The PI proposes two major themes of research for the next three years. One is the study of eigenvectors of sparse but large random graphs. Although such eigenvectors are of both theoretical and applied interest, very few results are rigorously known about them. The PI proposes to use techniques from Random Matrix Theory and combinatorics to continue his investigation of spectral properties of sparse random graphs. A particular goal will be to study the effect of cycles in the graph on the localization/ delocalization property of the eigenvectors. The other line of work is related to models used in mathematical finance. Particle system models from statistical physics have been recently successfully used to explain observed financial data and construct portfolios that can be thought of as arbitrage opportunities with respect to the market index. The PI proposes, on the one hand, to further study the properties of these processes as particle systems, and on the other, collaborate with industry groups to help them build more efficient portfolios.
The mathematics we study has the following broader impact. Random graphs and networks are popular in diverse areas such as social networks, models for the internet, computer vision, and number theory. Several natural optimization problems on graphs (e.g., figuring out clusters, or ranking algorithms such as Google PageRank) involve what are called eigenvectors of the graph. If the network grows randomly, its eigenvectors are random, and it is of interest how they behave. The proposed work in finance is useful in building portfolios that gain from the presence of volatility in financial market. As such it is of great interest to practitioners who would like to harvest growth out of market fluctuations. The proposed research is a further study in an area which is already being applied by portfolio managers, some of whom are in consultation with the PI and his students.
