Long-range dependence within point processes of arithmetic origins

Research Project, 2024 – 2027

The proposal describes two projects, both concerned with random measures on homogeneous spaces. The first project explores representation-theoretic aspects around (quasi-crystalline) random measures: When are there irreducible sub-representations? If they exist, how "many" are there? Can we say anything about what kind of sub-representations which appear? Are there explicit intertwiners?The second project deals with spherical diffraction (Bartlett spectrum) of point processes, with a view towards hyperuniformity (a notion first defined by Stillinger and Torquato). The Euclidean setting is rather well-understood by now, but there are plenty of new phenomena for random measures in Heisenberg nilmanifolds and symmetric spaces. To end up with a non-empty set of examples of hyperuniform processes in "curved" spaces, the classical definitions need to be modified. The second projects investigates some possibilities. One of the key difficulties is that the spherical spectrum tend to consist of very different parts and these parts interact in a highly non-trivial way when they make up observable quantities like the variance of point process. This is particularly subtle in the setting of quasi-crystalline point processes in the Heisenberg group.