Rémi Rhodes

Gaussian multiplicative chaos and applications: A review

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 2d-Liouville 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 nance, through the Kolmogorov-Obukhov model of turbulence to 2d-Liouville 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.

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of logcorrelated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.

Gaussian multiplicative chaos and KPZ duality

This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter γ2 beyond the transition phase (i.e. γ2 > 2d) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a simplified proof of the classical KPZ formula as well as the dual KPZ formula for atomic Gaussian multiplicative chaos. In particular, this framework allows to construct singular Liouville measures and to understand the duality relation in Liouville quantum gravity.

Glassy phase and freezing of log-correlated Gaussian potentials

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 so-called 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 \cite{Rnew7,Rnew12}. This could be seen as a first rigorous step in the renormalization theory of super-critical Gaussian multiplicative chaos.

Lognormal {\star} -scale invariant random measures

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with lognormal weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation; these measures fall under the scope of the Gaussian multiplicative chaos theory developed by Kahane in 1985 (or possibly extensions of this theory). As a by product, we also obtain an explicit characterization of the covariance structure of these measures. We also prove that qualitative properties such as long-range independence or isotropy can be read off the equation.

Log-correlated Gaussian fields: an overview

We survey the properties of the log-correlated Gaussian field (LGF), which is a centered Gaussian random distribution (generalized function) h on ℝ d , defined up to a global additive constant.

Liouville heat kernel: Regularity and bounds

Dans ce papier, nous initions l’etude des proprietes analytiques du noyau de la chaleur de Liouville. En particulier, nous etablissons des estimees de regularite pour le noyau et nous l’encadrons par des bornes inferieures et superieures non triviales.

KPZ formula derived from Liouville heat kernel

In this paper, we establish the Knizhnik--Polyakov--Zamolodchikov (KPZ) formula of Liouville quantum gravity, using the heat kernel of Liouville Brownian motion. This derivation of the KPZ formula was first suggested by F. David and M. Bauer in order to get a geometrically more intrinsic way of measuring the dimension of sets in Liouville quantum gravity. We also provide a careful study of the (no)-doubling behaviour of the Liouville measures in the appendix, which is of independent interest.

Local Conformal Structure of Liouville Quantum Gravity

In 1983 Belavin, Polyakov, and Zamolodchikov (BPZ) formulated the concept of local conformal symmetry in two dimensional quantum field theories. Their ideas had a tremendous impact in physics and mathematics but a rigorous mathematical formulation of their approach has proved elusive. In this work we provide a probabilistic setup to the BPZ approach for the Liouville Conformal Field Theory (LCFT). LCFT has deep connections in physics (string theory, two dimensional gravity) and in mathematics (scaling limits of planar maps, quantum cohomology). We prove the validity of the conformal Ward identities that represent the local conformal symmetry of LCFT and the Belavin–Polyakov–Zamolodchikov differential equations that form the basis for deriving exact formuli for LCFT. We prove several celebrated results on LCFT, in particular an explicit formula for the degenerate 4 point correlation functions leading to a proof of a non trivial functional relation on the 3 point structure constants derived earlier using physical arguments by Teschner. The proofs are based on exact identities for LCFT correlation functions which rely on the underlying Gaussian structure of LCFT combined with estimates from the theory of critical Gaussian Multiplicative Chaos and a careful analysis of singular integrals (Beurling transforms and generalizations). As a by-product, we give bounds on the correlation functions of LCFT when two points collide making rigorous certain predictions from physics on the so-called “operator product expansion” of LCFT. This paper provides themathematical basis for the proof of integrability results for LCFT (the DOZZ conjecture) and the construction of the Virasoro representation theory for LCFT.

Levy multiplicative chaos and star scale invariant random measures

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with infinitely divisible weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation. We obtain an explicit characterization of the structure of these measures, which reflects the constraints imposed by the continuous setting. In particular, we show that the continuous equation enjoys some specific properties that do not appear in the discrete star equation. To that purpose, we define a Levy multiplicative chaos that generalizes the already existing constructions. Keywords: Random measure, star equation, scale invariance, multiplicative chaos, uniqueness, infinitely divisible processes, multifractal processes.