November 5, 2010
Location: Kromme Nieuwegracht 80 (Utrecht), room 132
Condensation in stochastic particle systems
In recent years, condensation phenomena in interacting particle systems have attracted a lot of research interest in statistical mechanics and probability. Most results in that direction basically study a family of (grand canonical) product measures for particle configurations, indexed by a parameter that controls the particle density. If the range of this parameter is bounded condensation can occur, i.e., a macroscopic fraction of all the particles concentrates on a single lattice site.
Mathematically, this can be related to large deviation properties of heavy-tailed distributions, to a breakdown of the usual law of large numbers for triangular arrays, or to spatial inhomogeneities leading to different tails of the marginal distributions. All three cases have been studied in the context of zero-range processes, where I will give an overview of the main results and recent developments (including a collaboration with I. Armendariz and M. Loulakis). In recent work with Frank Redig and Kiamars Vafayi we establish condensation also for the inclusion process, which is a 'bosonic' analog of the exclusion process, and for the Brownian energy process, which provides an interesting example with continuous state space.
Stationary discrete processes with long-range one-sided dependencies
We consider one-dimensional stationary processes introduced by Doeblin and Fortet in 1937, specified by a regular g-function. In the first part we describe the original uniqueness result of Doeblin–Fortet, under summability of the variation of g, and the more recent result of Johannson–Oberg giving uniqueness under square summability.
In the second part we will consider the mechanism invented by Bramson and Kalikow, giving non-uniqueness for a class of attractive g-measures, and discuss a problem related to their example.