Exact values of Fourier dimensions of Gaussian multiplicative chaos on high dimensional torus

Yanqi Qiu; Yukun Chen; Zhaofeng Lin

REVIEW 2 major objections 4 minor 19 references

Exact values of Fourier dimensions of Gaussian multiplicative chaos on high dimensional torus

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that on the d-dimensional torus there exists a log-correlated Gaussian field whose sub-critical Gaussian multiplicative chaos measure has Fourier dimension exactly d−γ² for small γ and (√(2d)−γ)² for large γ, for every…

desk verdict Strong paper with a real but likely patchable gap: derivative estimates for the smoothing construction are stated only up to order d, while the high-frequency argument needs order 2d. read the letter →

arxiv 2507.23494 v1 pith:EZPKRF4W submitted 2025-07-31 math.PR math-phmath.FAmath.MP

Yukun Chen , Zhaofeng Lin , Yanqi Qiu This is my paper · ORCID

classification math.PRmath-phmath.FAmath.MP MSC 60G5742A6146B0960G46

keywords GaussianmultiplicativechaosFourierdimensioncorrelationlog-correlatedfieldFourier–Lebesguespacevector-valuedmartingalemethodsmoothpartitionofunitytorus

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper determines the exact Fourier dimension of a Gaussian multiplicative chaos (GMC) measure on the d-dimensional torus for every integer d≥1, a problem previously solved only in low dimensions or with partial parameter ranges. Fourier dimension measures how fast the Fourier coefficients of a measure decay, and the paper shows that for a suitably chosen log-correlated Gaussian field the GMC measure's Fourier dimension coincides exactly with its correlation dimension, given by $D_{\gamma,d}=d-\gamma^2$ when $\gamma<\sqrt{2d}/2$ and $D_{\gamma,d}=(\sqrt{2d}-\gamma)^2$ otherwise. The advance is a new construction of the Gaussian field as a sum of smooth, practically independent processes, which lets the proof localize the measure at dyadic scales and apply repeated integration by parts without boundary artifacts. Combined with a global martingale estimate, this yields matching upper and lower bounds. If correct, the result closes the dimensional gap for torus GMC and provides a template for similar Fourier-type decay statements on other boundary-free spaces.

What carries the argument

The proof runs through three mechanisms. First, a field decomposition: the log-correlated kernel is written as the sum of kernels of independent stationary Gaussian processes whose paths are $C^\infty$ and whose covariance has support within distance $3\cdot 2^{-j}$; this is obtained by mollifying a dyadic field with carefully chosen smoothing parameters. Second, a smooth partition of unity adapted to dyadic cubes, which avoids the spectral leakage that sharp cutoffs would introduce. Third, a local estimate controlling the $\ell^q$ Fourier–Lebesgue norm of each localized piece: repeated integration by parts through Green's identity uses the Laplacian eigenfunction equation $\Delta z^n=-4\pi^2|n|^2 z^n$ to trade high frequency $|n|$ for derivatives of the smooth random factors, and the martingale type inequality for vector-valued martingales is applied to sum the localized pieces globally. The precision comes from matching the derivative cost $(k^2 2^k)^A$ with the dyadic scale factor.

What would settle it

Compute, for the processes built in Proposition 3.1, $\mathbb{E}[|D^\alpha \psi_j(z)|^p]$ for a fixed multi-index $\alpha$ with $|\alpha|=2d$ and check whether it stays bounded by a constant times $j^{2|\alpha|p}2^{j|\alpha|p}$. If it grows faster, Proposition 4.4 and hence the lower bound in Theorem 1.1 fail.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for every integer $d\ge 1$, there exists a centered log-correlated Gaussian field on $\mathbb{T}^d$ with covariance $$K(z,w)=\log^+\left(\frac{1}{d_{\mathbb{T}^d}(z,w)}\right)+g(z\bar w),$$ where $g$ is bounded continuous, such that for every sub-critical $\gamma\in(0,\sqrt{2d})$, the GMC measure $\mathrm{GMC}^K_\gamma$ almost surely has Fourier dimension $D_{\gamma,d}$, with $D_{\gamma,d}=d-\gamma^2$ for $0<\gamma<\sqrt{2d}/2$ and $D_{\gamma,d}=(\sqrt{2d}-\gamma)^2$ for $\sqrt{2d}/2\le\gamma<\sqrt{2d}$. The proof establishes a matching lower bound by showing that the GMC measure almost surely belongs to a weighted Fourier–Lebesgue space, which forces the desired polynomial decay of its Fourier coefficients.

Load-bearing premise

The paper assumes, through property (P4), uniform $p$-th moment bounds for derivatives of the smooth approximating processes up to order $d$, yet Lemma 4.11 needs the same control for derivatives up to order $2d$; if that higher-order estimate is not derivable from the construction, the repeated integration by parts is not fully justified.

