Random geometric systems

Critical points of random spherical harmonics and isotropic stationary Gaussian fields are examples of points process showing a regular structure. In this talk, we will present some recent results aimed at quantifying how critical points differ from independently picked points. We will focus in particular on the limiting distribution, in the high energy limit, of critical points and extrema of random spherical harmonics. Moreover, by computing the main term in the asymptotic expansion of the two-point correlation function near the diagonal, we obtain that critical points neither repel nor attract each other. Our analysis also allows us to study how the short-range behaviour of critical points depends on their index. Finally we discuss the full correlation in the high frequency limit between critical points and other geometric functionals.
Based on joint works with D. Beliaev, D. Marinucci and I. Wigman.

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above, but there are many natural generalizations. In this talk I will give a survey of the main results and conjectures in this area.