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arxiv: 2512.19441 · v2 · submitted 2025-12-22 · 🧮 math.PR · math-ph· math.FA· math.MP

Fourier dimension of imaginary Gaussian multiplicative chaos

Benjamin Bonnefont , Hermanni Rajam\"aki , Vincent Vargas This is my paper

Pith reviewed 2026-05-16 20:26 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.FAmath.MP
keywords imaginary gaussian multiplicative chaosfourier dimensionlog-correlated fieldssubcritical phasefourier asymptoticswhite noise convergencesobolev regularityjack polynomials
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The pith

Imaginary Gaussian multiplicative chaos has Fourier dimension exactly 1 minus beta squared almost surely for beta between 0 and 1.

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

The paper shows that the imaginary Gaussian multiplicative chaos, formally exp of i beta times a log-correlated Gaussian field on the circle, has Fourier coefficients whose squared magnitudes decay polynomially with a precise optimal exponent. That exponent equals 1 minus beta squared almost surely in the subcritical range. This controls the high-frequency regularity of the chaos and determines how its Fourier modes behave at large frequencies. The result extends from the exact logarithmic covariance to a broad class where the covariance differs by a regular function.

Core claim

In the subcritical phase beta in (0,1), the Fourier dimension of M_i beta, defined via the optimal polynomial decay exponent of the squared Fourier coefficients, equals 1 minus beta squared almost surely. For the exact log-correlated field the chaos fails to lie in the critical Sobolev space H to the power of minus beta squared over 2, the rescaled coefficients converge in law to isotropic complex Gaussians, consecutive coefficients converge jointly to independent copies, and the rescaled and modulated chaos converges in negative Sobolev spaces to a complex white noise with explicit intensity kappa of beta.

What carries the argument

Moment identities obtained from Coulomb-gas integrals together with Jack-polynomial expansions, whose asymptotics are controlled by partitions with large gaps where the Pieri coefficients simplify to explicit leading terms.

If this is right

  • The chaos lies outside the critical Sobolev space H to the power minus beta squared over 2 almost surely.
  • The rescaled Fourier coefficients converge in law to independent isotropic complex Gaussians.
  • The modulated high-frequency content of the chaos converges in negative Sobolev spaces to a complex white noise whose intensity is given explicitly by a Gamma function and sine expression.
  • The same dimension and white-noise limit hold when the covariance is perturbed by a sufficiently regular function.

Where Pith is reading between the lines

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

  • The white-noise limit for high frequencies suggests that linear statistics of the chaos at fine scales behave like those of Gaussian white noise, which could be tested on other multiplicative chaos constructions.
  • The explicit intensity formula for the limiting noise may allow direct computation of the variance of integrals against test functions at high frequencies.
  • The result supplies a concrete rate that can be used to calibrate numerical schemes for sampling imaginary chaos or studying its multifractal spectrum.

Load-bearing premise

The covariance of the underlying log-correlated field differs from the exact logarithmic kernel by a sufficiently regular function, and the moment identities plus Jack-polynomial asymptotics extend to this wider class.

What would settle it

Numerical computation of the empirical decay rate of the squared Fourier coefficients for large frequencies in a discretized realization of the chaos at a fixed beta such as 0.5, checking whether the observed exponent is statistically consistent with 0.75.

Figures

Figures reproduced from arXiv: 2512.19441 by Benjamin Bonnefont, Hermanni Rajam\"aki, Vincent Vargas.

Figure 1
Figure 1. Figure 1: An example of partition ν = λ + σ obtained by adding σ of shape N1 = 3 and N2 = 2. The two following lemmas give the estimates of the Pieri coefficients in the regime where the spacings between the parts of λ become large (3.4) gap(λ) −→ +∞. We call this regime the large gap regime [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Young diagram of a partition λ with aλ(s) = 5 and lλ(s) = 2. If λ ⊂ µ (i.e. λi ≤ µi for all i), the skew diagram µ/λ is the set-theoretic difference of their Young diagrams. A skew diagram µ/λ is a horizontal strip if it contains at most one cell in each column, and a vertical strip if it contains at most one cell in each row. µ/λ [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The skew diagram obtained from two partitions [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
read the original abstract

We study the high-frequency Fourier asymptotics of imaginary Gaussian multiplicative chaos on the unit circle, a complex-valued random distribution formally given by $\mathrm M_{\mathrm i\beta}=\exp(\mathrm i\beta X)$, where $X$ is a log-correlated Gaussian field. In the subcritical phase $\beta\in(0,1)$, we prove that its Fourier dimension, defined by the optimal polynomial decay exponent of $|\widehat{\mathrm M_{\mathrm i\beta}}(n)|^2$, is almost surely equal to $1-\beta^2$. This result holds for a broad class of log-correlated fields whose covariance differs from the exact logarithmic kernel by a sufficiently regular function. For the exactly log-correlated field on the circle, we obtain the following results. We prove that the chaos almost surely fails to belong to $H^{-\beta^2/2}(\mathbb T)$, the critical Sobolev space left open by previous regularity results. We further establish a central limit theorem: the rescaled coefficients $n^{(1-\beta^2)/2}\widehat{\mathrm M_{\mathrm i\beta}}(n)$ converge in law to an isotropic complex Gaussian random variable, and finitely many consecutive coefficients converge jointly to independent copies. The high-frequency content of $\mathrm M_{\mathrm i\beta}$ behaves as a white noise: $n^{(1-\beta^2)/2}e^{\mathrm ii n\theta}\mathrm M_{\mathrm i\beta}$ converges in $H^s(\mathbb T)$, $s<-1/2$, to a complex white noise with explicit intensity $\kappa(\beta)=\frac{1}{\pi}\Gamma(1-\beta^2)\sin\big(\frac{\pi\beta^2}{2}\big)$. The proof relies on moment identities obtained from Coulomb-gas integrals and Jack-polynomial expansions. Their asymptotic analysis is governed by partitions with large gaps, where the Pieri coefficients appearing in these expansions simplify, and the leading contribution becomes explicit.

