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Renormalization of Critical Gaussian Multiplicative Chaos and KPZ relation

Vargas, Vincent; Sheffield, Scott; Duplantier, Bertrand; Rhodes, Rémi (2014), Renormalization of Critical Gaussian Multiplicative Chaos and KPZ relation, Communications in Mathematical Physics, 330, 1, p. 283-330. http://dx.doi.org/10.1007/s00220-014-2000-6

Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-00760405
Date
2014
Journal name
Communications in Mathematical Physics
Volume
330
Number
1
Publisher
Springer
Pages
283-330
Publication identifier
http://dx.doi.org/10.1007/s00220-014-2000-6
Metadata
Show full item record
Author(s)
Vargas, Vincent
Sheffield, Scott
Duplantier, Bertrand
Rhodes, Rémi
Abstract (EN)
Gaussian Multiplicative Chaos is a way to produce a measure on $\R^d$ (or subdomain of $\R^d$) 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 martingale, which allows us to establish the KPZ formula at criticality.
Subjects / Keywords
KPZ; Gaussian multiplicative chaos; Liouville quantum gravity; renormalization; derivative martingale

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