Gaussian multiplicative chaos and KPZ duality

Details Gaussian multiplicative chaos and KPZ duality Gaussian multiplicative chaos and KPZ duality

2012-02-23

arxiv.org/abs/1202.5296v2

Mathematics

Probability

The new version contains the convergence in probability of the atomic multiplicative chaos, simulations and conjectures

Scientific paper

This paper is concerned with the KPZ formula. On the first hand, we give a simplified (in comparison with the existing literature) proof of the classical KPZ formula. On the other hand, we construct purely atomic random measures corresponding to values of the parameter $\gamma^2$ beyond the transition phase (i.e. $\gamma^2>2d$). We prove the dual KPZ formula for these measures and check the duality relation. In particular, this framework allows to construct singular Liouville measures and to understand the duality relation in Liouville quantum gravity.

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