Editorial extensions

If this is right

The GMC measure's Fourier dimension equals its correlation dimension $D_{\gamma,d}$ for the constructed field, confirming the general phenomenon on the torus.
All dimensions $d\ge 1$ are covered by one argument, removing the earlier low-dimensional restriction.
Sharp cutoffs on the unit cube create $|n|^{-1}$ boundary decay; using smooth truncation transfers the torus result to the unit cube.
The method replaces the Fourier basis by Laplace–Beltrami eigenfunctions and should yield analogous abstract Fourier-type decay statements on general compact boundary-free Riemannian manifolds, as the paper announces as forthcoming work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same smooth-decomposition scheme could be tested on other log-correlated fields, such as circular or spherical models, where the exact Fourier dimension is not yet settled; the torus construction indicates what derivative regularity the kernel decomposition must supply.
A quantitative version of the proof would give explicit constants in the bound on $\mathbb{E}[\|\mu_\infty\|^p_{FL^{\tau/2,q}}]$, yielding non-asymptotic Fourier-coefficient decay rates; this is not stated in the paper.
Because the upper bound $\dim_F\le\dim_2$ is general, the equality suggests that any log-correlated field on a boundary-free space with bounded continuous remainder should also have Fourier dimension equal to its correlation dimension; verifying this would be a natural next step.
The spectral-leakage argument suggests a rule of thumb: sharp spatial cutoffs degrade Fourier dimension by boundary terms, so exact Fourier dimensions should be formulated on boundary-free manifolds or with smooth windows.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take on arXiv:2507.23494. The paper does two genuinely new things: it builds a log-correlated Gaussian field on T^d whose decomposition into smooth processes has derivative bounds at the right dyadic scale, and it uses a smooth dyadic partition of unity to kill the spectral leakage that blocked the cube argument in d≥3. The multi-resolution estimate with Pisier's martingale type inequality is well organized, and the final optimization over ζ(p) lands exactly on D_{γ,d}. If the technical gap below is fixed, this is a solid resolution of the high-dimensional torus case for a constructed field.

The soft spot the referee will hit is real. Proposition 3.1(P4) controls derivatives of ψ_j only up to order d. Lemma 4.6 needs control up to order 2d because Lemma 4.11 differentiates the product X_φ^I up to A=2d, and the Faà di Bruno expansion in Lemma 4.6 can produce a block β of length 2d. The proof of Lemma 4.6 cites (P4) for arbitrary α, but the verification in §3.2 (Lemma 3.9) stops at |α|≤d. So the repeated integration by parts in Lemma 4.10 and the bound on the high-frequency piece in Proposition 4.4 are not justified as written. The gap looks patchable: the mollifier P_j has derivatives of all orders, and the L∞ bound scales like j^{2(d+|α|)}2^{j(d+|α|)}; multiplying by the support volume j^{-2d}2^{-jd} leaves the required j^{2|α|}2^{j|α|}. But the extension is not in the paper, and without it the proof is incomplete.

A smaller issue: the abstract says “we determine exact values of Fourier dimensions for GMC measures on the torus,” while Theorem 1.1 asserts existence of a field with a specified kernel. It’s the difference between “there exists a GMC with this property” and “for every GMC.” The title is fine, but the abstract should be aligned with the theorem.

On balance, this is a serious paper. The construction is reusable, the structure is coherent, and the flaw is a missing order-of-derivative estimate, not a conceptual error. I would send it to a good referee and ask them to verify whether Lemma 3.9 can be extended to all orders; likely it can.

For you: yes, worth a reading group slot once the patch is checked.

Referee Report

2 major / 4 minor

Summary. The paper constructs, for every integer d≥1, a centered log-correlated Gaussian field on the d-dimensional torus with covariance log^+(1/d_Td(z,w))+g(z\bar w) for a bounded continuous function g, and proves that for every subcritical γ∈(0,√(2d)) the associated GMC measure almost surely has Fourier dimension exactly D_{γ,d}=d−γ² for 0<γ<√(2d)/2 and D_{γ,d}=(√(2d)−γ)² for √(2d)/2≤γ<√(2d). The proof combines a new mollified decomposition of the field into smooth independent Gaussian processes (Proposition 3.1), a smooth dyadic partition of unity, repeated integration by parts via Green's identity to obtain sharp localized Fourier estimates (Proposition 4.4), and Pisier's martingale type inequality to sum these local estimates into a global Fourier–Lebesgue bound (Proposition 1.2). The upper bound dim_F≤dim_2=D_{γ,d} is quoted from prior work, so the main work is the matching lower bound.

