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Asymptotics of the real eigenvalue distribution for the real spherical ensemble

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Pith's one-line read The paper determines the leading asymptotic form of the probability that the real spherical ensemble has exactly M real eigenvalues, in the large-deviation, intermediate-deviation, and no-real-eigenvalue regimes.

desk verdict Two solid unconditional results for the spherical ensemble; the headline large-deviation rate is an honest but unproved conjecture resting on a Coulomb-gas hypothesis that needs referee scrutiny. read the letter →

arxiv 2508.04139 v1 pith:NKRIX4Q4 submitted 2025-08-06 math-ph math.MPmath.PR

Peter J. Forrester This is my paper · ORCID
classification math-phmath.MPmath.PR MSC 60B2015B5260F10
keywords realsphericalensembleeigenvalueslargedeviationsCoulombgaselectrostaticenergyintermediategeneratingfunctionlocalcentrallimittheorem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper determines the leading asymptotic form of the probability $p^r_{N,M}$ that the real spherical ensemble — a random matrix $AB^{-1}$ with $A,B$ independent $N\times N$ real Gaussian matrices — has exactly $M$ real eigenvalues. Earlier work supplied the mean number (order $\sqrt N$) and a Gaussian local central limit theorem near that mean; this paper adds the large-deviation and intermediate-deviation regions. For $M=\alpha N$ it derives, conditional on an electrostatics hypothesis, an explicit $N^2$-scale decay rate equal to the minimum Coulomb energy of a sphere carrying charge on the equator and in two polar caps. For $M=x\sqrt N$ it unconditionally derives an intermediate-deviation formula from the exact factored generating function by steepest descent, and shows the formula's two tails match the large-deviation and local-CLT regimes respectively. The boundary case $M=0$ gives $p^r_{N,0}\sim e^{-\sqrt{\pi N/8}\,\zeta(3/2)}$, i.e. $e^{-\mu_N\zeta(3/2)/2}$ in terms of the mean real-eigenvalue count.

What carries the argument

The load-bearing object for the intermediate regime is the exact factorised generating function $Z_N(\xi)=\sum_M\xi^M p^r_{N,M}$: its log-asymptotics reduce to a single integral, and the contour formula $p^r_{N,M}=\frac{1}{2\pi i}\oint Z_N(\xi)\xi^{-M-1}d\xi$, after $\xi=e^{-\mu}$, becomes a Laplace integral with a unique saddle point. The companion object for the large-deviation regime is the conditioned Coulomb gas on a sphere of radius $1/2$: density $\alpha/\pi$ on the equator, uniform $1/\pi$ caps near the poles cut at $\cos\theta_0=\alpha$, neutralised by uniform $-1/\pi$ background. Its minimum energy $E_\alpha$, computed from spherical-cap potential integrals, is the conjectured limi

What would settle it

Use the exact product formula for $Z_N(\xi)$ to compute $p^r_{N,M}$ for several increasing $N$ with a fixed ratio, say $N=60,120,240$ and $M=N/2$, and compare $-N^{-2}\log p^r_{N,M}$ with the value of $E_\alpha$ obtained from (1.2). If the quantity does not converge to $E_\alpha$ as $N$ grows, or converges to a different constant, the electrostatics hypothesis is refuted.

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Extended reading notes

Core claim

A single generating function organises the whole distribution of $p^r_{N,M}$. Its exact factored form has log-asymptotics $\log Z_N(e^{-\mu})\sim\sqrt{N/2}\int_0^\infty\log(1-(1-e^{-2\mu})e^{-t^2})\,dt$; steepest descent on the contour integral for $p^r_{N,M}$ then gives, for $M=x\sqrt N$, $p^r_{N,M}\sim\exp(\sqrt N\,\min_\mu(x\mu+\chi(\mu)))$. For $M=\alpha N$ the paper assumes the GinOE-style electrostatics hypothesis that the rate equals the minimum energy $E_\alpha$ of a sphere of radius $1/2$ with equator charge density $\alpha/\pi$, uniform $1/\pi$ polar caps cut at $\cos\theta_0=\alpha$, and neutralising $-1/\pi$ background; computing $E_\alpha$ gives Proposition 1. The two tails of t

Load-bearing premise

The load-bearing premise is that the limiting large-deviation rate of $p^r_{N,M}$ at $M/N=\alpha$ equals the minimum electrostatic energy $E_\alpha$ of the charged-sphere configuration; the paper assumes this electrostatics hypothesis by analogy with the GinOE case, explicitly declines to prove it, and notes the conditioned eigenvalue PDF is not exactly a Coulomb gas.

