Gaussian multiplicative chaos and KPZ duality

Abstract

This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter ${\gamma }^2$ beyond the transition phase (i.e. ${\gamma }^2>2d$) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a simplified proof of the classical KPZ formula as well as the dual KPZ formula for atomic Gaussian multiplicative chaos. In particular, this framework allows to construct singular Liouville measures and to understand the duality relation in Liouville quantum gravity.

Typo fixed in the title. To appear in Communications in Mathematical Physics

References in corpus (5)

• KPZ in one dimensional random geometry of multiplicative cascades
• Duality and KPZ in Liouville Quantum Gravity
• Lognormal scale invariant random measures
• Hydrodynamic turbulence and intermittent random fields
• Exact asymptotics of the freezing transition of a logarithmically correlated random energy model

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• Basic properties of critical lognormal multiplicative chaos
• Multifractality Breaking from Bounded Random Measures
• Spectral dimension of Liouville quantum gravity
• Nesting statistics in the O(n) loop model on random planar maps
• Multifractal processes: Definition, properties and new examples
• Dynamical Freezing in a Spin Glass System with Logarithmic Correlations
• Statistical Mechanics of Confined Polymer Networks
• Weak Liouville quantum gravity metrics with matter central charge $\mathbf{c}\in (-\infty ,25)$