Renormalization of Critical Gaussian Multiplicative Chaos and KPZ formula

Abstract

Gaussian Multiplicative Chaos is a way to produce a measure on ${\mathbb{R}}^d$ (or subdomain of ${\mathbb{R}}^d$) of the form $e^{\gamma X(x)}dx$, where $X$ is a log-correlated Gaussian field and $\gamma \in [0,\sqrt{2d})$ is a fixed constant. A renormalization procedure is needed to make this precise, since $X$ oscillates between $-\infty$ and $\infty$ and is not a function in the usual sense. This procedure yields the zero measure when $\gamma =\sqrt{2d}$. Two methods have been proposed to produce a non-trivial measure when $\gamma =\sqrt{2d}$. The first involves taking a derivative at $\gamma =\sqrt{2d}$ (and was studied in an earlier paper by the current authors), while the second involves a modified renormalization scheme. We show here that the two constructions are equivalent and use this fact to deduce several quantitative properties of the random measure. In particular, we complete the study of the moments of the derivative multiplicative chaos, which allows us to establish the KPZ formula at criticality. The case of two-dimensional (massless or massive) Gaussian free fields is also covered.

The new version contains the proofs for Free Fields

References in corpus (5)

• Extreme local extrema of two-dimensional discrete Gaussian free field
• KPZ in one dimensional random geometry of multiplicative cascades
• Duality and KPZ in Liouville Quantum Gravity
• Gaussian multiplicative chaos and applications: a review
• Convergence in law of the maximum of the two-dimensional discrete Gaussian free field

Cited by in corpus (10)

• Extreme local extrema of two-dimensional discrete Gaussian free field
• Liouville Quantum Gravity on the complex tori
• KPZ formula derived from Liouville heat kernel
• Liouville heat kernel: regularity and bounds
• On the heat kernel and the Dirichlet form of Liouville Brownian Motion
• Liouville Quantum Gravity on the unit disk
• Liouville Quantum Gravity on the Riemann sphere
• Renormalizability of Liouville Quantum Gravity at the Seiberg bound
• Glassy phase and freezing of log-correlated Gaussian potentials
• Liouville Brownian motion at criticality