The last two decades have seen the emergence of a new area of probability theory concerned with certain random fractal structures characterized by their invariance under conformal transformations. The study of such structures has had deep repercussions on both mathematics and physics, generating tremendous progress in probability theory, statistical mechanics and conformal field theory. In this series of talks, I will give a personal perspective on some aspects of this new area, focusing for concreteness on three specific examples: the Ising model, percolation, Brownian loops. The three talks will be independent and self-contained.

In this talk, I will discuss the scaling limit of the critical and near-critical Ising model. In particular, I will present recent results on the scaling behavior and decay of correlations of the magnetization in the near-critical regime with an external magnetic field. These results provide a rigorous derivation of the behavior of the correlation length at the critical temperature as the external field tends to zero, and imply the existence of a mass gap in the particle spectrum of the Ising field theory with an external field. (Based on joint work with J. Jiang and C. M. Newman.)

A stationary point process in Euclidean space is said to be hyperuniform, if the variance of the number of points in a large ball grows more slowly than its volume. In a more physical language hyperuniformity can be described as an anomalous suppression of large-scale density fluctuations. We can strongly recommend to read a comprehensive survey given by Salvatore Torquato in 2018. By now many different hyperuniform point processes have been discovered, among them perturbed lattices, the Ginibre process, some other determinantal processes and the two-dimensional Coulomb gas. In the first part of the talk we shall introduce and discuss a few fundamental properties of hyperuniform point processes. Then we shall continue with a short discussion of the closely related concept of number rigidity. In the final part of the talk (based on recent joint work with Michael Klatt and Norbert Henze) we shall introduce and explain a genuine test of hyperuniformity that applies to a large class of point processes.