3.9 Probability Theory

Organizer F. Martinelli (Rome)

3.9.1Aging in the Infinite Volume REM-Like Trap Model at Low Temperature

Luiz Renato Fontes

IME-USP lrenato@ime.usp.br

We exhibit the scaling limit of the REM-like trap model at low temperature (with time scaled as the deepest traps) and show that this dynamics exhibits aging (in the proper macroscopic time limit). We also discuss extensions to infinite volume GREM-like trap models at low temperature.

3.9.2 From Planar Gaussian Zeros to Gravitational Allocation

Yuval Peres

UC Berkeley and Microsoft Research peres@stat.Berkeley.edu

The zeros of the power series with IID complex Gaussian coefficients form a determinantal (Fermionic) process, invariant under hyperbolic isometries. This allows for exact calculation of “hole” probabilities. Recently another Gaussian power series, with Euclidean symmetry, has been investigated in depth. Results of Sodin-Tsirelson reveal a surprising analogy with a four-dimensional Poisson process. We show how the analysis of gravitational allocation for the Euclidean planar Gaussian analytic function (Nazarov-Sodin-Volberg) has led to an analysis of gravitational allocation for the Poisson process in dimensions 3 and higher; each Poisson point is allotted a unit of volume in a translation invariant way, and the diameters of the allocated regions have exponental tails. The argument starts with the classical calculation by Chandrasekar of the total force acting on a point, which has a stable law. (Joint works with S. Chatterjee, R. Peled, D. Romik, B. Virag).

3.9.3 Random Walks in Random Environments in the Perturbative Regime

Ofer Zeitouni

Univ. of Minnesota zeitouni@math.umn.edu

I will describe recent results and technique used in analysing random walks (and diffusions) in random environment in the perturbative regime, when the environment satisfies certain isotropy conditions.