Convergence for Complex Gaussian Multiplicative Chaos on phase boundaries

TL;DR

This paper studies the convergence behavior of complex Gaussian Multiplicative Chaos on phase boundaries, showing different modes of convergence depending on the parameter region, and completing the phase diagram analysis.

Contribution

It extends the understanding of complex GMC convergence on phase boundaries, especially in regions previously uncharacterized, and describes the limit objects in these cases.

Findings

Convergence in probability and $L^p$ for certain parameters in $ ext{I/II}$ region.

Convergence in law to complex Gaussian white noise in $ ext{II/III}$ region.

Complete characterization of phase boundary behavior for complex GMC.

Abstract

The complex Gaussian Multiplicative Chaos (or complex GMC) is informally defined as a random measure $e^{\gamma X} \mathrm{d} x$ where $X$ is a log correlated Gaussian field on $\mathbb R^d$ and $\gamma=\alpha+i\beta$ is a complex parameter. The correlation function of $X$ is of the form $$ K(x,y)= \log \frac{1}{|x-y|}+ L(x,y),$$ where $L$ is a continuous function. In the present paper, we consider the cases $\gamma\in \mathcal P_{\mathrm{I/II}}$ and $\gamma\in \mathcal{P}'_{\mathrm{II/III}}$ where $$ \mathcal P_{\mathrm{I/II}}:= \{ \alpha+i \beta \ : \alpha,\beta \in \mathbb R \ ; |\alpha|>|\beta| \ ; \ |\alpha|+|\beta|=\sqrt{2d} \}, $$ and $$ \mathcal{P}'_{\mathrm{II/III}}:= \{ \alpha+i \beta \ : \alpha,\beta \in \mathbb R \ ; \ |\alpha|= \sqrt{d/2} \ ; \ |\beta|>\sqrt{2d} \},$$ We prove that if $X$ is replaced by an approximation $X_\epsilon$ obtained via mollification, then $e^{\gamma X_\epsilon} \mathrm{d} x$, when properly rescaled, converges when $\epsilon\to 0$. The limit does not depend on the mollification kernel. When $\gamma\in \mathcal P_{\mathrm{I/II}}$, the convergence holds in probability and in $L^p$ for some value of $p\in [1,\sqrt{2d}/\alpha)$. When $\gamma\in \mathcal{P}'_{\mathrm{II/III}}$ the convergence holds only in law. In this latter case, the limit can be described a complex Gaussian white noise with a random intensity given by a critical real GMC. The regions $\mathcal P_{\mathrm{I/II}}$ and $ \mathcal{P}'_{\mathrm{II/III}}$ correspond to phase boundary between the three different regions of the complex GMC phase diagram. These results complete previous results obtained for the GMC in phase I and III and only leave as an open problem the question of convergence in phase II.