Stochastic equations in infinite dimensions

We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

Shape Fluctuations and Random Matrices

Thank you for downloading an introduction to the theory of point processes. As you may know, people have search hundreds times for their chosen novels like this an introduction to the theory of point processes, but end up in infectious downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they juggled with some harmful virus inside their computer. an introduction to the theory of point processes is available in our digital library an online access to it is set as public so you can download it instantly. Our book servers hosts in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Merely said, the an introduction to the theory of point processes is universally compatible with any devices to read.

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

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

In this article, we extend the theory of multiplicative chaos for positive definite functions in Rd of the form f(x) = 2 ln+ T|x|+ g(x) where g is a continuous and bounded function. The construction is simpler and more general than the one defined by Kahane in 1985. As main application, we give a rigorous mathematical meaning to the Kolmogorov-Obukhov model of energy dissipation in a turbulent flow.

/pdf/gaussian-multiplicative-chaos-revisited-5cec5a0x2e.pdf

Gaussian multiplicative chaos revisited

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.

Vincent Vargas

Papers

Gaussian multiplicative chaos and applications: A review

Liouville Quantum Gravity on the Riemann Sphere

Gaussian multiplicative chaos revisited

Renormalization of Critical Gaussian Multiplicative Chaos and KPZ Relation

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale