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

Abstract: We study the asymptotic distribution of level crossings for random matrix pencils A_n+λB_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.

• The paper establishes a potential-theoretic framework showing that level-crossing measures arise from the Laplacian of normalized logarithmic discriminant energies.
• For complex i.i.d. pencils, it proves a conditional convergence to the Fubini–Study uniform law on \(\mathbb{CP}^1\), assuming uniform circular laws, small eigenvalue repulsion, and logarithmic tail control.
• For real and Hermitian pencils, it identifies major obstacles and conditional limits, including pseudocovariance effects, possible concentration near \(\mathbb{RP}^1\), and elliptic-ensemble densities away from the real axis.

The level-crossing problem for random matrix pencils

This paper studies the asymptotic distribution of level crossings of a random one-parameter spectral family $A_n + \lambda B_n$, where $\lambda \in \mathbb{C}$ and $A_n, B_n$ are independent random matrices. A level crossing is a value of $\lambda$ at which the pencil acquires a multiple eigenvalue; equivalently, it is a zero of the discriminant $\Delta_n(\lambda)$ of the characteristic polynomial in $t$ of $A_n + \lambda B_n - tI$. Since $\deg_\lambda \Delta_n = n(n-1)$, there are $n(n-1)$ crossings (with multiplicity), and the empirical level-crossing measure is

$$\mu_n = \frac{1}{2\pi n(n-1)}\Delta_\lambda \log|\Delta_n(\lambda)|$$

by the Poincaré–Lelong formula. The paper, which continues prior work on Gaussian pencils (2604.25785), asks whether the exact finite-$\lambda \in \mathbb{C}$0 uniform law known for complex Gaussian pencils persists asymptotically beyond the Gaussian setting — positioning itself as the level-crossing analogue of Girko's circular law and Wigner's semicircle law.

The method throughout is potential-theoretic: one writes the normalized log-discriminant as a deterministic scaling term plus the pairwise logarithmic energy of the eigenvalues of the normalized matrix $\lambda \in \mathbb{C}$1, proves convergence of that energy to a deterministic limit, and applies $\lambda \in \mathbb{C}$2 distributionally. Convergence is necessarily in $\lambda \in \mathbb{C}$3 in probability rather than locally uniformly, because $\lambda \in \mathbb{C}$4 has logarithmic poles at the crossings themselves.

Exact Gaussian results and the conditional circular law

For complex Ginibre-type ensembles, the starting point is an exact identity: if $\lambda \in \mathbb{C}$5 are drawn independently from the generalized complex Gaussian ensemble $\lambda \in \mathbb{C}$6, the level crossings are distributed with density $\lambda \in \mathbb{C}$7 — the Fubini–Study measure on $\lambda \in \mathbb{C}$8 — independently of both $\lambda \in \mathbb{C}$9 and $A_n, B_n$0. This holds more generally for any complex linear subspace $A_n, B_n$1 adjusted to the natural $A_n, B_n$2-action $A_n, B_n$3, covering symmetric, Toeplitz, band, and diagonal complex Gaussian matrices.

The main result for non-Gaussian entries is a conditional asymptotic uniform law. Under three hypotheses on the eigenvalue process of $A_n, B_n$4 — a uniform circular law (UCL), a small-repulsion condition (SR) controlling the contribution of near-coincident eigenvalue pairs to the log-energy, and a logarithmic tail condition (LT) — the empirical level-crossing measure converges weakly in probability to the uniform measure on $A_n, B_n$5. The proof truncates the log kernel two-sidedly, uses (UCL) plus the diagonal correction between the two-point measure $A_n, B_n$6 and $A_n, B_n$7, removes the truncations via (SR) and (LT), and obtains

$A_n, B_n$8

whose Laplacian yields exactly the Fubini–Study density.

The paper is explicit about the status of these inputs. (UCL) follows from uniform Girko hermitization estimates available via local circular-law technology (Bourgade–Yau–Yin); (SR) reduces, via a dyadic decomposition and a uniform two-point Wegner estimate, to local eigenvalue repulsion. For complex Ginibre pencils, fixed-parameter (SR) is proved from the determinantal kernel bound $A_n, B_n$9, but the compact-uniform version requires an additional modulus-of-continuity estimate in $\lambda$0 that is not established here. Outside the Gaussian case, (SR) is presented as a universality prediction, not a theorem.

