The Puzzle of Normal Distribution

Aron Wennman will receive his doctoral degree in mathematics from KTH Royal Institute of Technology in 2018. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Mikhail Sodin, Faculty of Exact Sciences, Tel Aviv University, Israel.

The aim of the project is to gain new insights into the concept of universality, one of the great puzzles of probability and mathematical physics. Universality implies that many complex systems exhibit common features, which are often independent of the underlying principles governing the systems.

One such law of universality is the central limit theorem, a key concept in probability theory. This says that if a large number of mutually independent parameters are measured, the distribution of their sum approaches the normal distribution with high probability. The theorem explains why so many observed distributions are normal. Examples include many biological measurements, such as the logarithm of human height, weight, or blood pressure, as well as results of standardized high school tests.

One way to further investigate the cause of the universality phenomenon is to study a large number of randomly scattered particles that repel each other pairwise while held together in a finite space by an external force. Physicists call such a setting a Coulomb gas. Mathematicians encounter such a model in a number of circumstances, for example when analyzing eigenvalues of random matrices.

The stability of the behavior of such a system, even when the external field or the particles’ properties evolve, is puzzling. The aim of the study is to investigate Coulomb gas models to elucidate further the properties of universality and set out limits when it breaks down.