The principal investigator is interested in the issues surrounding the notion of universality in systems with large degrees of freedom, specifically in two contexts-2d critical Ising model and the random matrix models. We would like to investigate rigorously at least three directions - 1. Rotational invariance of various correlation functions of the 2d critical Ising model. 2. We would like to investigate more fully certain aspects of universality in the 2d critical Ising model. 3. We would like to understand or derive a certain asymptotic expansion for a class of the random band matrix models using the supersymmetric random matrix formalism.
In a nutshell, we are interested in obtaining precise macroscopic information while we are only give some microscopic descriptions. This situation is ubiquitous in life ranging from biology to physics where we only know some local laws, yet we would like to know what happens at a larger scale. Aside from such general considerations, we hope that the results and methods obtained from this investigation may prove useful for other outstanding conjectures such as establishing universality for the general 2d critical Potts models and the various forms of universality phenomena associated with the higher dimensional random Schroedinger operator.
