Stochastic equations in infinite dimensions

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An Introduction To The Theory Of Point Processes

Diffusion processes and their sample paths

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 finance, 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.

/pdf/gaussian-multiplicative-chaos-and-applications-a-review-2rzt0sb95w.pdf

Gaussian multiplicative chaos and applications: A review

Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫
 D
 ∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum gravity measure on D is the weak limit as e→0 of the measures $$\varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)}dz,$$
 where dz is Lebesgue measure on D and h
 e
 (z) denotes the mean value of h on the circle of radius e centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (Mod. Phys. Lett. A, 3:819–826, 1988) relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of ∂D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.

Liouville quantum gravity and KPZ

In this paper, we rigorously construct 2d Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov Quantum Geometry of bosonic strings. We also establish some of its fundamental properties like conformal covariance un-der P SL 2 (C)-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula for Liouville Quantum Gravity.

https://hal-cea.archives-ouvertes.fr/cea-01509870/document

Liouville Quantum Gravity on the Riemann Sphere

Gaussian Multiplicative Chaos is a way to produce a measure on ${\mathbb{R}^d}$ (or subdomain of ${\mathbb{R}^d}$) 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 −∞ and ∞ and is not a function in the usual sense. This procedure yields the zero measure when ${\gamma = \sqrt{2d}}$.

/pdf/renormalization-of-critical-gaussian-multiplicative-chaos-190hnmsu07.pdf

Renormalization of Critical Gaussian Multiplicative Chaos and KPZ Relation

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 log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

/pdf/critical-gaussian-multiplicative-chaos-convergence-of-the-4y6u4u9v05.pdf

Rémi Rhodes

Papers

Gaussian multiplicative chaos and applications: A review

Liouville Quantum Gravity on the Riemann Sphere

Renormalization of Critical Gaussian Multiplicative Chaos and KPZ Relation

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale