The principal investigator works on problems in Random Matrix Theory and Probability Theory with the main emphasis on statistical properties of the eigenvalues of large random matrices. The P.I. intends to continue his studies of the distribution of the largest eigenvalues in Wigner, sample covariance, and related ensembles of random matrices. The proposed research is closely related to the Tracy-Widom distribution of the largest eigenvalues of Hermitian and real symmetric random matrices that appears in many important problems in physics, statistics, and theoretical computer science: crystal shapes, exclusion processes, directed polimers in random media, principal component analysis. In particular, the P.I. proposes to establish Tracy-Widom distribution in some important ensembles of random matrices. Recently, the P.I. discovered Poisson statistics of the largest eigenvalues in random matrices with power-law tailed elements. Remarkably, such matrices very often appear in financial applications. The P.I. intends to continue his studies of heavy-tailed random matrices and, in particular, to study in detail the transition form the Tracy-Widom regime to the Poisson regime.
The random matrix models studied in the project come from or have applications in nuclear physics (statistics of energy levels of heavy nuclei), mathematical statistics (principal component analysis), theoretical computer science (computational complexity, statistical analysis of errors, linear numerical algorithms), population biology, mathematical finance, and solid state physics (modeling transport properties of small metallic particles and quantum dots). The importance of the field has increased significantly in the last ten years as many different areas of mathematics and physics including combinatorics, representation theory, number theory, quantum gravity, integrable systems, and random growth models have been shown to possess deep and fruitful connections to random matrix theory.