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

1 major / 2 minor

Summary. The manuscript proves that the Fourier dimension of imaginary Gaussian multiplicative chaos M_{iβ} = exp(iβ X), where X is a log-correlated Gaussian field, equals 1 - β² almost surely in the subcritical regime β ∈ (0,1). This holds for a broad class of fields whose covariance is the logarithmic kernel plus a sufficiently regular perturbation. For the exact logarithmic kernel on the circle, the paper additionally shows that M_{iβ} fails to belong to the critical Sobolev space H^{-β²/2} almost surely, establishes a central limit theorem in which n^{(1-β²)/2} M̂_{iβ}(n) converges in law to an isotropic complex Gaussian, proves joint convergence for finitely many consecutive coefficients to independent copies, and shows that the rescaled high-frequency content converges in H^s (s < -1/2) to complex white noise with explicit intensity κ(β) = (1/π) Γ(1-β²) sin(π β² / 2). The proof uses Coulomb-gas moment identities and asymptotic analysis of Jack-polynomial expansions, with leading contributions from partitions having large gaps.

Significance. If the central claims hold, the work supplies a sharp, explicit description of the high-frequency Fourier decay of imaginary GMC, establishing white-noise behavior after optimal rescaling and furnishing an explicit intensity constant. The combinatorial approach via Jack polynomials and Pieri coefficients, which simplifies for large-gap partitions, is a technical strength that yields parameter-free leading asymptotics and connects GMC regularity to classical orthogonal-polynomial techniques.

major comments (1)
  1. [Abstract and moment-identity derivation] Abstract and the section deriving the moment identities: the claim that the Fourier-dimension result extends to covariances differing from the exact logarithmic kernel by a regular function R rests on an unverified transfer of the leading n^{-(1-β²)} scaling. The Coulomb-gas integrals and Jack expansions are derived for the pure log kernel; the perturbation alters the joint law of the field values, and no uniform error bound is supplied showing that the difference in the relevant Pieri coefficients or gap asymptotics remains o(1) in the exponent for the high-n regime. Without such control, the optimal polynomial decay exponent could shift.
minor comments (2)
  1. [White-noise convergence statement] The statement of the white-noise convergence in H^s for s < -1/2 would benefit from an explicit reference to the precise Sobolev norm used and a short remark on why the intensity κ(β) is independent of the particular regular perturbation.
  2. [Notation paragraph] Notation for the Fourier coefficients M̂_{iβ}(n) is introduced without an immediate reminder of the normalization convention (e.g., whether the circle is equipped with Lebesgue measure normalized to 1); a single clarifying sentence would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a point that requires clarification in the extension of our Fourier-dimension result to perturbed kernels. We address this major comment below and will incorporate additional justification in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and moment-identity derivation] Abstract and the section deriving the moment identities: the claim that the Fourier-dimension result extends to covariances differing from the exact logarithmic kernel by a regular function R rests on an unverified transfer of the leading n^{-(1-β²)} scaling. The Coulomb-gas integrals and Jack expansions are derived for the pure log kernel; the perturbation alters the joint law of the field values, and no uniform error bound is supplied showing that the difference in the relevant Pieri coefficients or gap asymptotics remains o(1) in the exponent for the high-n regime. Without such control, the optimal polynomial decay exponent could shift.

    Authors: We agree that the explicit Coulomb-gas integrals and Jack-polynomial expansions are derived for the exact logarithmic kernel, and that the manuscript does not currently supply uniform error bounds quantifying the effect of a regular perturbation R on the leading asymptotics. The extension to covariances of the form log + R (with R sufficiently smooth) is justified heuristically by the fact that such an R induces a continuous additive perturbation to the underlying Gaussian field; for the high-frequency moments governing the Fourier dimension, this perturbation contributes only a multiplicative factor whose logarithm remains bounded uniformly in the large-n regime, preserving the exponent 1-β². Nevertheless, to make the argument fully rigorous, we will revise the manuscript by adding a dedicated subsection (or appendix) that derives explicit error estimates. These estimates will show that, for partitions with large gaps, the difference in the relevant Pieri coefficients and gap asymptotics is O(n^{-δ}) for some δ>0 independent of the partition, which is negligible compared with the leading exponential term. With this control, the optimal polynomial decay exponent remains unchanged. We thank the referee for highlighting the need for this quantitative transfer. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives the Fourier dimension result 1-β² from moment identities obtained via Coulomb-gas integrals applied to the log-correlated field, followed by asymptotic analysis of Jack-polynomial expansions where partitions with large gaps make Pieri coefficients explicit and leading terms computable. These steps use standard external combinatorial identities and do not reduce the target exponent to a fitted parameter, self-definition, or load-bearing self-citation by construction. The extension to covariances perturbed by regular functions is asserted to preserve the leading scaling, but this is an analytic claim rather than a definitional reduction; no equation equates the claimed decay directly to an input quantity. The argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of log-correlated Gaussian fields and on the analytic continuation of Coulomb-gas integrals; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The covariance of the underlying Gaussian field differs from the exact logarithmic kernel by a sufficiently regular function.
    Invoked to extend the result beyond the exactly log-correlated case.
  • standard math Moment identities for the chaos can be obtained from Coulomb-gas integrals and Jack-polynomial expansions.
    Stated as the basis for the asymptotic analysis.

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Reference graph

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