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The probability that a large elliptic Ginibre matrix has a given number of real eigenvalues obeys explicit rate functions in the intermediate deviation regime, with a universal form in the strong-asymmetry case.

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2026-08-03 22:40 UTC pith:PVMRWO4P

load-bearing objection Solid paper with explicit intermediate-deviation rate functions for eGinOE real eigenvalues and a reusable o(n)-speed exponential-profile theorem; the alleged τ-scaling inconsistency is a red herring. the 3 major comments →

arxiv 2511.09191 v2 pith:PVMRWO4P submitted 2025-11-12 math.PR cond-mat.stat-mechmath-phmath.MP

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

Sung-Soo Byun , Jonas Jalowy , Yong-Woo Lee , Gr\'egory Schehr This is my paper
classification math.PR cond-mat.stat-mechmath-phmath.MP MSC 60B2060F10
keywords real Ginibre ensembleelliptic Ginibre ensemblereal eigenvaluesmoderate deviationslarge deviationsrandom matricesrate functionpolylogarithm
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper pinpoints the rare-event statistics of the number of real eigenvalues in the elliptic real Ginibre ensemble, the matrix model interpolating between real Ginibre and GOE. It proves that in two intermediate scaling windows—the strong-asymmetry regime with a fixed non-Hermiticity parameter and the weak-asymmetry regime where the matrix is almost Hermitian—the logarithm of the probability of observing m real eigenvalues is asymptotically an explicit rate function times the appropriate scale. The formulas close the gap between the Gaussian fluctuations of the CLT scale and the extreme large-deviation tails studied earlier, and they recover the known mean, variance, left-tail, and right-tail limits. The results are new even for the standard real Ginibre ensemble.

Core claim

Writing N_{2n} for the number of real eigenvalues of a 2n×2n eGinOE matrix and p_{2n,2m} for the probability that N_{2n} equals 2m, the paper establishes that if 2m/E N_{2n}→x∈(0,∞) in the strong-asymmetry regime (fixed τ∈[0,1)), then lim log p_{2n,2m}/E N_{2n} = -1/2 sup_{u∈R}{xu + Li_{3/2}(1-e^u)}; if instead m/n→x∈(0,1) in the weak-asymmetry regime τ=1-α²/(2n), then lim log p_{2n,2m}/(2n) = -1/2 sup_{u∈R}{xu - (4/π)∫_0^1 log(1-(1-e^u)e^{-α²s²})√(1-s²) ds}. The keystone is that the generating function ∑ z^k p_{2n,2k} has a known asymptotic logarithmic limit in each regime, and the authors prove a general theorem that turns such a limit for a polynomial with non-positive real roots into a l

What carries the argument

The engine is the determinantal representation ∑_{k=0}^n z^k p_{2n,2k} = det(I_n+(z-1)M_n), inherited from the Pfaffian structure of the eGinOE point process; M_n is an n×n symmetric matrix whose spectrum lies in (0,1). Because of that spectral bound, the logarithm of the generating function expands in a power series in Tr(M_n^k), whose known n-asymptotics give the limit of the log-generating function in both regimes. The second main device is a new 'exponential profile' theorem: for any sequence of polynomials with non-positive real roots whose logarithms grow with speed c_n→∞ and whose limit Ψ satisfies strict convexity of z↦Ψ(e^z), the coefficients have a limiting exponential profile on c

Load-bearing premise

The derivation relies on the inherited structural fact that the generating polynomial ∑ z^k p_{2n,2k} has only non-positive real roots (with the generating function equal to det(I_n+(z-1)M_n) and the spectrum of M_n contained in (0,1)); if that representation or the spectral bound failed, the coefficients would no longer be Bernoulli sums and the exponential-profile theorem would not apply.

What would settle it

Take the real Ginibre case (τ=0), compute P(N_n=m) exactly for small n via the determinantal formula or by high-precision simulation for n=64,128,256 and m≈c√n, and plot (1/√n) log P against c. The strong-asymmetry formula predicts a single universal curve; any dependence on τ when repeating at τ=1/2, or a systematic drift that does not vanish as n grows, would falsify the result.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • At the minimum of each rate function, its value and curvature reproduce the known mean and variance of N_{2n} on the relevant scale, so the new result continuously contains the Gaussian CLT as the typical case.
  • In the strong-asymmetry regime the rate function is independent of τ; the same function appears for the spherical ensemble, supporting a universality across ensembles with a real-eigenvalue Pfaffian or determinantal structure.
  • The x→0 limit recovers the known extreme left-tail rate for very few real eigenvalues; in the weak case it fills the gap in a previously conjectured inequality, giving the exact constant.
  • The x→1 (weak) and x→∞ (strong) limits match the known right-tail decays, including the cubic growth of the strong rate that is expected to connect to the macroscopic Coulomb-gas regime.
  • The generating-function asymptotics also provide explicit formulas for all cumulants of N_{2n}, beyond the mean and variance used in the CLT.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the strong-asymmetry rate is τ-independent and the weak-asymmetry rate tends to it after rescaling when α→∞, one may expect a single universal crossover profile across the whole parameter plane; the paper leaves this unification implicit.
  • The same Legendre-transform machinery could apply to other Pfaffian point processes, for instance the number of real roots of random polynomials, where similar rate functions have been conjectured.
  • A direct numerical test is cheap: for fixed τ and several n, plot (1/√n) log P(N_n=m) against m/√n; the curves should collapse to a single universal curve, and the same collapse should occur for τ=0 and τ=1/2.
  • The explicit rate functions may help predict finite-size corrections in counting stationary points of random dynamical systems, where real eigenvalues determine the number of equilibria.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper studies the probability p_{n,m} that an n×n elliptic real Ginibre matrix has exactly m real eigenvalues. In the strong-asymmetry regime (τ fixed), for m=O(√n), and in the weak-asymmetry regime (1−τ=O(1/n)), for m=O(n), it derives explicit exponential asymptotics: log p_{2n,2m} normalized by √(2n) or 2n converges to a rate function given by the Legendre transform of a log-generating function, involving Li_{3/2} in the strong case and an integral of log in the weak case. The proof uses the determinantal representation, trace-moment asymptotics, and a new theorem on exponential profiles of coefficients of real-rooted polynomials. It also proves generating-function/cumulant asymptotics and verifies matching with left/right tails.

Significance. If correct, these are the first complete moderate-to-large deviation results for the number of real eigenvalues, connecting the CLT regime with extreme large deviations, and they are new even for the real Ginibre ensemble. The rate functions are explicit, and the matching with known extreme tails is a strong feature. The proof strategy is coherent and the paper contains an independently interesting exponential-profile theorem (Theorem 4.1). The main caveat is the constant-factor issue noted below, which must be resolved before the results can be used as stated.

major comments (3)
  1. [Eqs. (2.4), (2.8), (3.4), (2.18)] In the version I reviewed, the displayed prefactors in the strong-asymmetry formulas appear to be missing a square root. As printed, (2.4) gives −a/(2π)Li_{3/2}(1−z), so (2.13) yields a/(2π)ζ(3/2), contradicting the stated recovery of (1.5), which has a/√(2π)ζ(3/2). Similarly, (3.4) as printed has 1/(2πk); via (3.2) this would lead to a Li_2 term, not the Li_{3/2} in (2.25). Finally, (2.18) asserts x_s = (2/π)√((1+τ)/(1−τ)) = lim EN_{2n}/√(2n), but (1.1) gives √(2/π)√((1+τ)/(1−τ)). These inconsistencies are load-bearing: if they are not typesetting artifacts, the constants in the central theorem are wrong. Please correct the factors throughout and ensure (2.4), (2.8), (3.4), and (2.18)–(2.19) are consistent with (1.1), (1.5), and (2.22).
  2. [Section 3.2, proof of Proposition 2.3(ii)] The proof of the weak-asymmetry generating-function limit establishes (2.27) for z∈(0,1) and then says the general case follows by the identity theorem. However, (0,1) has empty interior in C, so the identity theorem alone cannot extend convergence from that interval. One needs the same normal-family/local-boundedness argument used in part (i): for compact subsets of C\(−∞,0] the log-determinant is bounded by a constant times Tr M_n, and Tr M_n ≤ n, giving the required local boundedness. This is easily repaired, but as written the proof of a key step is incomplete.
  3. [Theorem 2.1(ii) and Proposition 2.3(ii) scaling] A possible concern is that Theorem 2.1(ii) states τ=1−α²/(2n) while the introduction uses τ=1−α²/n for n×n matrices. This is not an inconsistency: Theorem 2.1 is formulated for 2n×2n matrices, so τ=1−α²/(2n) is precisely the same weak-asymmetry scaling as the introduction's τ=1−α²/n for n×n matrices. The formula (2.3) and the trace limit (3.5) are consistent with this interpretation, and the right-tail match (2.17) with (1.8) then holds. I do not regard this as an error.
minor comments (4)
  1. [Section 3.2, proof of Proposition 2.2] In the weak-asymmetry part, the derivatives are written ψ′_s and ψ″_s; they should be ψ′_w and ψ″_w. Also, “x_w = 2ψ_w(0)” should read “x_w = 2ψ′_w(0)”.
  2. [Abstract and title] The abstract/header contains a typo: “MODERA TE-TO-LARGE” should be “MODERATE-TO-LARGE.” Please proofread the final version.
  3. [Remark 4] The condition “1≪α≪√(2n)” appears in the text; since the weak-asymmetry parameter is τ=1−α²/n for n×n matrices (and τ=1−α²/(2n) for the 2n×2n matrices in the theorem), the condition should probably be α≪√n rather than α≪√(2n). Please clarify.
  4. [Figure 3] The caption refers to panels (A)–(C) and (D)–(F), but the panels are labeled (a)–(f). The labels should be aligned.

