Technical Summary: Moderate-to-Large Deviation Asymptotics for Real Eigenvalues of Elliptic Ginibre Matrices

1. Problem Statement

The paper investigates the statistical properties of the number of real eigenvalues, denoted $N_n$, in the elliptic deformation of the real Ginibre ensemble (eGinOE). While the typical behavior of $N_n$ is well-established—specifically, the Law of Large Numbers (LLN) and the Central Limit Theorem (CLT) for both strong asymmetry (fixed asymmetry parameter $\tau \in [0,1)$) and weak asymmetry ($\tau \to 1$ as $n \to \infty$)—the probabilities of rare events (large deviations) remain largely unexplored in intermediate regimes.

Existing literature has addressed extreme cases:

• Left Tail: Exceptionally small numbers of real eigenvalues ($m \ll \mathbb{E}[N_n]$).
• Right Tail: Nearly all eigenvalues being real ($m \approx n$).

However, two critical intermediate regimes have remained open:

1. Strong Asymmetry: The intermediate regime where $m = O(\sqrt{n})$, bridging the Gaussian fluctuation scale and the extreme left tail.
2. Weak Asymmetry: The macroscopic regime where $m = O(n)$, bridging the Gaussian fluctuations and the extreme right tail.

The primary objective is to derive the asymptotic probabilities $p_{n,m} = \mathbb{P}(N_n = m)$ in these moderate-to-large deviation regimes, thereby establishing a natural connection between the known Gaussian fluctuation regime and the extreme large deviation regimes.

2. Methodology

The authors employ a rigorous analytic approach centered on the generating function of the number of real eigenvalues.

2.1. Generating Function and Pfaffian Structure

The eGinOE forms a Pfaffian point process. For even dimensions $2n$, the generating function for the probability distribution of real eigenvalues is expressed as a determinant:
$$\sum_{k=0}^n z^k p_{2n,2k} = \det \left[ I_{2n} + (z-1)M_{2n} \right]$$
where $M_{2n}$ is a specific symmetric matrix with spectrum in $(0,1)$. This representation allows the log-generating function to be expanded in terms of the traces of powers of $M_{2n}$:
$$\log \left( \sum_{k=0}^n z^k p_{2n,2k} \right) = -\sum_{k=1}^\infty \frac{(1-z)^k}{k} \text{Tr}(M_{2n}^k)$$

2.2. Asymptotic Analysis of Moments

The authors utilize known asymptotic results for $\text{Tr}(M_{2n}^k)$ in both regimes:

• Strong Asymmetry: $\text{Tr}(M_{2n}^k) \sim \sqrt{2n} \cdot C_k(\tau)$.
• Weak Asymmetry: $\text{Tr}(M_{2n}^k) \sim 2n \cdot C_k(\alpha)$, where $\tau = 1 - \alpha^2/n$.

These limits allow the derivation of the limiting log-generating functions, $\Psi_s(z)$ and $\Psi_w(z)$, which are expressed in terms of polylogarithms and integrals involving Bessel functions.

2.3. From Generating Functions to Coefficients (The Core Technical Step)

The central challenge is recovering the asymptotic behavior of the coefficients $p_{2n,2k}$ from the asymptotic behavior of the generating function. The authors apply a recent result by Borodin, Corwin, and Gorin [47] regarding the "exponential profiles" of polynomials with real roots.

• Weak Asymmetry: Since the scaling is $O(n)$, the standard theorem from [47] applies directly, establishing that the coefficients follow a large deviation principle with a rate function derived via the Legendre transform of the limiting log-generating function.
• Strong Asymmetry: The scaling is $O(\sqrt{n})$. The standard theorem does not apply because the leading asymptotic order is smaller than the polynomial degree. The authors prove a new extension (Theorem 4.1) to handle this case, showing that even when only a small portion of the polynomial contributes to the leading order, the exponential profile of the coefficients still exists and is determined by the Legendre transform.

3. Key Contributions and Results

3.1. Main Theorems (Moderate-to-Large Deviation Probabilities)

The paper derives the rate functions $\phi_s(x)$ and $\phi_w(x)$ governing the probabilities:
$$\lim_{n \to \infty} \frac{\log p_{2n,2m}}{\text{scaling}} = -\phi(x)$$
where $x$ is the normalized number of real eigenvalues.

• Strong Asymmetry ($\tau$ fixed, $m \sim \sqrt{n}$):
  The rate function is given by the Legendre transform of a function involving the polylogarithm $\text{Li}_{3/2}$:
  $$\phi_s(x) = \sup_{u \in \mathbb{R}} \left\{ \frac{x}{2}u - \Psi_s(e^u) \right\}$$
  where $\Psi_s(z) = -\sqrt{\frac{1+\tau}{1-\tau}} \frac{1}{2\pi} \text{Li}_{3/2}(1-z)$.

• Weak Asymmetry ($\tau = 1 - \alpha^2/n$, $m \sim n$):
  The rate function is given by the Legendre transform of an integral involving the parameter $\alpha$:
  $$\phi_w(x) = \sup_{u \in \mathbb{R}} \left\{ \frac{x}{2}u - \Psi_w(e^u) \right\}$$
  where $\Psi_w(z) = \frac{2}{\pi} \int_0^1 \log(1 - (1-z)e^{-\alpha^2 s^2}) \sqrt{1-s^2} \, ds$.

3.2. Consistency and Matching

The authors demonstrate that their results rigorously connect with existing literature:

• Gaussian Regime: The minima of the rate functions $\phi_s$ and $\phi_w$ occur at the known limiting means ($\mathbb{E}[N_n]$), and their curvatures match the known variances, recovering the Central Limit Theorem.
• Extreme Left Tail: As $x \to 0$, the derived rate functions recover the previously known asymptotic probabilities for vanishingly small numbers of real eigenvalues (matching results from [15, 50]).
• Extreme Right Tail: In the weak asymmetry regime, as $x \to 1$, the rate function recovers the known large deviation result for $m \approx n$ (matching [4]).
• Universality: The rate function form in the strong asymmetry regime is shown to be independent of $\tau$ (up to a scaling factor), suggesting a universal structure similar to that found in the spherical ensemble [31] and the simple symmetric exclusion process (SSEP) [18].

3.3. Cumulants and Generating Functions

The paper also establishes the asymptotic behavior of the cumulants of $N_n$ for all orders $\ell$, providing a complete description of the distribution's shape beyond just the mean and variance.

4. Significance and Claims

The paper claims to provide the first systematic derivation of moderate-to-large deviation asymptotics for the number of real eigenvalues in the elliptic Ginibre ensemble.

• Bridging Regimes: The primary contribution is filling the gap between the Gaussian fluctuation regime and the extreme large deviation regimes, providing a unified description of the probability distribution across all scales.
• New Results for Classical Cases: The results are noted to be new even for the classical real Ginibre ensemble ($\tau=0$), where the intermediate regime $m=O(\sqrt{n})$ was previously uncharacterized.
• Methodological Advancement: The paper extends the "exponential profile" technique from [47] to handle scaling regimes where the speed of the large deviation ($c_n$) is sublinear relative to the polynomial degree ($n$), a technical extension required for the strong asymmetry case.
• Physical Relevance: The authors note that understanding these statistics is crucial for characterizing spectral properties in non-Hermitian systems, including phase transitions, stability of dynamical systems, and connections to annihilating Brownian particles.

The paper concludes by acknowledging that the analysis relies on the determinant formula available for even dimensions; extending these results to odd dimensions remains an open technical challenge due to the lack of a tractable determinant representation for the generating function in that case.