MARK KAC SEMINAR

Knuth's parking scheme is a model in computer science for hashing with linear probing. One may imagine a circular parking with n sites; cars arrive at each site with unit rate. When a car arrives at a vacant site, it parks there; otherwise it turns clockwise and parks at the first vacant site which is found. It is known from a work by Chassaing and Louchard that the formation of large occupied blocks is governed by the so-called additive coalescent. We incorporate fires to this model by throwing Molotov cocktails on each site at a smaller rate n^-α. When a car is hit by a Molotov cocktail, it burns and the fire propagates to the entire occupied interval which turns vacant. We show that with high probability when the size of the parking is large, the parking becomes saturated at a time close to 1 (i.e., as in the absence of fire) for α>2/3, whereas for α<2/3, the mean occupation approaches 1 at time 1 but then quickly drops to 0 before the parking is ever saturated.

We study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between neighboring particles of the same type. We start the dynamics from the empty box and compute the transition time to the full box. This transition is triggered by a critical droplet appearing somewhere in the box.

We identify the region of parameters for which the system is metastable. For this region, in the limit as the temperature tends to zero, we show that the first entrance distribution on the set of critical droplets is uniform, compute the expected transition time up to and including a multiplicative factor of order one, and prove that the transition time divided by its expectation is exponentially distributed. These results are derived for a certain subregion of the metastable region. The proof involves three model-dependent quantities: the energy, the shape and the number of the critical droplets.

The main motivation is to understand metastability of multi-type particle systems. It turns out that for two types of particles the geometry of subcritical and critical droplets is more complex than for one type of particle. Consequently, it is a somewhat delicate matter to capture the proper mechanisms behind the growing and the shrinking of subcritical droplets until a critical droplet is formed.