Fourier dimension of imaginary Gaussian multiplicative chaos

Vortrag von Dr. Benjamin Bonnefont

Sprecher eingeladen von: Prof. Dr. Jean Bertoin

Datum: 18.03.26  Zeit: 17.15 - 18.45  Raum: Y27H12

Recent works have established sharp Fourier decay for subcritical real Gaussian multiplicative chaos (GMC) on the circle, and in this talk I will discuss the corresponding harmonic picture for imaginary GMC. Gaussian multiplicative chaos is obtained by exponentiating log-correlated Gaussian fields; on the unit circle, one may take the trace of the two-dimensional Gaussian free field with covariance $\log 1/|e^{i\theta}-e^{i\theta'}|$. For purely imaginary parameters $\gamma=i\beta$ with $\beta\in(0,1)$, the resulting object $M_{i\beta}$ exists as a complex-valued random distribution and enjoys strong integrability properties.

The Fourier dimension captures the decay of the Fourier coefficients $c_n$ of a distribution. It is defined as the supremum of $s\in(0,1)$ such that $|c_n|^2 = O(|n|^{-s})$. We prove that the Fourier dimension of $M_{i\beta}$ is almost surely $1-\beta^2$ and establish a joint CLT for the rescaled coefficients.

The proof uses the method of moments specific to the imaginary regime. The moments of $c_n$ (and mixed moments of nearby modes) are rewritten as Coulomb-gas integrals on the circle, and then analysed via the Selberg inner product and Jack polynomial expansions, which convert the moment integrals into positive partition sums amenable to sharp asymptotic analysis.

Joint work with Hermanni Rajamäki and Vincent Vargas.