## Periodic Reporting for period 4 - SPECTRUM (Spectral theory of random operators)

Reporting period: 2019-10-01 to 2020-08-31

The project is devoted to an area of mathematical physics called the spectral theory of random operators. Mathematical physics is a branch of mathematics concerned with the mathematical description of physical phenomena and the mathematical analysis of the laws of nature. One of the parts of mathematical physics, called the theory of disordered systems, is devoted to the study of phenomena appearing in the presence of randomness. For example, an ideal wire is a conductor. However, a material wire contains impurities; how is the conductivity influence the impurities? Such questions are of inherent theoretical interest and also of importance in applications.

Spectral theory provides a mathematical framework in which such questions can be posed and studied. The spectral theory of random operators is an active field of research since the work of P.W.Anderson and I.M.Lifshitz in the 1950-s, however, some of the central problems are still open. A challenging and interesting aspect of these problems is the fascinating interplay between the randomness and the geometry of the medium.

As part this project, we have developed new mathematical methods which allowed to address some of these questions. In particular, we have gained new understanding of the Wegner orbital operator, introduced in the 1980s by F.J.Wegner to describe the motion of a quantum particle with many internal degrees of freedom in a disordered medium.

Some of the methods developed as part of the project have found applications significantly exceeding the original scope, and ranging from number theory to statistical mechanics. We have also discovered new connections between the theory of random operators and other fields of mathematics, including classical analysis, probability theory and representation theory. It even turned out that a randomised version of the fifteen puzzle, a sliding puzzle from the XIX century, enjoys a close connection to the random operators studied by Lifshitz and Anderson!

Spectral theory provides a mathematical framework in which such questions can be posed and studied. The spectral theory of random operators is an active field of research since the work of P.W.Anderson and I.M.Lifshitz in the 1950-s, however, some of the central problems are still open. A challenging and interesting aspect of these problems is the fascinating interplay between the randomness and the geometry of the medium.

As part this project, we have developed new mathematical methods which allowed to address some of these questions. In particular, we have gained new understanding of the Wegner orbital operator, introduced in the 1980s by F.J.Wegner to describe the motion of a quantum particle with many internal degrees of freedom in a disordered medium.

Some of the methods developed as part of the project have found applications significantly exceeding the original scope, and ranging from number theory to statistical mechanics. We have also discovered new connections between the theory of random operators and other fields of mathematics, including classical analysis, probability theory and representation theory. It even turned out that a randomised version of the fifteen puzzle, a sliding puzzle from the XIX century, enjoys a close connection to the random operators studied by Lifshitz and Anderson!

During the 5 years of the project, we managed to make significant progress in several directions, of which we mention just a few.

The first group of results pertains to a model called the Wegner orbital model, introduced by F. Wegner in the 1970-s to model the motion of a quantum particle with many internal degrees of freedom in a disordered medium. We managed to quantify some the dependence of some of the characteristics of the model on the number of orbitals.

Another important group of results, obtained by the Magazinov as part of his work on the project, pertains to the hard sphere model in statistical mechanics, going back to the work of Boltzmann from the XIX century. In this model, rigid unit spheres are randomly placed in space, with the condition that they can not overlap. Answering a long-standing open question, Magazinov showed the following: for a typical configuration, if each sphere is inflated by an arbitrary small amount, then an infinite component appears, provided that the density of spheres is sufficiently close to the maximally possible.

Finally, we have managed to develop several new analytic tools, which have found applications in the original scope of the project and beyond it. For example, Kopelevitch used methods from complex analysis to resum, i.e. to attach rigorous mathematical meaning, to a divergent series (i.e. an infinite sum such as 1 - 2 + 4 - 8 + 16 ... which approaches no particular limit point) describing one of the classical random matrix ensembles. Such a resummation was previously thought to be impossible.

All the results were published in international journals and also uploaded to an open repository (arxiv.org). They were also presented at a number of international conferences and seminars. I have also lectured on the results at several summer schools directed at young researchers (PhD students and post-docs). To further disseminate the outcomes and to stimulate the interaction with other research groups, I have coörganised several conferences and workshops, particularly, the international conference "Classical and Quantum Motion in Random Environment". I have also contributed to a popular article devoted to the project.

The first group of results pertains to a model called the Wegner orbital model, introduced by F. Wegner in the 1970-s to model the motion of a quantum particle with many internal degrees of freedom in a disordered medium. We managed to quantify some the dependence of some of the characteristics of the model on the number of orbitals.

Another important group of results, obtained by the Magazinov as part of his work on the project, pertains to the hard sphere model in statistical mechanics, going back to the work of Boltzmann from the XIX century. In this model, rigid unit spheres are randomly placed in space, with the condition that they can not overlap. Answering a long-standing open question, Magazinov showed the following: for a typical configuration, if each sphere is inflated by an arbitrary small amount, then an infinite component appears, provided that the density of spheres is sufficiently close to the maximally possible.

Finally, we have managed to develop several new analytic tools, which have found applications in the original scope of the project and beyond it. For example, Kopelevitch used methods from complex analysis to resum, i.e. to attach rigorous mathematical meaning, to a divergent series (i.e. an infinite sum such as 1 - 2 + 4 - 8 + 16 ... which approaches no particular limit point) describing one of the classical random matrix ensembles. Such a resummation was previously thought to be impossible.

All the results were published in international journals and also uploaded to an open repository (arxiv.org). They were also presented at a number of international conferences and seminars. I have also lectured on the results at several summer schools directed at young researchers (PhD students and post-docs). To further disseminate the outcomes and to stimulate the interaction with other research groups, I have coörganised several conferences and workshops, particularly, the international conference "Classical and Quantum Motion in Random Environment". I have also contributed to a popular article devoted to the project.

The results obtained so far constitute significant progress beyond the state of art:

- The work on random band matrices and other quasi-one-dimensional systems. We have made significant progress on several outstanding open problems in this area.

- The study of the connections between random operators, random matrices, and other fields -- particularly, representation theory and number theory.

- The study of the interplay between the problems pertaining to quantum and classical particles.

- The work on random band matrices and other quasi-one-dimensional systems. We have made significant progress on several outstanding open problems in this area.

- The study of the connections between random operators, random matrices, and other fields -- particularly, representation theory and number theory.

- The study of the interplay between the problems pertaining to quantum and classical particles.