Soshnikov The principal investigator works on several problems in random matrix theory and random point processes. The main emphasis of the research is on the statistical properties of the eigenvalues of large random matrices and fundamental questions about an important class of random point processes (namely determinantal and pfaffian random point processes) appearing in random matrix ensembles and applications of random matrix technique in random growth models, representation theory and combinatorics. The problems of a special interest to the principal investigator in random matrix theory include local statistical properties of the spectrum of large sample covariance matrices, spectral properties of the adjacency matrices of random graphs and Poisson statistics at the edge of the spectrum for random matrices with heavy tails of marginal distributions. The principal investigator expects to collaborate in this research with a theoretical physicist Yan Fyodorov and a combinatorialist Benny Sudakov. The problems in random point processes include, among others, the study of the ergodic properties of the translation-invariant pfaffian random point processes and the analysis of the Janossy densities in determinantal and pfaffian ensembles. The random matrix models and random point processes that are proposed to study come from, or have applications in multivariate statistical analysis (principal component analysis), nuclear physics (statistics of energy levels of heavy nuclei), solid state physics (modeling transport properties of small metallic particles and quantum dots), quantum chaos (spectral properties of the quantum analogues of strongly chaotic classical systems) and theoretical computer science (computational complexity, statistical analysis of errors and linear numerical algorithms). The importance of the field increases as many different areas of mathematics and physics including combinatorics, representation theory, number theory, integrable systems, random growth models, quantum gravity appear to have deep and fruitful connections to random matrices.
