Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

TL;DR

This paper investigates the critical case of Gaussian multiplicative chaos, proving the almost sure convergence of the derivative martingale to a full-support, non-atomic measure, and discusses implications for log-correlated Gaussian fields.

Contribution

It establishes the almost sure convergence of the derivative martingale in Gaussian chaos at criticality and characterizes the properties of the limiting measure.

Findings

Derivative martingale converges almost surely in all dimensions.

Limiting measure has full support and no atoms.

Connections to the maximum of log-correlated Gaussian variables.

Abstract

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.