Statistical Topology -- Distribution and Density Correlations of Winding Numbers in Chiral Systems

TL;DR

This paper reviews recent work on the statistical properties of winding numbers and densities in chiral systems, emphasizing the universality and spectral mappings in topological invariants using random matrix models.

Contribution

It introduces statistical topology in the context of chiral systems and reviews new results on winding number distributions derived from random matrix models.

Findings

Winding number statistics can be mapped to spectral problems.

Universal behaviors are observed in the distribution of topological invariants.

Random matrix models effectively capture topological correlations.

Abstract

Statistical Topology emerged since topological aspects continue to gain importance in many areas of physics. It is most desirable to study topological invariants and their statistics in schematic models that facilitate the identification of universalities. Here, the statistics of winding numbers and of winding number densities are addressed. An introduction is given for readers with little background knowledge. Results that my collaborators and I obtained in two recent works on proper random matrix models are reviewed, avoiding a technically detailed discussion. A special focus is on the mapping of topological problems to spectral ones as well as on the first glimpse on universality.