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arxiv: 2606.22940 · v1 · pith:OLI7OTXTnew · submitted 2026-06-22 · 🧮 math.PR

From smooth to discontinuous kernels: a variance transfer principle for hyperuniform processes

Rapha\"el Lachi\`eze-Rey (MAP5 - UMR 8145 , MATHNET) This is my paper

Pith reviewed 2026-06-26 07:58 UTC · model grok-4.3

classification 🧮 math.PR
keywords variance transfer principlehyperuniformityCoulomb gasesGirko matricesnumber variancelinear statisticsparticle systemsrandom matrices
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The pith

A transfer principle moves polynomial variance decay from smooth kernels to counting kernels for particle systems.

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

The paper develops a method to transfer variance bounds on linear statistics from smooth test functions to the discontinuous indicator functions that count particles inside regions. If the variance for smooth kernels decays polynomially with exponent a relative to the independent case, then the number variance over balls decays with exponent min(1,a), including a log factor when a is 1. This transfer applies to non-periodic systems over certain scale ranges and is used to establish hyperuniformity properties for random matrix eigenvalues and Coulomb gases. The results show optimal surface-order fluctuations in two dimensions and linear-order in three dimensions, which in turn imply finite interaction energy and controlled Wasserstein distance to the uniform measure.

Core claim

We give a transfer principle for the fluctuations of linear statistics of finite particle systems around Lebesgue measure: if for a smooth kernel the variance decays polynomially with some exponent a compared to independent particles, then the number variance over balls centred at almost every point decays with exponent min(1,a) times a log term if a=1, over a possibly reduced range of scales for non-periodic systems. Applications yield that 2D-Coulomb gases are 2-hyperuniform and 3D ones are 1-hyperuniform.

What carries the argument

The variance transfer principle relating smooth kernel fluctuations to number variance for indicator functions over balls.

If this is right

  • Optimal perimeter-like number variance for eigenvalues of random Girko matrices on microscopic and some mesoscopic scales after local averaging.
  • 2D Coulomb gases exhibit surface order number variance, which is optimal.
  • 3D Coulomb gases exhibit linear order number variance.
  • In 2D, the hyperuniformity implies finite Coulomb energy and finite Wasserstein distance to Lebesgue measure.

Where Pith is reading between the lines

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

  • The principle may extend to other models of interacting particles where smooth variance decay is established.
  • It could connect hyperuniformity results across different dimensions and interaction potentials.
  • Local averaging might be removable in periodic cases or with stronger assumptions.

Load-bearing premise

The base polynomial variance decay for smooth kernels is already known in the Girko matrix and Coulomb gas settings and transfers directly without further loss.

What would settle it

An explicit counterexample particle system where smooth kernel variance decays with exponent a greater than 1 but the number variance over balls decays slower than the surface area would disprove the transfer principle.

read the original abstract

We give a transfer principle for the fluctuations of linear statistics of finite particle systems around Lebesgue measure: if for a smooth kernel the variance decays polynomially with some exponent a compared to independent (non-interacting) particles, then the number variance over balls centred at almost every point decays with exponent min(1,a) times a log term if a=1, over a possibly reduced range of scales for non-periodic systems. We apply this principle to eigenvalues of random N*N Girko matrices, leveraging results of Cipolloni et al., and obtain the optimal perimeter-like number variance, on the microscopic and some mesoscopic scales range, after local averaging. We also apply the results to Coulomb gases, by transferring the results of Serfaty: we prove that 2D-Coulomb gases are 2-hyperuniform, i.e. they have surface order number variance, which is optimal, and that 3D Coulomb gases are 1-hyperuniform. In 2D, it allows to prove finite Coulomb energy and Wasserstein distance to Lebesgue measure.

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

3 major / 2 minor

Summary. The paper establishes a variance transfer principle for fluctuations of linear statistics in finite particle systems: polynomial variance decay of exponent a for smooth kernels implies number variance decay of exponent min(1,a) (with an extra log factor when a=1) for the discontinuous counting kernel, at almost every center and over a possibly reduced range of scales when the system is non-periodic. The principle is applied to Girko matrices (via Cipolloni et al.) to obtain optimal perimeter-like number variance on microscopic and some mesoscopic scales after local averaging, and to 2D/3D Coulomb gases (via Serfaty) to conclude that 2D gases are 2-hyperuniform (surface-order variance, optimal) and 3D gases are 1-hyperuniform; the 2D case is further used to deduce finite Coulomb energy and Wasserstein convergence to Lebesgue measure.

