Probability and Stochastic Analysis - University of Bonn

Research activity

The main focus in my research activity is the question of universality of fluctuations under appropriate scaling limits. The probabilistic models which we study are often, but not exclusively, motivated by statistical mechanics in and out of equilibrium. Here are some of these models, some of them being tightly related, others having only connections after taking the scaling limits.

On the right, a snapshot of a 2+1 dimensional growth model studied in this paper: the projection to fixed time leads to a random tiling model, while the projection to a lower space dimension is a 1+1 growth models in the KPZ universality class.

  • Interacting particle systems, like the (asymmetric) exclusion process: universality of current fluctuations for large times, with dependence on class of initial conditions.
  • Stochastic growth models of interfaces, like the polynuclear growth model: universality of the interface process for large times, with dependence on the curvature of the limit shape.
  • Random matrix models, like the Gaussian Ensembles: universality in the bulk and for the largest eigenvalues, for large matrices.
  • Random tilings, like the Aztec diamond: universality of the north-polar region process and in the temperate region for large system.
  • Percolation, like directed last passage: universality of the last passage time fluctuations and generic slow-decorrelation phenomenon (dependence on the class: point-to-point, point-to-line, point-to-random line).
  • Equilibrium crystals: fluctuations of the facets ledges for crystals (below the roughtening temperature) and in the rounded part of the crystal.

Below you can find animation of (some of) these models.

See W3Schools for a good guide.

A growth model in the 2+1 dimensional anisotropic KPZ class Javascript

Animations

The continuous time polynuclear growth (PNG) model Java Applet
The totally asymmetric simple exclusion process (TASEP) Java Applet
A growth model in the 2+1 dimensional anisotropic KPZ class Java Applet
A 2+1 dimensional particle dynamics and the Aztec diamond Java Applet
A visualization of the slow decorrelation phenomena Java Applet
A visualization of the qTASEP Java Applet
A visualization of GOE-GOE shock in TASEP Java Applet