A valószinűségszámitás egyik klasszikus problémája független valószinűségi változók részletösszegei határeloszlásainak leirása, melyet Kolmogorov, Lévy, Hincsin, Feller, Gnedenko és Doeblin az 1930-40 évtizedben teljesen megoldottak. Lévy legfontosabb eredményeit e területeten egy 1935-ben irt cikke tartalmazza, mely - többek között - a stabilis es szemistabilis eloszlások konstrukcióját, valamint a normális eloszlás vonzási es részleges vonzási tartományának teljes leirását adja. Előadásunkban e cikk néhány megjegyzésével foglalkozunk, melyek az utókor figyelmét elkerülték, és melyek olyan eredményeket tartalmaznak, mint  erős invariancia-elv, Szkorohod reprezentáció, kvantilis transzformáció, stabilis eloszlások sorfejtése, valamint a szentpétervári paradoxonban fellépő határeloszlás, eredmények, melyek évtizedekkel előzik meg korukat, és számos ma is új információt szolgáltatnak.

We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph G are either open or closed and refresh their status at rate mu, while at the same time a random walker moves on G at rate 1, but only along edges which are open. On the d-dimensional torus with side length n, when the bond parameter is subcritical, we determined (with Y. Peres and A. Stauffer) the mixing times for both the full system and the random walker. The supercritical case is harder, but using evolving sets we were able (with Y. Peres and P. Sousi) to analyze it.

For stationary KPZ growth in 1+1 dimensions the height fluctuations are governed by the Baik-Rains distribution. Using the totally asymmetric single step growth model, alias TASEP, we investigate height fluctuations for a general class of spatially homogeneous random initial conditions. We prove that for TASEP there is a one-parameter family of limit distributions, labeled by the roughness of the initial conditions. The distributions are defined through a variational formula.

We introduce the multiplicative coalescent with linear deletion, a continuous-time Markov process describing the evolution of a collection of blocks. This process arises in connection with  random graph models which exhibit self-organised criticality.  We focus on results describing states of the process in terms of collections of excursion lengths of random functions, in which the coalescence of blocks is related to a "tilt" of the random function and deletion of blocks is related to a "shift" of the random function. Joint work with James Martin.

In 1964, V. Arnold constructed an example of a nearly integrable deterministic system exhibiting instabilities. In the 1970s, physicist B. Chirikov coined the term for this phenomenon "Arnold diffusion", where diffusion refers to stochastic nature of instability. One of the most famous examples of stochastic instabilities for nearly integrable systems is dynamics of Asteroids in Kirkwood gaps in the Asteroid belt. They were discovered numerically by astronomer J. Wisdom.
During the talk we describe a class of nearly integrable deterministic systems, where we prove stochastic diffusive behavior. Namely, we show that distributions given by deterministic evolution of certain random initial conditions weakly converge to a diffusion process. This result is conceptually different from known mathematical results, where existence of "diffusing orbits" is shown. This work is based on joint papers with O. Castejon, M. Guardia, J. Zhang, and K. Zhang.