Description: |
There is a transdisciplinar idea known as "Universality" which aims to understand how the macroscopic laws of nature can be obtained from those ruling the microscopic world. The following phenomenon seems to be pervasive: often, a system of interacting objects (for instance, electrons) can display an extreme complexity even at small dimensions. However, if one increases the dimension of the system further enough, the system organizes itself, leading to the simple laws of the macroscopic world. A simple example is the central limit theorem in statistics. Percy Deift has coined the term “macroscopic mathematics” to describe this relatively new field of mathematical research. We will explain some basic ideas about how to random distribute points in a plane, using determinantal point processes and illustrating with the model case of the Ginibre ensemble, where, for high dimensions, the distribution of the points approaches the uniform distribution of a disc ( the famous "circle law"). We will present a new point process, based in time-frequency representations, which contains the Ginibre and other processes as special cases. We have found a law of large numbers which go beyond the circle law: the limit distributions are uniform over domains wich can be quite irregular in shape and frontier regularity. If time allows, some applications in physics, statistics and time-frequency analysis will be outlined. This is joint work with Karlheinz Gröchenig and José Luis Romero from the University of Vienna.
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