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.
