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Non-Gaussian multiplicative chaoses

Let Pn(z)=∑0≤k≤n1kξkzkP_n(z) = \sum_{0\le k\le n} \frac1{\sqrt{k}} \xi_k z^k with ξk\xi_k iid centered random variables of variance one, and consider the random measures μn\mu_n on S1S^1 with density

exp⁡(γPn(z)−γ22Var(Pn(z)))\exp( \gamma P_n(z) - \frac{\gamma^2}2 Var(P_n(z)))

with respect to Lebesgue measure, where γ>0\gamma>0 is a parameter.

When the ξk\xi_k are Gaussian it is known that for γ\gamma sufficiently small these random measures converge weakly to a limiting random measure μ∞γ\mu_\infty^\gamma (with respect to the vague topology) known as the Gaussian multiplicative chaos.

Prove convergence to a limiting random measure μ∞γ\mu_\infty^\gamma in the general non-Gaussian case, and show that the (γ\gamma-dependent) Hausdorff dimension is the same as in the Gaussian case.

References

Primary source

AimPL: Zeros of random polynomials (posed by Ofer Zeitouni)

probability
InAIM Problem Lists
Posted

Progress summary

Refreshed
Claimed progress

The limiting measure is known to exist for a broad class of non-Gaussian coefficients, and recent work shows it has the same measure-theoretic structure as the Gaussian limit, but the requested dimension statement is not explicit in the retrieved sources.

The problem asks for convergence of the multiplicative-chaos measures beyond Gaussian coefficients and for universality of their Hausdorff dimension. Retrieved work treats this Fourier-series model under stronger tail assumptions than those stated here.

Known results

  • Junnila, 2020: almost-sure convergence to a nondegenerate limit for γ∈(0,2)\gamma\in(0,\sqrt{2}); degeneration for γ≥2\gamma\ge\sqrt{2}.
  • Webb and collaborators, 2016: convergence, moment bounds, and parameter analyticity for non-Gaussian log-correlated fields, including Fourier series, throughout the subcritical range.

February 2025 universality theorem

A 2025 theorem constructs a coupling under which, for every γ∈(0,2)\gamma\in(0,\sqrt{2}), the Gaussian and non-Gaussian limiting measures are mutually absolutely continuous: μγ,g≪μγ,a≪μγ,g\mu_{\gamma,g}\ll\mu_{\gamma,a}\ll\mu_{\gamma,g}. This is stronger than convergence and supports equality of Gaussian and non-Gaussian dimension, but the retrieved source does not state that dimension conclusion explicitly; the claim is therefore unverified here.

Current status (as of August 2026): Convergence is established under stronger exponential-moment assumptions for γ∈(0,2)\gamma\in(0,\sqrt{2}), and mutual absolute continuity with the Gaussian chaos is claimed, while the exact Hausdorff-dimension conclusion and the stated minimal hypotheses remain unconfirmed in the retrieved evidence.

Sources

Solutions 0

No solutions have been posted yet.