MARK KAC SEMINAR

Bootstrap percolation models are Cellular Automata describing growth processes, in which starting from a random initial condition, empty sites are filled or not, depending on the states of the sites in their neighborhood. They have been used as tools in various contexts, e.g. for the study of metastability in Ising models, in glass models and Kinetically Constrained Models, and they have also been studied for their own sake. In the first half of the talk I review what is known about these models, including some general results on the classification of 2-dimensional models, and some open questions.

In the second half I discuss a sharp result for a 2-dimensional anisotropic model in which up to three terms of the expansion of the finite-size percolation threshold can be computed.

Work with Hugo Duminil-Copin, Tim Hulshof and Rob Morris.

Often one wishes to describe the collective behaviour of an infinite number of particles by the differential equation governing the evolution of their joint density. The theory of hydrodynamic limits addresses this problem. In this talk, the focus will be on linking the particles with the geometry of the macroscopic evolution. In particular, gradient flows will be explained; they can be thought of as steepest descent/ascent of an entropy. Over the last two decades the associated theory of metric gradient flows has flourished and the talk will summarise some now-classic key results and present an extension to a class of nonlinear diffusion models. Large deviations will be used to provide a way to describe the many-particle dynamics by partial differential equations (PDEs); the PDE is a minimiser of the so-called rate function. However, the underlying microscopic process contains more information, notably fluctuations around the minimum state described by the deterministic PDE. Can stochastic terms be derived which model this additional information, in a way that is compatible to the limit passage via large deviations (and the geometric structure, such as the Wasserstein setting)? This will be discussed in the second part.