The real case and the pseudocovariance reduction

Real pencils introduce a genuinely new phenomenon: by conjugation symmetry, level crossings may lie on or near $\lambda$1, so a singular component in the limit cannot be excluded a priori. Two structural results address this. First, a no-concentration theorem shows that if the expected number of crossings within spherical distance $\lambda$2 of $\lambda$3 is $\lambda$4 uniformly as $\lambda$5, then every subsequential weak limit assigns zero mass to $\lambda$6; the weaker condition $\lambda$7 only forces vanishing finite-$\lambda$8 mass on the real line itself. Supporting evidence comes from the fact that real generalized eigenvalues number only $\lambda$9 in expectation, but the authors do not verify the full no-concentration hypothesis for general real i.i.d. pencils.

Second, for GOE pencils the paper achieves a sharp reduction. Normalizing $\Delta_n(\lambda)$0 produces centered complex Gaussians whose entire $\Delta_n(\lambda)$1-dependence enters through the pseudocovariance $\Delta_n(\lambda)$2, or equivalently $\Delta_n(\lambda)$3 where $\Delta_n(\lambda)$4 is the height coordinate on the sphere. Consequently the expected normalized log-discriminant has the form $\Delta_n(\lambda)$5, and the GOE uniformity conjecture of the earlier work collapses to a single analytic assertion: the pseudocovariance does not contribute to the leading $\Delta_n(\lambda)$6-term of the log-discriminant. If this pseudocovariance-independence conjecture holds, the limiting measure is uniform on $\Delta_n(\lambda)$7. The special point $\Delta_n(\lambda)$8, where $\Delta_n(\lambda)$9 and circular symmetry holds, suggests a Gaussian interpolation route. Notably, the naive statement fails for real symmetric matrices under arbitrary diagonal rescaling — the resulting branch-point distribution is not radial — although numerics suggest uniformity survives for diagonal scaling parameters in $t$0, with the location of the phase transition left open.

For general real i.i.d. matrices, the uniform law is stated as a conjecture, consistent with numerical evidence for $t$1 at sizes up to $t$2, together with a refined conjecture allowing $t$3-invariant corrections of the form $t$4 for structured real ensembles.

Hermitian pencils and the elliptic energy

The Hermitian case is qualitatively different: the limit is not expected to be uniform. Already for $t$5 the density is $t$6, which vanishes on the real axis and equals $t$7 in spherical coordinates. The key observation is that for GUE pencils the normalized matrix $t$8 is a Gaussian elliptic ensemble with parameter $t$9, so the problem reduces to the logarithmic energy $A_n + \lambda B_n - tI$0 of the elliptic law. Under a uniform log-energy convergence assumption for the elliptic ensemble away from the equator $A_n + \lambda B_n - tI$1 — motivated by determinantal kernel bounds and quadratic repulsion, though stated as an assumption because weak convergence alone does not control the logarithmic singularity — the paper proves that the GUE level-crossing measures converge locally on $A_n + \lambda B_n - tI$2 to

$A_n + \lambda B_n - tI$3

This establishes the proposed Wigner-type limiting picture off the real axis for the Gaussian case. A conditional Hermitian Wigner universality theorem extends the conclusion to general Wigner ensembles given uniform elliptic-law convergence and uniform pair-energy convergence, plus a no-concentration condition near $A_n + \lambda B_n - tI$4 for the global statement. The remaining open analytic difficulty is uniform control as $A_n + \lambda B_n - tI$5, where the elliptic ensemble degenerates to the Hermitian regime.

Limitations and open questions

The paper's results are largely conditional outside the exactly solvable finite-dimensional Gaussian cases. Specifically: the compact-uniform small-repulsion input (SR) is unproved even for complex i.i.d. entries and is delicate for discrete entries; the no-concentration condition near $A_n + \lambda B_n - tI$6 is assumed rather than verified for general real ensembles; the uniform elliptic log-energy assumption underlying the Hermitian theorem is justified heuristically from determinantal kernels but not proven in the required uniform form; and the pseudocovariance-independence conjecture for GOE pencils remains open, with Gaussian interpolation at $A_n + \lambda B_n - tI$7 proposed as a possible approach. The phase transition for diagonal rescaling of real symmetric pencils is also unresolved.

Conclusion

The paper organizes the study of level crossings in random matrix pencils around a potential-theoretic framework in which the limiting crossing measure is the Laplacian of a deterministic potential built from the logarithmic energy of the associated eigenvalue ensemble. It proves a conditional Girko-type uniform law for complex i.i.d. pencils, reduces the real symmetric case to a single pseudocovariance-independence assertion, rules out singular mass on $A_n + \lambda B_n - tI$8 under explicit non-concentration hypotheses, and derives the Wigner-analogue limiting density for Hermitian pencils off the real axis via the elliptic ensemble. The remaining gaps are precisely identified: uniform small-spacing control in the pencil parameter, concentration control near the real projective line, and the leading-order effect of the pseudocovariance in the real Gaussian case.