Significance. If the transfer holds with controlled scale loss, the principle supplies a systematic way to upgrade existing smooth-kernel variance bounds to hyperuniformity statements for counting statistics, directly yielding optimality results for Coulomb gases and their global consequences. The approach is independent of the base derivations and credits the external inputs from Cipolloni et al. and Serfaty explicitly.

major comments (3)
  1. [§3, §5] §3 (transfer principle statement) and §5 (Coulomb-gas application): the abstract and the optimality claims for 2-hyperuniformity assert surface-order number variance, yet the principle only guarantees the transferred exponent over a 'possibly reduced range of scales for non-periodic systems.' No explicit quantification of the reduced range appears, so it is unclear whether the mesoscopic window needed for the global consequences (finite Coulomb energy, Wasserstein distance) remains covered after transfer.
  2. [§4, §5] §4 (Girko application) and §5: the local-averaging step that restores the full microscopic-to-mesoscopic range for Girko matrices is not shown to carry over to the non-periodic Coulomb setting; if the averaging is unavailable there, the 1-hyperuniformity claim for 3D gases rests on an unverified scale interval.
  3. [Theorem 1.1] Theorem 1.1 (main transfer statement): the almost-everywhere centering and the precise dependence of the scale reduction on the modulus of continuity of the density are stated, but the proof sketch does not indicate how the reduction factor is bounded in terms of the base exponent a when a>1; this affects whether the min(1,a) exponent is achieved without further loss.
minor comments (2)
  1. [§2] Notation for the smooth kernel K_ε and the counting kernel 1_B is introduced without an explicit comparison table; a short display equating the two variance expressions would improve readability.
  2. [Theorem 1.1] The log factor when a=1 is mentioned in the abstract but its precise form (log N or log(1/r)) is not restated in the theorem statement; consistency would help.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Below we respond point-by-point to the major comments and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§3, §5] §3 (transfer principle statement) and §5 (Coulomb-gas application): the abstract and the optimality claims for 2-hyperuniformity assert surface-order number variance, yet the principle only guarantees the transferred exponent over a 'possibly reduced range of scales for non-periodic systems.' No explicit quantification of the reduced range appears, so it is unclear whether the mesoscopic window needed for the global consequences (finite Coulomb energy, Wasserstein distance) remains covered after transfer.

    Authors: We agree that an explicit quantification of the scale reduction is needed for clarity. The reduction factor is controlled by the modulus of continuity of the limiting density and by the base exponent a; for the Coulomb-gas applications the input bounds of Serfaty extend over a range wide enough that the transferred scales still cover the mesoscopic window required for the global consequences. In the revision we will state the precise dependence of the reduction on the modulus of continuity inside Theorem 1.1 and add a remark in §5 confirming coverage of the needed scales. revision: yes

  2. Referee: [§4, §5] §4 (Girko application) and §5: the local-averaging step that restores the full microscopic-to-mesoscopic range for Girko matrices is not shown to carry over to the non-periodic Coulomb setting; if the averaging is unavailable there, the 1-hyperuniformity claim for 3D gases rests on an unverified scale interval.

    Authors: Local averaging is used only for the Girko matrices to recover the full range. For Coulomb gases we do not invoke averaging; the transfer is applied directly. The base smooth-kernel estimates of Serfaty are sufficiently uniform that the reduced interval still yields the claimed 1-hyperuniformity for 3D gases. We will add an explicit verification of the resulting scale interval in the revised §5. revision: yes

  3. Referee: [Theorem 1.1] Theorem 1.1 (main transfer statement): the almost-everywhere centering and the precise dependence of the scale reduction on the modulus of continuity of the density are stated, but the proof sketch does not indicate how the reduction factor is bounded in terms of the base exponent a when a>1; this affects whether the min(1,a) exponent is achieved without further loss.

    Authors: The full proof in §3 derives the bound on the reduction factor, which for a>1 is mild enough that the exponent min(1,a)=1 is attained without extra loss beyond the stated reduction. To make this transparent we will expand the proof sketch in the revision to display explicitly the dependence of the reduction on a when a>1. revision: yes

Circularity Check

0 steps flagged

No circularity: transfer principle is independently derived and applied to external results

full rationale

The paper's core contribution is a new transfer principle mapping smooth-kernel variance decay (exponent a) to counting-kernel number variance decay (min(1,a), with log for a=1) at a.e. centers, possibly over reduced scales for non-periodic systems. This principle is stated as proven in general and then applied to pre-existing external results (Cipolloni et al. for Girko matrices; Serfaty for Coulomb gases). No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear. The derivation chain is self-contained against the cited external benchmarks, which are independent of the present work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters or invented entities; it relies on standard mathematical assumptions about kernels, linear statistics, and the validity of the cited base-case variance decays.

axioms (1)
  • domain assumption Polynomial variance decay holds for the smooth-kernel linear statistics in the Girko and Coulomb settings as established by the cited prior works.
    The transfer principle takes these decays as input and produces the discontinuous-kernel conclusions.

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