Let gamma∈(1/√2,√2), let mu_gamma be the Gaussian multiplicative chaos measure on [0,1] with parameter gamma, and let c_n be its n-th Fourier coefficient. Subcritical Fourier coefficient limit conjecture. The random variables (log n)^3 gamma/(2√2)n^(√2-gamma)^2/2|c_n| should converge in distribution, as n→∞, to a non-trivial limiting random variable. This is a heuristic prediction for the subcritical range gamma∈(1/√2,√2); the surrounding discussion explicitly says that the heuristics are not made rigorous there.

Subcritical Gaussian multiplicative chaos Fourier coefficient limit conjecture

About 1 year old · traced to 2025

Let $\gamma\in(1/\sqrt{2},\sqrt{2})$ , let $\mu_\gamma$ be the Gaussian multiplicative chaos measure on $[0,1]$ with parameter $\gamma$ , and let $c_n$ be its $n$ -th Fourier coefficient. Subcritical Fourier coefficient limit conjecture. The random variables

(\log n)^{3\gamma/(2\sqrt{2})}n^{(\sqrt{2}-\gamma)^2/2}|c_n|

should converge in distribution, as $n\to\infty$ , to a non-trivial limiting random variable. This is a heuristic prediction for the subcritical range $\gamma\in(1/\sqrt{2},\sqrt{2})$ ; the surrounding discussion explicitly says that the heuristics are not made rigorous there.

References

Primary source

Louis-Pierre Arguin and Jad Hamdan, “On the Fourier coefficients of critical Gaussian multiplicative chaos”, arXiv:2510.24424 (2026).

fourier-analysis gaussian-multiplicative-chaos

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Open

No proof or counterexample has been found for the conjectured subcritical limit; only weaker decay and convergence results are known.

Arguin and Hamdan formulate the conjecture that the stated normalization of the Fourier coefficient of subcritical Gaussian multiplicative chaos converges to a non-trivial random limit. Their source explicitly says the supporting heuristics are not rigorous in the range $\gamma\in(1/\sqrt{2},\sqrt{2})$ .

Known results

Garban and Vargas proved the Rajchman property: $c_n\to0$ almost surely for $\gamma<\sqrt{2}$ .
Garban and Vargas obtained a convergence-in-law result in the smaller range $\gamma<1/\sqrt{2}$ .
Subsequent work on the Fourier dimension of one-dimensional GMC addresses a different question and does not establish the conjectured normalized limit.

Current status (as of August 2026): the normalized convergence remains open for $\gamma\in(1/\sqrt{2},\sqrt{2})$ ; no proof, counterexample, or claimed settlement was found.

Sources

No solutions have been posted yet.