Gaussian multiplicative chaos and KPZ duality

Julien Barral (1) , Xiong Jin (2) , Rémi Rhodes (3) , Vincent Vargas (3)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 Mathematics Institute
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: Gaussian multiplicative chaos; KPZ formula; duality; Liouville quantum gravity; Gaussian free field; Hausdorff dimension; atomic measure

Domaines: Probabilités [math.PR]; Physique mathématique [math-ph]

Julien Barral, Xiong Jin, Rémi Rhodes, Vincent Vargas. Gaussian multiplicative chaos and KPZ duality. 2012. ⟨hal-00673629v2⟩