MARK KAC SEMINAR

Interfaces are much studied in mathematics and in the applications, I will discuss examples coming from biology and physics. In the first part I will discuss the so called N - BBM model. This is a system of N independent branching Brownian motions with annihilation. At each branching time a new particle is created and the leftmost one is deleted. This system fits in a class of models proposed by Brunet and Derrida (Phys. Rev. E, 56, 1997) to study selection mechanisms in biological systems: particles are individuals in a population, the position of a particle is ``its degree of fitness'', the larger the position the higher the fitness. The environment supports only populations of a given size so that to each birth there must correspond a death and the removal of the leftmost and hence less fitted particle, implements a Darwinian selection rule. The position of the leftmost particle may be regarded as the interface which separate the region with no particles and the one with particles. This is the``microscopic" interface and the goal is to study its motion in the hydrodynamic limit when N diverges. In this limit we deal with interesting free boundary problems.

Long-range Ising models with polynomially decaying interactions to some extent simulate the behaviour of higher-dimensional short-range models. However, the analogy is imperfect. In the last few years there have been a number of developments around these models, mostly based on the behaviour of low-temperature excitations and interfaces. I plan to review a number of those. They include the difference between the spatial and time-like stochastic processes, the stability of interfaces, metastability and possible metastates for these models.