7 research outputs found
Small ball probability, Inverse theorems, and applications
Let be a real random variable with mean zero and variance one and
be a multi-set in . The random sum
where are iid copies of
is of fundamental importance in probability and its applications.
We discuss the small ball problem, the aim of which is to estimate the
maximum probability that belongs to a ball with given small radius,
following the discovery made by Littlewood-Offord and Erdos almost 70 years
ago. We will mainly focus on recent developments that characterize the
structure of those sets where the small ball probability is relatively
large. Applications of these results include full solutions or significant
progresses of many open problems in different areas.Comment: 47 page
The polyanalytic Ginibre ensembles
We consider a polyanalytic generalization of the Ginibre ensemble. This
models allowing higher Landau levels (the Ginibre ensemble corresponds to the
lowest Landau level). We study the local behavior of this point process under
blow-ups.Comment: 31 page