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arxiv: 2508.04386 · v5 · submitted 2025-08-06 · 🧮 math.PR · math-ph· math.MP

Universality for fluctuations of counting statistics of random normal matrices

J. Marzo , L. D. Molag , J. Ortega-Cerd\`a This is my paper

Pith reviewed 2026-05-19 00:32 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords random normal matricescounting statisticseigenvalue fluctuationsdeterminantal point processdropletvariance asymptoticscorrelation kernelharmonic measure
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0 comments X

The pith

For Borel sets strictly inside the droplet, the variance of the number of eigenvalues in random normal matrices scales as sqrt(n) times a boundary integral of sqrt(Delta Q).

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

This paper proves a limiting formula for the variance of the eigenvalue counting function N_A^n in random normal matrices with potential Q. When A lies strictly inside the droplet, the scaled variance converges to an explicit constant times the integral of sqrt(Delta Q) along the measure-theoretic boundary of A. The proof relies on strengthened boundary asymptotics for the correlation kernel. The authors also treat the case of microscopic dilations around the droplet boundary, replacing the Hausdorff measure by harmonic measure at infinity.

Core claim

When A is a Borel set strictly inside the droplet, lim (n to infinity) of (1/sqrt(n)) Var(N_A^(n)) equals (1/(2 pi sqrt(pi))) times the integral over partial_* A of sqrt(Delta Q(z)) d H^1(z). For A a microscopic dilation of the droplet the same limit holds after replacing d H^1 by the harmonic measure at infinity associated to the exterior of the droplet. Both statements are obtained by strengthening the boundary asymptotics of the correlation kernel beyond the results of Hedenmalm-Wennman and Ameur-Cronvall.

What carries the argument

The strengthened boundary asymptotics of the correlation kernel near the droplet, which permit explicit computation of the double integral that yields the variance.

If this is right

  • The variance is determined solely by the geometry of the boundary of A and the local value of sqrt(Delta Q).
  • The result holds for general potentials obeying the mild conditions, not merely quadratic ones.
  • When the test set touches the droplet boundary the variance formula switches to one involving harmonic measure from infinity.
  • The same kernel asymptotics control both bulk interior fluctuations and near-boundary fluctuations.

Where Pith is reading between the lines

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

  • The boundary integral form suggests that interior counting fluctuations are ultimately controlled by local density gradients even far from the edge.
  • Numerical checks on non-Gaussian potentials such as quartic ones could confirm the universality of the constant prefactor.
  • The same technique may extend to other determinantal processes whose kernels admit comparable boundary expansions.

Load-bearing premise

The potential Q must satisfy mild conditions that force the eigenvalues to accumulate on a compact droplet and that allow the correlation kernel to admit the required strengthened boundary asymptotics.

What would settle it

Direct numerical computation, for the Gaussian potential and a disk A well inside the unit disk, of (1/sqrt(n)) Var(N_A^n) for increasing n, checking convergence to the explicit integral value (1/(2 pi sqrt(pi))) times 2 pi r sqrt(2) or its analogue.

read the original abstract

We consider the fluctuations of the number of eigenvalues of $n\times n$ random normal matrices depending on a potential $Q$ in a given set $A$. These eigenvalues are known to form a determinantal point process, and are known to accumulate on a compact set called the droplet under mild conditions on $Q$. When $A$ is a Borel set strictly inside the droplet, we show that the variance of the number of eigenvalues $N_A^{(n)}$ in $A$ has a limiting behavior given by \begin{align*} \lim_{n\to\infty} \frac1{\sqrt n}\operatorname{Var } N_A^{(n)} = \frac{1}{2\pi\sqrt\pi}\int_{\partial_* A} \sqrt{\Delta Q(z)} \, d\mathcal H^1(z), \end{align*} where $\partial_* A$ is the measure theoretic boundary of $A$, $d\mathcal H^1(z)$ denotes the one-dimensional Hausdorff measure, and $\Delta = \partial_z \overline{\partial_z}$. We also consider the case where $A$ is a microscopic dilation of the droplet and fully generalize a result by Akemann, Byun and Ebke for arbitrary potentials. In this result $d\mathcal H^1(z)$ is replaced by the harmonic measure at $\infty$ associated with the exterior of the droplet. This second result is proved by strengthening results due to Hedenmalm-Wennman and Ameur-Cronvall on the asymptotic behavior of the associated correlation kernel near the droplet boundary.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes universality for fluctuations of the counting statistics N_A^{(n)} for the determinantal point process of eigenvalues of n x n random normal matrices with potential Q. Under mild conditions ensuring a compact droplet with positive interior density, it proves that for Borel sets A strictly inside the droplet, lim_{n→∞} (1/√n) Var(N_A^{(n)}) equals (1/(2π √π)) ∫_{∂_* A} √(ΔQ(z)) dℋ¹(z), where ∂_* A is the measure-theoretic boundary. It also generalizes the Akemann-Byun-Ebke result to arbitrary potentials for microscopic dilations of the droplet boundary, by strengthening the correlation-kernel asymptotics of Hedenmalm-Wennman and Ameur-Cronvall near the edge.