Editorial extensions

If this is right

  • If the electrostatics hypothesis is correct, Proposition 1 fixes the leading $N^2$ decay of $p^r_{N,M}$ for every fixed $\alpha=M/N\in(0,1]$, and the limit $\alpha\to1^-$ recovers the known all-real leading order.
  • The intermediate-deviation formula is unconditional: for $M=x\sqrt N$ it gives $p^r_{N,M}$ to leading exponential order on the $\sqrt N$ scale.
  • The two tail matchings establish consistency between all three regimes: the $x\to\infty$ tail of the intermediate formula reproduces the small-$\alpha$ large-deviation tail, and the $x\to0$ tail reproduces the local CLT exponential.
  • The probability of no real eigenvalues decays as $e^{-\sqrt{\pi N/8}\,\zeta(3/2)}$, which equals $e^{-\mu_N\zeta(3/2)/2}$, and fixed nonzero $M$ has the same leading decay.
  • The no-real-eigenvalue exponent is proportional to the expected number $\mu_N\sim\sqrt{\pi N/2}$ of real eigenvalues, echoing the form found for GinOE.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: if the electrostatics hypothesis is eventually proved, the same spherical-cap variational scheme should supply large-deviation rates for other conditioned ensembles with spherical symmetry, because the minimising charge configuration remains explicit.
  • Beyond the paper: the factored generating function with negative real zeros suggests that refined saddle-point or cumulant methods could promote the intermediate-deviation formula to a full asymptotic expansion in powers of $N^{-1/2}$.
  • Beyond the paper: the shared $\zeta(3/2)/2$-times-mean formula for the no-real-eigenvalue probability across spherical and Ginibre-type ensembles is a concrete universality candidate; testing it on elliptic and product ensembles with the same $\mu_N\sim c\sqrt N$ scaling would either extend or break it.
  • Beyond the paper: the mismatch of scales ($N^{-2/3}$ where $E_\alpha$ is order one versus $N^{-3/4}$ for the local CLT) points to a crossover distribution not captured by either formula; an expansion of the intermediate-deviation expression near $x\sim N^{-1/12}$ should reveal it.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the probability p^r_{N,M} that the real spherical ensemble has M real eigenvalues. For M/N = alpha fixed in (0,1], Proposition 1 gives an N^2 large-deviation formula for log p^r_{N,M} in terms of a closed-form Coulomb energy E_alpha, under an explicit electrostatics hypothesis. For M = x sqrt(N), Proposition 3 derives the intermediate-deviation asymptotics exp(sqrt(N) min_mu (x mu + chi(mu))) from the exact generating function Z_N(xi), and Proposition 4 obtains the leading probability of no real eigenvalues, p^r_{N,0} ~ exp(-sqrt(pi N/8) zeta(3/2)). Tail matchings are exhibited between the large-deviation, intermediate-deviation, and local-CLT regimes.

Significance. If the conditional large-deviation result holds, the paper provides a complete leading-order picture across all relevant scales for the real-eigenvalue count in the spherical ensemble. The unconditional parts are solid: Proposition 2 derives the generating-function asymptotics directly from the exact product formula (2.24), the steepest-descent argument in Proposition 3 is checkable, Proposition 4 evaluates the resulting integral explicitly, and the two tail matchings are nontrivial and correct. The paper also gives an explicit numerically computable prediction (1.2) that can be tested against exact finite-N data from (2.24). The main weakness is that the headline large-deviation formula rests on an unproved electrostatics identity, as the manuscript itself acknowledges.

