Results 1  10
of
23
V.: Gaussian multiplicative chaos and applications: a review, arxiv
"... In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane’s seminal work in 1985. Though this beautiful paper faded from memory until recently, it already contains ideas and results that are nowadays under active investigation, like the construction of th ..."
Abstract

Cited by 26 (0 self)
 Add to MetaCart
In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane’s seminal work in 1985. Though this beautiful paper faded from memory until recently, it already contains ideas and results that are nowadays under active investigation, like the construction of the Liouville measure in 2dLiouville quantum gravity or thick points of the Gaussian Free Field. Also, we mention important extensions and generalizations of this theory that have emerged ever since and discuss a whole family of applications, ranging from finance, through the KolmogorovObukhov model of turbulence to 2dLiouville quantum gravity. This review also includes new results like the convergence of discretized Liouville measures on isoradial graphs (thus including the triangle and square lattices) towards the continuous Liouville measures (in the subcritical and critical case) or multifractal analysis of the measures in all dimensions.
V.: Renormalization of critical Gaussian multiplicative chaos and KPZ formula
"... Gaussian Multiplicative Chaos is a way to produce a measure on R d (or subdomain of R d) of the form e γX(x) dx, where X is a logcorrelated Gaussian field and γ ∈ [0, √ 2d) is a fixed constant. A renormalization procedureis needed to make this precise, since X oscillates between − ∞ and ∞ and is n ..."
Abstract

Cited by 22 (5 self)
 Add to MetaCart
Gaussian Multiplicative Chaos is a way to produce a measure on R d (or subdomain of R d) of the form e γX(x) dx, where X is a logcorrelated Gaussian field and γ ∈ [0, √ 2d) is a fixed constant. A renormalization procedureis needed to make this precise, since X oscillates between − ∞ and ∞ and is not a function in the usual sense. This procedure yields the zero measure when γ = √ 2d. Two methods have been proposed to produce a nontrivial measure when γ = 2d. Thefirstinvolves takingaderivative at γ = 2d(and was studiedin 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.
V.: Gaussian multiplicative chaos and KPZ duality
"... 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 γ2 beyond the transition phase ..."
Abstract

Cited by 19 (6 self)
 Add to MetaCart
(Show Context)
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 γ2 beyond the transition phase (i.e. γ2> 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. 1.
2013): Extreme local extrema of twodimensional discrete Gaussian free field. arXiv preprint arXiv:1306.2602
"... ar ..."
Basic properties of critical lognormal multiplicative chaos. ArXiv eprints
, 2013
"... ar ..."
(Show Context)
V.: Glassy phase and freezing of logcorrelated Gaussian potentials, arXiv:1310.5574
"... In this paper, we consider the Gibbs measure associated to a logarithmically correlated random potential (including two dimensional free fields) at low temperature. We prove that the energy landscape freezes and enters in the socalled glassy phase. The limiting Gibbs weights are integrated atomic r ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
(Show Context)
In this paper, we consider the Gibbs measure associated to a logarithmically correlated random potential (including two dimensional free fields) at low temperature. We prove that the energy landscape freezes and enters in the socalled glassy phase. The limiting Gibbs weights are integrated atomic random measures with random intensity expressed in terms of the critical Gaussian multiplicative chaos constructed in [10, 11]. This could be seen as a first rigorous step in the renormalization theory of supercritical Gaussian multiplicative chaos.
Complex Gaussian multiplicative chaos
"... In this article, we study complex Gaussian multiplicative chaos. More precisely, we study the renormalization theory and the limit of the exponential of a complex logcorrelated Gaussian field in all dimensions (including Gaussian Free Fields in dimension 2). Our main working assumption is that the ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
In this article, we study complex Gaussian multiplicative chaos. More precisely, we study the renormalization theory and the limit of the exponential of a complex logcorrelated Gaussian field in all dimensions (including Gaussian Free Fields in dimension 2). Our main working assumption is that the real part and the imaginary part are independent. We also discuss applications in 2D string theory; in particular we give a rigorous mathematical definition of the socalled Tachyon fields, the conformally invariant operators in critical Liouville Quantum
V.: Limiting laws of supercritical branching random
 I 350 (2012), 535–538, and arXiv:1203.5445v2
"... walks ..."
(Show Context)
Maximum of a logcorrelated Gaussian field
"... Abstract. We study the maximum of a Gaussian field on [0, 1]d (d ≥ 1) whose correlations decay logarithmically with the distance. Kahane [22] introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas [19] [2 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We study the maximum of a Gaussian field on [0, 1]d (d ≥ 1) whose correlations decay logarithmically with the distance. Kahane [22] introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas [19] [20] extended Kahane’s construction to the critical case and established the KPZ formula at criticality. Moreover, they made in [19] several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in [19]: we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale. 1