arxiv: 2604.25785 · v1 · submitted 2026-04-28 · 🧮 math-ph · hep-th· math.MP

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Level Crossing in Random Matrices. III. Analogs of Girko's circular and Wigner's semicircle laws

B.Shapiro

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Pith reviewed 2026-05-07 14:12 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MP

keywords random matrix pencilslevel crossingscircular lawempirical measurelogarithmic potentialGirko lawWigner semicircleuniversality

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The pith

Under assumptions of uniform circular law plus repulsion and tail controls, the empirical measure of level crossings in random matrix pencils converges to a deterministic limit.

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

The paper studies the points at which eigenvalues of the pencil A_n + λ B_n coincide as the scalar λ varies. It rewrites the normalized log-discriminant as a sum of pairwise logarithmic interactions among the eigenvalues and shows that this quantity is governed by a deterministic potential once a uniform circular law, logarithmic tail bounds, and small-spacing repulsion estimates are in place. Under these conditions the empirical distribution of the crossing points converges weakly to an explicit deterministic measure. The assumptions hold outright for complex Gaussian matrices after a uniformity verification and remain conditional for general i.i.d. ensembles, where they are motivated by universality. Separate arguments show that any limiting measure in the real case stays away from the real projective line and that analogous limits appear for elliptic and Hermitian pencils.

Core claim

Under assumptions combining a uniform circular law, logarithmic tail control, and small-spacing repulsion estimates, the empirical measure of level crossings for the random matrix pencil A_n + λ B_n converges weakly to an explicit deterministic limit whose density is determined by the equilibrium measure of a logarithmic energy functional built from the circular law.

What carries the argument

The normalized log-discriminant, rewritten as a sum of pairwise eigenvalue interactions, whose large-n limit is controlled by a deterministic potential derived from the circular law.

If this is right

In the complex Gaussian case the limiting measure is unconditional once uniformity is checked.
Any limiting measure in the real i.i.d. case does not concentrate on the real projective line.
The same convergence statement extends to elliptic and Hermitian ensembles under the corresponding circular-law and repulsion hypotheses.
The results are conditional in the general i.i.d. setting and rely on universality to motivate the assumptions.

Where Pith is reading between the lines

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

Verification of the three assumptions in additional ensembles would immediately upgrade the conditional statements to unconditional ones.
The representation via pairwise interactions suggests that similar energy functionals govern level crossings in other parameter-dependent random-matrix families.
The deterministic limit supplies a concrete prediction for the typical locations of degeneracies that can be tested numerically without ensemble averaging.

Load-bearing premise

The combination of a uniform circular law for the matrices, logarithmic tail control on the eigenvalues, and small-spacing repulsion estimates.

What would settle it

A direct Monte-Carlo computation of the empirical measure of crossings for large n in the complex Gaussian ensemble that visibly fails to approach the predicted deterministic density.

Figures

Figures reproduced from arXiv: 2604.25785 by B.Shapiro.

**Figure 1.** Figure 1: GER n -distributions of the branch points, for n = 2, 5, 15, 25 approaching the uniform distribution on CP 1 . in probability on CP1 . Equivalently, the limiting law is the uniform spherical measure. Remark 10. The conjecture is meant for genuinely full real i.i.d. matrices, not for real symmetric matrices. It is consistent with the numerical evidence in [GrShZa], where the real Gaussian ensemble GER n ap… view at source ↗

**Figure 2.** Figure 2: Density of GUE2 in C. for every η > 0. Let µnk ⇒ µ in probability. Passing to a further subsequence if necessary, assume the convergence is almost sure. Choose continuous cutoffs 0 ≤ χj ≤ 1 on CP1 such that χj = 1 on RP1 and suppχj ⊂ (RP1 )εj , where εj ↓ 0. By another diagonal extraction, using the preceding probability estimate, we may assume that lim sup k→∞ µnk ((RP1 )εj ) = 0 almost surely for every f… view at source ↗

**Figure 3.** Figure 3: Empirical distributions of ∣Y ∣ for (1.1) taken from GUEn with n = 2, 3, 4, 5, 6. (Curves corresponding to the increasing values of n lie one below the other; the blue straight line corresponds to n = 2.) We do not have an exact formula for the density P GUE n (x, y), for n > 3. However these densities have simpler formulas in cylindrical or spherical coordinates on CP1 . In particular density 1 π 1 (1+… view at source ↗

read the original abstract

We study the asymptotic distribution of level crossings for random matrix pencils A_n+\lambda B_n in several ensembles, including complex and real i.i.d. matrices and Gaussian/Hermitian settings. We derive a representation of the normalized log-discriminant in terms of pairwise eigenvalue interactions and formulate conditions under which its limit is governed by a deterministic potential. Under assumptions combining a uniform circular law, logarithmic tail control, and small-spacing (repulsion) estimates, we prove convergence of the empirical measure of level crossings to an explicit deterministic limit. In the complex Gaussian case these assumptions are verified (modulo a uniformity step), while in the general i.i.d. setting the results are conditional and motivated by universality theory. We further analyze the real case, showing that any limiting measure does not concentrate on the real projective line under suitable hypotheses, and discuss analogous phenomena for elliptic/Hermitian ensembles. Our results highlight the role of logarithmic energy and universality in governing spectral degeneracies of random matrix pencils.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends circular and semicircle laws to level crossings in matrix pencils via a pairwise-interaction representation, but the convergence is conditional on assumptions that are only partially checked.