major comments (2)
  1. [Section 2.1, Eq. (2.3) and Proposition 1] The central claim (1.2) is conditional on the hypothesis that -N^{-2} log p^r_{N,M}|_{M/N=alpha} equals the minimum Coulomb energy E_alpha of the specific equator/cap charge configuration. The manuscript explicitly states that the correspondence between the conditioned eigenvalue PDF and the Coulomb gas is not exact due to additional one-body terms, and it declines to prove that these terms are subleading at the N^2 scale. Since Proposition 1 is the main advertised result, this is a load-bearing gap. The numerical check at N=60, M=30 gives -37.873 versus the predicted -38.513, a 1.7% discrepancy in the rate, which is not a precision confirmation. I recommend that the authors either (a) supply a proof or a rigorous large-deviation argument for the avoided one-body terms, or (b) substantially strengthen the numerical evidence by computing the N^{-2} log p^r_{N,M} slope for several fixed al
  2. [Section 2.1, minimizer of E_alpha] Even within the electrostatic hypothesis, the paper asserts without proof that the minimizing configuration must be a uniform equator charge density alpha/pi plus two uniform polar caps of density 1/pi with boundary cos(theta_0)=alpha. The argument 'rotational symmetry plus repulsion' is plausible but does not exclude other symmetric minimizers, e.g. configurations with a non-uniform equatorial strip or with additional neutral pairs. Since E_alpha is used exactly in (1.2), a different minimizer would change the formula at leading order. Please clarify whether the minimization is over all rotationally invariant signed measures with the stated total charges, and either prove the cap/equator form or state it as an additional part of the hypothesis.
minor comments (5)
  1. [Abstract] There is a typo: 'p_{N.M}' should be 'p_{N,M}'. Also, 'when N is proportional to sqrt N' should read 'when M is proportional to sqrt N'.
  2. [Proposition 2, Eq. (3.5)] The logarithm in (3.5) is not single-valued for arbitrary complex xi. Please specify the branch and the domain of xi for which the asymptotic formula is claimed, especially since (3.4) requires a contour integral in the complex xi-plane.
  3. [Proposition 3 proof] The statement that there is a unique real stationary point and that the contour can be chosen parallel to the negative real axis is made without proof or a reference to a detailed steepest-descent lemma. Since this is the basis of (3.9), a few lines of justification or a cited theorem would help.
  4. [References] Reference [56] is a Wikipedia page. For a binomial local CLT, a standard textbook reference would be more appropriate.
  5. [Section 3.3, after Eq. (3.14)] The sentence 'either of the asymptotic expressions in (3.14)' is slightly confusing because the two expressions are not alternatives but successive simplifications; consider rewording.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's main large-deviation result is explicitly conditional on an unproved electrostatics hypothesis, and the intermediate-deviation results are derived from an exact generating function.

full rationale

The paper does not derive Proposition 1 from first principles; it states it as conditional on an electrostatics hypothesis (Section 2.1). The hypothesis asserts that the large-deviation rate equals the minimum energy E_alpha of a specific charge configuration, and the rest of Section 2 computes E_alpha analytically. This is a transparent conditional statement, not a hidden definition or a fitted parameter. The one numerical comparison and the alpha=1 limit are checks, not fitting inputs. Propositions 2-4 are derived from the exact generating function (2.24) taken from prior work [34], which is an independent exact result; no parameters are fit and no circular reduction occurs. Self-citations appear but only to exact formulas or earlier proofs, not to unverified claims. The unproved hypothesis and the weak numerical support for the large-deviation formula are legitimate concerns about rigor or correctness, but they are not circularity. Hence the score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

Everything in the large-deviation claim rests on a postulated electrostatic model that the paper declines to prove (the load-bearing unknown). The intermediate-deviation and p_{N,0} claims rest on the exact generating function (2.24) from [34] plus standard saddle-point machinery, with no fitted constants. No numbers in the paper are fitted to data: the normalization E_alpha(0) = 0 is a stated convention, and the 1/pi charge densities and cap angle theta_0 follow from charge balance within the hypothesis, not from data.

