abstract:
Suppose at time 0 each site of the d-dimensional cubic lattice is occupied by a particle. As long as particles don't meet, they perform independent random walks. However, if two particles meet, they stick together and behave like one new particle.
This is the basic Coalescing Random Walk model, which was studied around 1980 by Bramson and Griffeath, who obtained interesting results for the asymptotic behaviour (at large times) of the particle density.
In joint work with Harry Kesten we studied (around 2000) natural modifications of the model and introduced (for dimension at least 3) more robust methods. Recently it turned out that these methods are also useful for a seemingly quite different Coalescing Random Walk problem.
In this talk I will start by an introduction to the basic model and then discuss the results and ideas of the more recent work.
abstract:
We study metastability in large
volumes at low temperatures for Ising spins subject to Glauber spin-flip
dynamics. We run the dynamics starting from a random initial configuration
where all the droplets are small. In the low temperature regime, we
investigate how the transition from the metastable state (with small
droplets) to the stable state (with large droplets) takes place under the
dynamics. This transition is triggered by the occurrence of a single
critical droplet, occurring somewhere in the lattice. Using
potential-theoretic methods, we derive sharp estimates on the average
transition time. Since it is inversely proportional to the volume of the
lattice, this type of behaviour is called homogeneous nucleation.