Determinantal point processes (DPP's) form a popular class of models for mutually repulsive random points. In this talk, I will consider only determinantal processes in R^d, with the focus on d=2. The defining feature of DPP's is that all statistical information is contained in a function K, which is an integral kernel of a locally trace class operator with spectrum in [0,1]. A common way to study such point processes is through linear statistics tr(f):=∑_{x ∈ X} f(x), where X is the point process and f is a suitable test function, for example smooth and compactly supported.  A well-known theorem of Soshnikov asserts the following: letting f_R(x)= f(x/R) and assuming that the variance of tr(f_R) tends to infinity rapidly enough, the variables tr(f_R) converge (after suitable normalizations) to a normal variable as R tends to infinity. Soshnikov's theorem has wide applicability, but the assumption on the variance growth excludes some natural DPP's in dimension two, in particular those whose kernel is reproducing and has rapid off-diagonal decay. Our recent work with José Luis Romero fills this gap. Our theorem builds on works of Rider and Virag as well as Ameur, Hedenmalm and Makarov.

https://univienna.zoom.us/j/67922750549?pwd=Ulh5L1QxNFhBOC9PUjlVdG9hc0tmUT09