Significance. If the claims hold, the work supplies explicit, universal surface-integral expressions for variance fluctuations of linear statistics in both the bulk and near the droplet boundary for normal-matrix ensembles. The bulk result derives a parameter-free surface term from standard determinantal variance formulas combined with microscopic kernel scaling; the boundary result extends prior work via strengthened kernel asymptotics. These are falsifiable predictions that could be checked numerically and may transfer to other determinantal processes with varying density.

major comments (2)
  1. [Abstract and §1] Abstract and §1: the bulk limit is stated to follow from the determinantal variance formula plus bulk scaling asymptotics of the kernel, but the manuscript should include an explicit one-paragraph derivation showing how the volume terms cancel at order n and the remaining surface contribution produces precisely the prefactor 1/(2π √π) after rescaling by the local density factor √(ΔQ).
  2. [Boundary result (abstract)] The boundary result in the abstract invokes strengthened versions of the Hedenmalm-Wennman and Ameur-Cronvall kernel asymptotics; the precise error bounds and the range of validity of the strengthening must be stated explicitly (including the microscopic dilation scale) so that it is clear they suffice to control the variance integral up to o(√n).
minor comments (2)
  1. [Abstract] Clarify the notation Δ = ∂_z ∂_¯z at first use and confirm it coincides with the standard Laplacian (up to the usual factor of 4).
  2. [Introduction] Add a short sentence in the introduction recalling the exact statement of the Akemann-Byun-Ebke theorem being generalized.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major comments point by point below, and we will incorporate the suggested clarifications in the revised version.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the bulk limit is stated to follow from the determinantal variance formula plus bulk scaling asymptotics of the kernel, but the manuscript should include an explicit one-paragraph derivation showing how the volume terms cancel at order n and the remaining surface contribution produces precisely the prefactor 1/(2π √π) after rescaling by the local density factor √(ΔQ).

    Authors: We agree that providing an explicit derivation would improve the clarity of the argument for the bulk limit. In the revised manuscript, we will add a one-paragraph derivation in Section 1, immediately after the statement of the main bulk theorem. This paragraph will detail how the volume terms cancel at order n in the determinantal variance formula, leaving a surface contribution that, after rescaling by the local density factor √(ΔQ), produces the prefactor 1/(2π √π). revision: yes

  2. Referee: [Boundary result (abstract)] The boundary result in the abstract invokes strengthened versions of the Hedenmalm-Wennman and Ameur-Cronvall kernel asymptotics; the precise error bounds and the range of validity of the strengthening must be stated explicitly (including the microscopic dilation scale) so that it is clear they suffice to control the variance integral up to o(√n).

    Authors: We thank the referee for pointing this out. In the revised manuscript, we will explicitly state the precise error bounds and the range of validity for the strengthened kernel asymptotics from Hedenmalm-Wennman and Ameur-Cronvall, including the specific microscopic dilation scale used. These bounds will be verified to ensure that the variance integral is controlled up to o(√n), thereby justifying the boundary result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external kernel results

full rationale

The central variance limit for Borel sets A strictly inside the droplet is obtained from the standard variance formula for determinantal processes together with bulk scaling asymptotics of the correlation kernel; the surface integral coefficient arises directly from the local microscopic kernel after conformal adjustment by sqrt(Delta Q). The strengthened boundary asymptotics are taken from independent prior works (Hedenmalm-Wennman, Ameur-Cronvall) and used only for the separate microscopic-dilation case, which generalizes an external result of Akemann-Byun-Ebke. No step reduces by definition or by fitting to the paper's own inputs, no self-citation is load-bearing, and the mild conditions on Q are the usual ones guaranteeing a compact droplet. The derivation chain therefore stands on independent external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the existence of the droplet under mild conditions on Q, the determinantal structure of the eigenvalue process, and the availability of boundary asymptotics for the correlation kernel that are strengthened in the paper.

axioms (2)
  • domain assumption Eigenvalues of the random normal matrix form a determinantal point process with a known correlation kernel.
    Invoked in the abstract as the starting point for counting statistics.
  • domain assumption Under mild conditions on Q the eigenvalues accumulate on a compact droplet.
    Stated explicitly as the setting in which the interior and edge results hold.

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