Activities This Week

The zeros of the Gaussian Entire Function and the infinite Ginibre ensemble are two natural examples of two-dimensional random point configurations whose distribution is invariant under rigid motions of the plane. Due to non-trivial correlations, the features of these two processes are quite different from the ones of the homogeneous Poisson point process. For this reason, these processes are of interest to analysts, probabilists, and mathematical physicists.

I will describe some of the things that we know about the structure and the statistics of these processes. Of particular interest are rare events, when the number of points in a certain domain is very different from its ‘typical’ value. An important example is the ‘hole’ event, when there are no zeros in a large disk. Conditioned on the hole event, the zeros exhibit a large forbidden region, outside the hole, where there are very few zeros asymptotically. This is a new phenomenon, which is in stark contrast to the corresponding result known to hold for the Ginibre ensemble.

Based on a joint work with S. Ghosh (arXiv:1609.00084).

By a flow I mean a one-parameter point-norm continuous group of automorphisms of a C*-algebra. In 1996, Kishimoto introduced a concept of the Rokhlin property for flows, which is analogous to the Rokhlin property for a single automorphism. I’ll discuss a generalization of this, Rokhlin dimension. The results parallel to a great extent results previously obtained in the discrete settings for actions of Z, Z^n, finite groups and certain residually finite groups.

The main results are that crossed products by flows with finite Rokhlin dimension preserve finite nuclear dimension and D-absorption (the latter with an additional technical assumption), crossed products by flows with finite Rokhlin dimension are stable, and any free flow on a commutative C*-algebra with a finite dimensional spectrum has finite Rokhlin dimension. In particular, this shows that crossed products of commutative algebras with finite dimensional spectrum by minimal flows fall under the Elliott’s classification program, provided the have non-zero projections (which follows, e.g., if the flow has a transversal).

This is joint work with Szabo, Winter and Wu, to appear in Comm. Math. Phys.