read the letter

This paper gives a representation of the normalized log-discriminant for random matrix pencils as a sum of pairwise eigenvalue interactions, then uses that to prove convergence of the empirical measure of level crossings to a deterministic limit under three assumptions: a uniform circular law, logarithmic tail control, and small-spacing repulsion estimates. In the complex Gaussian case they verify most of this modulo a uniformity step, which is concrete progress. They also show that in the real case the limiting measure does not concentrate on the real projective line under their hypotheses, and they sketch the elliptic and Hermitian analogs. That is the actual new content, and it sits on top of existing circular-law and repulsion results without introducing fitted parameters or circular definitions. The soft spot is that the general i.i.d. claims remain conditional; the paper motivates them by universality but does not close the uniformity or repulsion steps for non-Gaussian entries. Any gap in those inputs directly limits how far the explicit deterministic limit can be applied. The work is aimed at people already working inside random matrix theory who care about spectral degeneracies in pencils. A reader who knows the Girko and Wigner literature will see the extension clearly; someone looking for unconditional theorems on arbitrary i.i.d. matrices will notice the remaining conditions. It deserves a serious referee because the Gaussian verification is explicit and the representation is clean, even though the general case needs more work to become unconditional.

Referee Report

1 major / 0 minor

Summary. The manuscript studies the asymptotic distribution of level crossings for random matrix pencils A_n + λ B_n in i.i.d., complex/real Gaussian, and Hermitian/elliptic ensembles. It derives a representation of the normalized log-discriminant in terms of pairwise eigenvalue interactions and proves, under the combined assumptions of a uniform circular law, logarithmic tail control, and small-spacing (repulsion) estimates, that the empirical measure of level crossings converges to an explicit deterministic limit. These assumptions are verified only modulo a uniformity step in the complex Gaussian case and remain conditional (motivated by universality) in the general i.i.d. setting; additional results show that any limiting measure does not concentrate on the real projective line in the real case.

Significance. If the assumptions can be fully verified, the work supplies direct analogs of Girko's circular law and Wigner's semicircle law for the distribution of spectral degeneracies in random pencils, emphasizing the governing role of logarithmic energy and pairwise interactions. The parameter-free derivation from the log-discriminant representation and the explicit deterministic limit (when assumptions hold) are strengths that could advance the study of non-Hermitian random matrices and universality phenomena.

major comments (1)

[Abstract and main results section] The central convergence claim (stated in the abstract and derived in the main results section) is conditional on three assumptions (uniform circular law, logarithmic tail control, small-spacing estimates) that are only partially verified: modulo an uncompleted uniformity step for complex Gaussians and left conditional for general i.i.d. ensembles. Since the explicit deterministic limit holds if and only if these inputs are satisfied, the gap in verification is load-bearing and weakens the applicability of the main theorem; the manuscript should either complete the uniformity argument or provide a precise roadmap with explicit error bounds.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive feedback. We address the major comment point by point below.

read point-by-point responses

Referee: The central convergence claim (stated in the abstract and derived in the main results section) is conditional on three assumptions (uniform circular law, logarithmic tail control, small-spacing estimates) that are only partially verified: modulo an uncompleted uniformity step for complex Gaussians and left conditional for general i.i.d. ensembles. Since the explicit deterministic limit holds if and only if these inputs are satisfied, the gap in verification is load-bearing and weakens the applicability of the main theorem; the manuscript should either complete the uniformity argument or provide a precise roadmap with explicit error bounds.

Authors: We agree that the main theorem is conditional on the three assumptions, as already stated explicitly in the abstract, introduction, and results section. This conditional formulation is deliberate: it isolates the contribution of the log-discriminant representation and the resulting deterministic limit under inputs that are standard in non-Hermitian random matrix theory and widely expected to hold by universality. For complex Gaussians the uniformity step remains open, while for general i.i.d. ensembles the results are presented as conditional. We do not claim unconditional convergence. In the revised manuscript we will add a dedicated subsection that supplies a precise roadmap, including explicit error bounds and references to existing techniques for completing the uniformity argument. This addresses the referee's request without overstating the current verification status. revision: partial

standing simulated objections not resolved

Completing the uniformity step for the complex Gaussian case requires new technical estimates on the circular law that are currently beyond the scope of this work.

Circularity Check

0 steps flagged

No circularity: limit derived from pairwise interactions under external assumptions.

full rationale

The manuscript derives a representation of the normalized log-discriminant via pairwise eigenvalue interactions and proves convergence of the empirical measure of level crossings to an explicit deterministic limit under the stated assumptions (uniform circular law, logarithmic tail control, small-spacing estimates). These assumptions are cited from prior literature or partially verified in the Gaussian case, but the derivation chain does not reduce the target limit to a fitted parameter, self-definition, or self-citation by construction. The central result therefore retains independent content from the interaction representation and is not equivalent to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from random matrix theory without introducing new free parameters or invented entities.

axioms (2)

domain assumption Uniform circular law holds for the matrix ensembles under consideration
Invoked to control eigenvalue distributions and enable the deterministic potential limit.
domain assumption Logarithmic tail control and small-spacing repulsion estimates hold
Required to ensure the normalized log-discriminant converges to the deterministic potential.

pith-pipeline@v0.9.0 · 5474 in / 1250 out tokens · 59590 ms · 2026-05-07T14:12:38.741104+00:00 · methodology

discussion (0)

Reference graph

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