Significance. If the proof is completed, the result resolves the open problem left in [LQT24, LQT25] for d≥3 and confirms the Garban–Vargas phenomenon—Fourier dimension equal to correlation dimension—for GMC measures on tori of every dimension. The paper contains several genuine contributions: a clean construction of a log-correlated Gaussian field whose decomposition has explicit derivative scale bounds, a smooth partition-of-unity localization that avoids spectral leakage from sharp cutoffs, and a transparent final optimization over the exponent p that yields exactly D_{γ,d}. The overall strategy is convincing and the martingale bookkeeping is coherent. However, the derivative estimates in §4 require a regularity order that the construction in §3 does not currently supply, and one key lemma statement is inconsistent with its proof; these issues need repair before the main theorem is fully established.

major comments (2)

[§3, Proposition 3.1(P4) and §4.2, Lemma 4.6] Property (P4), as stated in Proposition 3.1 and verified in Lemma 3.9, controls only derivatives of order |α|≤d, with the normalization j^{2|α|p}2^{j|α|p}. Lemma 4.6, however, is stated for arbitrary multi-indices α, and its proof via the multivariate Faà di Bruno formula applies (P4) to every block βℓ in a partition of α. Since Lemma 4.11 is later used with A=2d, a partition of a multi-index of length 2d can contain a block of length 2d, so the required estimate sup_j sup_z E[|D^β X_j(z)|^p]/(j^{4dp}2^{2jdp})<∞ for |β|=2d is not supplied by (3.1). This is load-bearing: Lemma 4.10 applies Δ^d, creating derivatives of order 2d, and Lemma 4.11 needs derivative control up to that order. The gap is probably repairable by extending (P4) to all orders using the extra room from the mollifier ε_j=j^{-2}2^{-j}, but that extension is not written in the manuscript and the proof of Proposition 4.4 is incomplete as it stands.
[§4.3, Lemma 4.11 and Eq. (4.6)] The statement of Lemma 4.11 bounds sup_{|α|=A} (E[|∂^α X^φ_I(t)|^p])^{1/p} by a factor involving ∏_{j=1}^k (E[|D^α X_j(z)|^p])^{1/p}, with the same multi-index α of length A on the right-hand side. The proof, however, concludes with ∏_{j=1}^k (E[|X_j(z)|^p])^{1/p}: the derivative factors E[|D^{α_j}X_j|^p] are bounded separately via Lemma 4.6 and absorbed into the normalization (k^2 2^k)^A, and the product of undifferentiated X_j terms is then bounded by sup_z. Consequently the displayed bound (4.6), which contains E[|D^α X_j|^p], cannot be combined with Lemma 4.5 to produce the factor 2^{kp(p−1)γ²/2} used in (4.2). The statement of Lemma 4.11 and the display (4.6) should refer to E[|X_j(z)|^p], matching the proof and the conclusion in §4.3.4.

minor comments (4)

[§4.1, Proposition 1.2] Proposition 1.2 states the equality E∥µ∞∥^p_{FL}=sup_{m≥1}E∥µm∥^p_{FL}, but the proof only establishes finiteness of the supremum. The inequality E∥µ∞∥^p≤sup_m E∥µm∥^p follows from Fatou's lemma and weak convergence, and this weaker statement is all that is needed for Theorem 1.1. Please either prove the asserted equality (for example by L^p convergence of the approximate densities) or reformulate the proposition with the inequality.
[§4.2, proof of Lemma 4.6] In the product estimate after the display 'By (P4) in Proposition 3.1', the exponent in the factor 2^{j|α|mℓpℓ} should be 2^{j|βℓ|mℓpℓ}; the displayed formula as written would not give the subsequent product 2^{j|α|p} after summing Pℓ mℓ|βℓ|=|α|.
[Lemma 2.3 and Appendix A] The support condition in Lemma 2.3(2) is written as E([−2^{−k},2^k]^d), which appears to be a typo for E([−2^{−k},2^{−k}]^d) in view of the construction in Appendix A; please correct the notation.
[General] There are several minor typos, including 'avarage' in §1.2.2 and 'Thereofre' in the proof of Lemma 4.11; a careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the exact Fourier dimension emerges from an explicit optimization, not from an assumed input; the main weakness is an unproved derivative-order extension, which is a correctness gap rather than circularity.

