_n)$ and $m \mapsto n\,\phi_{\mathrm{w}}(m/n)$ (full solid line). Here, we set $\tau = 1/2$ for the strong asymmetry (blue) and $\alpha = 1$ for the weak asymmetry (orange). The formula \ref{['eq for det formula of gen function']} is used for the numerical evaluations.
• Figure 4: The diagram illustrates the interrelations between our main results and the existing literature, the details of which can be found in Remarks \ref{['Rem_strong left right']}, \ref{['Rem_weak left right']}, and \ref{['Rem_strong weak']}.

Theorems & Definitions (16)

• Theorem 2.1: Moderate and large deviation probabilities
• Remark 1: Universal form in the strong asymmetry regime, cf. Fo25
• Remark 2: Extremal case; matching to the left and right tail large deviations in the strong asymmetry regime
• Remark 3: Extremal case; matching to the left and right tail large deviations in the weak asymmetry regime
• Proposition 2.2: Minimiser and curvature of the rate functions
• Remark 4: Interpolating properties of the rate functions
• Proposition 2.3: Asymptotic behaviour of the generating function and cumulants
• Lemma 3.1
• proof
• proof : Proof of Proposition \ref{['Prop_minimisers and curvature']}