Random matrices are used for the statistical analysis of large samples but find also application in various fields of physics. In particular they became of interest around 1990 in string theory as a theory of random surfaces and 2d quantum gravity. Indeed they generate the Brownian sphere as recently proven rigorously. From this perspective it is a natural question whether these results generalize to dimension d>2 and over the last 10 years this was accomplished generalizing from matrices to tensors. In this seminar we will review first the main results of matrix models with a special focus on their combinatorics, the surface topologies they generate and the geometries at criticality. We will then study how these results generalize to tensors with a specific U(N) invariance both from a perturbative as well as a constructive perspective. We will look furthermore at its physics interpretation as a field theory of space(time), its critical behaviour and the possibility of a phase transition from discrete to continuum due to symmetry breaking.

Kurs im HIS-LSF

Semester: WiSe 2019/20