full rationale

The central value D_{γ,d} is not assumed as an input. In Section 4.1 the proof defines ζ(p) = 2d + γ^2 − (2d/p + pγ^2) and computes sup_{1≤p≤2} ζ(p), obtaining d−γ^2 in the first regime and (√(2d)−γ)^2 in the second; these numbers emerge from the variance limit log 2, the choice of q, and the convergence condition for the series in (4.2), so the lower-bound argument does not substitute the desired conclusion into its hypotheses. The upper bound invokes the known correlation-dimension formula, but the matching lower bound is proved independently through Proposition 1.2, Proposition 4.4, and the vector-valued martingale decoupling. The construction in Section 3 is explicit: the processes ψ_j are defined by mollification of ξ_j with P_j, and properties (P0)–(P4) are verified directly; the citation to [LQT25] for the auxiliary bump Φ and for the log-sum identity concerns deterministic ingredients and does not inject the theorem's conclusion. There is a genuine proof gap noted by the reader: Proposition 3.1(P4) is stated only for |α| ≤ d, while Lemma 4.6 is claimed for all multi-indices and Lemma 4.11 applies it with A = 2d, so the required higher-order derivative estimates are not supplied by the manuscript as written. This is a correctness and completeness concern, not a circularity: the missing estimate is not derived by assuming the Fourier decay being proved, and the gap appears repairable from the mollifier scaling. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported to force the construction, and no known result is repackaged as a new derivation. The paper therefore has no significant circular structure.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No constants are fitted to data. The threshold D_{γ,d} emerges from optimizing ζ(p) after the construction fixes the limiting variance at log 2. The paper's own construction parameters (ε_j = j^{-2}2^{-j}, bump functions Φ and Q) are design choices, not free parameters of the result. All other assumptions are standard or explicitly cited.

assumptions (6)

standard math Kahane's T-martingale convergence and uniqueness of GMC for σ-positive kernels.
Used in Section 1.2 to define μ∞ and to assert independence from the particular decomposition.
standard math Pisier's martingale type inequality for ℓ^q with 1<p<=2<=q<∞.
Used in Section 2.2.2 and Section 4.1 to decouple the global Fourier-Lebesgue norm into localized pieces.
domain assumption Known formula for the correlation dimension dim_2(GMC_γ^K)=D_{γ,d} for kernels with bounded continuous remainder g.
Cited from [Ber23], [RV14], [GV23], and [LQT25, Lemma 3.6]; used for the upper bound and as the target value.
domain assumption Known inequality dim_F(μ) <= dim_2(μ) for the Fourier and correlation dimensions.
Used in Lemma 4.1; adapted from [LQT25, Lemma 3.6] from the cube to the torus setting.
domain assumption Existence of a smooth positive definite isotropic bump Φ with support in B(0,1), Φ(0)=1, and C_Φ<∞, plus the log-sum identity (3.4) for the kernels L_j.
Taken from [LQT25, Lemma 3.9 and Proposition 3.8]; these are inputs to the smooth decomposition construction in Section 3.
standard math Kolmogorov continuity theorem and Bochner's theorem for constructing Gaussian processes and checking positive definiteness.
Used in Section 3.1 to construct continuous Gaussian processes with prescribed covariance kernels.

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Pith. "Pith review of Exact values of Fourier dimensions of Gaussian multiplicative chaos on high dimensional torus." pith.science (2026). https://pith.science/paper/EZPKRF4W

@misc{pith2026250723494,
  author       = {Pith},
  title        = {Pith review of: Exact values of Fourier dimensions of Gaussian multiplicative chaos on high dimensional torus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EZPKRF4W}},
  note         = {Machine review of arXiv:2507.23494}
}

abstract

We determine the exact values of the Fourier dimensions for Gaussian Multiplicative Chaos measures on the $d$-dimensional torus $\mathbb{T}^d$ for all integers $d \ge 1$. This resolves a problem left open in previous works [LQT24,LQT25] for high dimensions $d\ge 3$. The proof relies on a new construction of log-correlated Gaussian fields admitting specific decompositions into smooth processes with high regularity. This construction enables a multi-resolution analysis to obtain sharp local estimates on the measure's Fourier decay. These local estimates are then integrated into a global bound using Pisier's martingale type inequality for vector-valued martingales.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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work page 2011

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Federico Bertacco. Multifractal analysis of Gaussian multiplicative chaos and applications. Electron. J. Probab. 28, Paper No. 2, 36 pp, 2023

work page 2023

[4] [4]

Functional analysis, Sobolev spaces and partial differential equations

Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011

work page 2011

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Harmonic analysis of mandelbrot cascades---in the context of vector-valued martingales

Xinxin Chen, Yong Han, Yanqi Qiu, and Zipeng Wang. Harmonic analysis of mandelbrot cascades---in the context of vector-valued martingales. arXiv preprint. arXiv: 2409.13164, 2024

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Ciprian Demeter. Fourier restriction, decoupling, and applications. Cambridge Stud. Adv. Math., 184. Cambridge University Press, Cambridge: xvi+331 pp, 2020

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