Derivatives of Gaussian multiplicative chaos

Abstract

Consider a logarithmically-correlated Gaussian field $X$ in $d$ dimensions. For all $\gamma \in (-\sqrt{2d},\sqrt{2d})$, we show that the derivatives $\frac{\partial^k}{\partial\gamma^k} :e^{\gamma X_\epsilon}:$ of the regularised Gaussian multiplicative chaos $:e^{\gamma X_\epsilon}:$ converge as $\epsilon \to 0$. By deriving optimal bounds on their growth as $k\to\infty$, we control the power expansion of $:e^{\gamma X_\epsilon}:$ about each $\gamma\in(-\sqrt{2d},\sqrt{2d})$. This yields an alternative approach to complex Gaussian multiplicative chaos in the whole subcritical regime, based entirely on real-valued quantities. One of our key technical contributions is to provide a truncated second moment approach to the uniform integrability of the derivatives of multiplicative chaos and its associated complex variant.