Circularity Check

0 steps flagged

No circularity: the derivation is self-contained given the cited structural inputs; the weak-asymmetry scaling typo is a correctness issue, not a circular reduction.

full rationale

The claimed derivation chain is: (i) the determinantal representation (3.1) and the spectral bound (0,1) for M_n taken from [15]; (ii) trace-moment asymptotics (3.4)-(3.5) from [15]; (iii) log-generating-function limits in Proposition 2.3 obtained by expanding log det(I+(z-1)M_n) and using those trace limits; (iv) the exponential-profile theorem (Theorem 4.1), proved in the paper via Bernoulli tilting and the local CLT, which extracts coefficient asymptotics from the generating-function asymptotics; and (v) Legendre inversion to obtain the rate functions in Theorem 2.1. At no point is the target probability p_{n,m} inserted as an assumption. The rate functions are Legendre transforms of already-derived log-generating functions, and the matchings to the earlier tails (1.5)-(1.8) in Remarks 2 and 3 are consistency checks, not inputs. The self-citations to [15] and [47] concern real structural results: [15] supplies the determinantal formula and moment limits independently of the present conclusions, and the part of [47] invoked for the weak-asymmetry case is replaced here by a fully written proof of the required generalization (Theorem 4.1). The remark that (1.7) is only conjectured as an equality is explicitly identified in the paper and is used only as a consistency check, with the paper noting it gives 'stronger conjectural evidence' rather than a proof. The only substantive concern in the manuscript is non-circular: Theorem 2.1(ii) states τ=1-α^2/(2n), whereas the trace limit (3.5), the cumulant formula (2.28), and the introduction's weak-asymmetry scaling correspond to τ=1-α^2/n. Under the literal stated scaling the exponential in (2.3) would become e^{-α^2 s^2/2} and the right tail would be α^2/32 instead of α^2/8. This is a correctness/consistency flaw in the statement, not a self-referential derivation of the result from itself, and therefore does not raise the circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters are fitted; τ and α are fixed model parameters of the ensemble, and x is a scaling variable. The paper's contribution is a derivation from prior structural results, not a fit. No new physical entities are introduced.

axioms (5)
  • domain assumption eGinOE eigenvalue point process is Pfaffian and the generating function of p_{2n,2k} equals det(I_n+(z-1)M_n) with Spec(M_n)⊂(0,1).
    Imported from [15, Prop. 2.1 and Lemma 2.3]; used in Eqs. (3.1)-(3.2) and gives real-rootedness of the generating polynomial, needed for Theorem 4.1.
  • domain assumption Asymptotic moment formulas: lim 1/√(2n) Tr(M_n^k) (strong) and lim 1/(2n) Tr(M_n^k) (weak), as in [15, Props. 2.5 and 2.6].
    These input asymptotics determine the limiting generating functions Ψ_s and Ψ_w in Proposition 2.3; the central result inherits their validity.
  • standard math Local central limit theorem for sums of independent Bernoulli variables with diverging variance, [10, Theorem 2].
    Used in the proof of Theorem 4.1 to show that the tilted coefficient mass is asymptotically uniform on the relevant scale.
  • standard math Gärtner-Ellis / convex Legendre-transform inversion under strict convexity.
    Used in Sections 2.3 and 4.2 to recover the exponential profile of coefficients from the log-moment generating function.
  • standard math Polylogarithm and Bessel integral identities, e.g. (2.12) and (3.12).
    Required to evaluate the Legendre transforms and to connect the rate functions to Li_{3/2} and modified Bessel functions.

reviewed 2026-08-03 · how reviews work

0 comments
read the original 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

Figures reproduced from arXiv: 2511.09191 by Gr\'egory Schehr, Jonas Jalowy, Sung-Soo Byun, Yong-Woo Lee.

Figure 1
Figure 1. Figure 1: Eigenvalues of 100 realisations of the eGinOE with n = 20 for a given τ . We now introduce our object of study. By definition, the real Ginibre ensemble (GinOE) is the ensemble of n × n matrices G with i.i.d. standard normal entries, see [11] for a review. Its elliptic extension, designed to interpolate between GinOE and the symmetric Gaussian orthogonal ensemble (GOE), is known as the elliptic GinOE (eGin… view at source ↗
Figure 2
Figure 2. Figure 2: A schematic sketch of m 7→ pn,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. Beyond the mean and variance of Nn, a central limit theorem has also been established for both the strong and weak asymmetry regimes. More prec… view at source ↗
Figure 3
Figure 3. Figure 3: Figures (A)–(C) show the numerical plots of m 7→ pn,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 7→ − log pn,m for both regimes (dotted line), together with their comparison to m 7→ ENn ϕs(m/ENn) and m 7→ n ϕw(m/n) (full solid line). Here, we set τ = 1/2 for the st… view at source ↗
Figure 4
Figure 4. Figure 4: The diagram illustrates the interrelations between our main results and the existing literature, the details of which can be found in Remarks 2, 3, and 4. where µγ is the minimiser of the energy functional I[µ] := 1 2  Z |z| 2 dµ(z) − Z Z log |z − w| dµ(z) dµ(w)  − 3 8 among all probability measures that place mass (1 − γ) on C \ R, and mass γ on R. See also [31] for recent progress on this electrostatic… view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 3, 2026.