Gaussian multiplicative chaos and KPZ duality

Vargas, Vincent; Rhodes, Rémi; Jin, Xiong; Barral, Julien

Vargas, Vincent; Rhodes, Rémi; Jin, Xiong; Barral, Julien (2013), Gaussian multiplicative chaos and KPZ duality, Communications in Mathematical Physics, 323, 2, p. 451-485. http://dx.doi.org/10.1007/s00220-013-1769-z

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00673629

Date

2013

Journal name

Communications in Mathematical Physics

Volume

323

Number

Publisher

Springer

Pages

451-485

Abstract (EN)

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.

Subjects / Keywords

Hausdorff dimension; Gaussian free field; Liouville quantum gravity; duality; KPZ formula; Gaussian multiplicative chaos