April 5, 2013Location: Janskerkhof 15a (Utrecht), room 101
Many evolutionary systems described by parabolic partial differential equations can be written as a gradient flow of some energy with respect to some metric. When present, this gradient-flow structure provides both high-level insight into the behaviour of the system, and low-level, practical tools for the analysis of the system and its solutions.
Since the pioneering work of Jordan, Kinderlehrer, and Otto (1998) an impressive collection of evolutionary PDEs has been formulated as a gradient flow of some energy with respect to the Wasserstein metric.
In this talk I will briefly introduce the Wasserstein metric and the concept of a Wasserstein gradient flow, and then turn to the main topic of the talk. This is the question: how can we understand why the Wasserstein metric appears in so many systems?
I will show how this gradient-flow structure is intimately connected to the large-deviation behaviour of systems of interacting particles. This connection gives us an explanation for the prevalence of Wasserstein gradient flows, and suggests directions for the derivation of a large number of related systems.
I will also describe more recent work, that illustrates how the same principles apply to non-particle stochastic systems, for instance systems in which energy is transported.
The incipient infinite cluster (IIC) of percolation is the random subgraph (of Zd) that you get when you condition the critical percolation measure on the event that the origin is part of an infinite percolation cluster. This object was first constructed for two-dimensional percolation by Harry Kesten. My talk will be about the IIC for high-dimensional percolation (typically, when d > 6).
The IIC contains a random subgraph that is called the backbone. The backbone contains all the vertices in the IIC that have disjoint paths to the origin and to infinity. In high dimensions the backbone resembles a random singly infinite path.
In the first part of the talk I will give some historical context about the IIC, and I will discuss some of the most important results about the IIC so far. This discussion will be about the construction of the IIC, its geometry, its scaling limit and the behavior of random walk on the IIC. Then, in the second part I will discuss recent work in which we prove that the scaling limit of the backbone is a d-dimensional Brownian motion.
(The second part is based on joint work with Markus Heydenreich, Remco van der Hofstad, and Gregory Miermont)