## 12th session, June 23rd 2015

__Tuesday 23rd__

*EHESS, 190-198*

*Avenue de France, room 466, 4th floor - 75013 Paris*

**

10h30 : Lenya Ryzhik

*(Stanford*)

**The weakly random Schroedinger equation: a consumer report.**Consider a Schroedinger equation with a weakly random time-independent potential. When the correlation function of the potential is, roughly speaking, of the Schwartz class, it has been shown by Spohn (1977), and Erdos and Yau (2001) that the kinetic limit holds -- the expectation of the phase space energy density of the solution converges to the solution of a kinetic equation. We "extend" this result to potentials whose correlation functions satisfy (in some sense) "sharp" conditions, and also prove a parallel homogenization result for slowly varying initial conditions.

I will explain the quotation marks above and make some speculations on the genuinely sharp conditions on the random potential that separate

various regimes. This talk is a joint work with T. Chen and T. Komorowski

13h30 : James Nolen (

*Duke*)

*Fluctuations of the effective conductance in a random conductor*I will talk about solutions to a linear, divergence-form elliptic PDE with conductivity coefficient that is random. It has been known for some time that homogenization may occur when the coefficients are scaled suitably; this talk is about fluctuations of the solution around its mean behavior. Suppose an electric potential is imposed at the boundary of some heterogeneous conducting material. Some current will flow through the material. What is the net current? For a finite sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe recent results about fluctuations of this quantity. In particular, I'll explain a central limit theorem for the effective conductivity. I'll also talk about a scaling limit for the potential field itself.