assumptions (5)
  • ad hoc to paper Electrostatics hypothesis: lim -N^{-2} log p_{N,M}|_{M/N=alpha} equals the minimum Coulomb energy E_alpha of the conditioned charge configuration (charge density alpha/pi on the equator, symmetric caps of density 1/pi about the poles, neutralizing background).
    Section 2.1 states this as a hypothesis, citing the proved GinOE analog [38]; for the real spherical ensemble it is unproved, and the paper declines to prove it ('To attempt to give a proof would change the emphasis of this work...'). Proposition 1 (Eq. (1.2)) depends entirely on it.
  • domain assumption The minimizing charge configuration is determined by rotational symmetry, like-charge repulsion (cap support about the poles), and neutrality (density 1/pi in the caps, zero field in the caps), giving cos theta_0 = alpha (Eq. (2.6)).
    Section 2.1: 'so that there is no electric field produced by this charge density when superimposed on the uniform negative background charge density, the charge density in each spherical cap must equal 1/pi'. Standard Coulomb gas equilibrium reasoning, but conditional on the hypothesis above.
  • domain assumption The exact generating function Z_N(xi) with coefficients (2.25), from [34].
    Input for Propositions 2, 3, and 4; cited from prior work (Forrester and Mays, 2012) and not reproved here.
  • domain assumption The binomial-coefficient local CLT, stated for (N-4l) = o(N^{2/3}), is applied over the full range of l; off-peak terms are asserted to be negligible at leading order.
    Proof of Proposition 2, Eqs. (3.6)-(3.7). The paper notes the validity condition but does not justify the truncation. The simplification is correct at the stated leading order because exponentially suppressed terms do not affect the leading log, but the step is not shown.
  • standard math Unique real saddle of x mu + chi(mu), convexity, negative value at the saddle, and validity of the steepest-descent deformation of the contour in (3.10).
    Proof of Proposition 3, Section 3.2: 'Straightforward considerations show...' asserted without proof. Standard one-dimensional saddle-point machinery, though the paper's description of the descent direction (parallel to the negative real axis) is inconsistent with a local minimum of a holomorphic integrand, for which the descent direction is normal to the real axis.
invented entities (1)
  • Conditioned Coulomb-gas charge configuration (equator charge density alpha/pi, symmetric polar caps of density 1/pi, neutralizing background -1/pi on the sphere of radius 1/2)
    purpose: To evaluate the large-deviation rate -(1/N^2) log p_{N,M} as the minimum electrostatic energy of this configuration, yielding Eq. (1.2) via (2.7).
    Postulated in Section 2.1 by analogy with the GinOE case proved in [38]. The paper notes the correspondence between the conditional eigenvalue PDF and this gas is 'not exact' and does not derive the configuration from the PDF. Indirect support only: the alpha -> 1 limit matches (1.1) and one N=60 check agrees to about 2% in the exponent. This is the load-bearing invented object.

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Pith. "Pith review of Asymptotics of the real eigenvalue distribution for the real spherical ensemble." pith.science (2026). https://pith.science/paper/NKRIX4Q4

@misc{pith2026250804139,
  author       = {Pith},
  title        = {Pith review of: Asymptotics of the real eigenvalue distribution for the real spherical ensemble},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NKRIX4Q4}},
  note         = {Machine review of arXiv:2508.04139}
}
abstract

The real Ginibre spherical ensemble consists of random matrices of the form $A B^{-1}$, where $A,B$ are independent standard real Gaussian $N \times N$ matrices. The expected number of real eigenvalues is known to be of order $\sqrt{N}$. We consider the probability $p_{N.M}^{\rm r}$ that there are $M$ real eigenvalues in various regimes. These are when $M$ is proportional to $N$ (large deviations), when $N$ is proportional to $\sqrt{N}$ (intermediate deviations), and when $M$ is in the neighbourhood of the mean (local central limit theorem). This is done using a Coulomb gas formalism in the large deviations case, and by determining the leading asymptotic form of the generating function for the probabilities in the case of intermediate deviations (the local central limit regime was known from earlier work). Moreover a matching of the left tail asymptotics of the intermediate deviation regime with that of the right tail of the large deviation regime is exhibited, as is a matching of the right tail intermediate deviation regime with the leading order form of the probabilities in the local central limit regime. We also give the leading asymptotic form of $p_{N,0}^{\rm r}$, i.e. the probability of no real eigenvalues.

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