15:00 – Nina Gantert: Einstein relation and monotonicity of the speed for random walk among random conductances.
Many applications, such as porous media or composite materials, involve heterogeneous media which are modeled by random fields. These media are locally irregular but are “statistically homogeneous” in the sense that their law has homogeneity properties. Considering random motions in such a random medium, it turns out often that they can be described by their effective behaviour. This means that there is a deterministic medium, the effective medium, whose properties are close to the random medium, when measured on long space-time scales. In other words, the local irregularities of the random medium average out over large space-time scales, and the random motion is characterized by the “macroscopic” parameters of the effective medium. How do the macroscopic parameters depend on the law of the random medium?
As an example, we consider the effective diffusivity (i.e. the covariance matrix in the central limit theorem) of a random walk among random conductances. It is interesting and non-trivial to describe this diffusivity in terms of the law of the conductances. The Einstein relates this diffusivity with the derivative of the speed of a biased random walk among random conductances. We explain the Einstein relation and we also discuss monotonicity questions for the speed of a biased random walk among random conductances.
The talk is based on joint work (in progress) with Noam Berger, Xiaoqin Guo and and Jan Nagel.
16:30 – Davoud Cheraghi: On a conjecture of Marmi, Moussa, and Yoccoz on the sizes of Siegel disks.
Quasi-periodic dynamics in one complex variable reveal fascinating interplays between complex analysis and Diophantine approximations. The question of whether a quasi-periodic dynamic is conjugate to an irrational rotation dates back to more than a century ago, with remarkable contributions by C. Siegel, A. Brjuno, and J.-C. Yoccoz. The maximal domain on which a conjugacy exists is called the Siegel disk of the map, and it is known that the size of the Siegel disk is given by an arithmetic function of the rotation, up to an error function. In 1992, Marmi, Moussa, and Yoccoz conjectured that the error function is 1/2-holder continuous. In this talk, I will discuss a major advance on this conjecture, using a renormalization operator acting on an infinite dimensional space of maps. This is based on a joint work with Arnaud Cheritat (Toulouse).