MARK KAC SEMINAR

Suppose we have a collection of blocks, which gradually split apart as time goes on. Each block waits an exponential amount of time with parameter given by its size to some power alpha, independently of the other blocks. Every block then splits randomly, but according to the same distribution. This is a rough description of the dynamics of a class of processes which were introduced by Jean Bertoin and which are called self-similar fragmentations. In this talk, I will give an introduction to these processes before focussing in on the case where alpha is negative, which means that smaller blocks split (on average) faster than larger ones. This gives rise to the phenomenon of loss of mass, whereby the smaller blocks split faster and faster until they are reduced to “dust”. Indeed, it turns out that the whole state is reduced to dust in a finite time, almost surely (we call this the extinction time). A natural question is then: how do the block sizes behave as the process approaches its extinction time? The answer turns out to involve a somewhat unusual “spine” decomposition for the fragmentation, and Markov renewal theory.

This is joint work with Bénédicte Haas (Paris–Dauphine).

I shall present recent results obtained in collaboration with Dmitry Ioffe and Senya Shlosman. We are considering a class of effective interface models approximating a variety of physical situations:

• Critical prewetting in the 2d Ising model,
• Interfacial adsorption in 2d systems (e.g. the 2d Blume–Capel model on the +/− coexistence line),
• Statistics of the innermost macroscopic level line of a 2+1-dimensional SOS model above a hard wall,
• etc.

Our main result provides a derivation of the scaling limits of these effective models (and presumably, of all the above situations and many others) in the limit of vanishing penalization. These scaling limits are given by stationary, ergodic, reversible diffusions with drifts given by the logarithmic derivative of the ground state of associated singular Sturm–Liouville operators.

In the special case of a log-Airy drift, such a diffusion was obtained by Ferrari and Spohn in the context of Brownian bridges conditioned to stay above parabolic or circular barriers.