Eigenvectors of random graphs, random matrices and triple collisions

PI提出了未来三年的两大研究主题。一是稀疏大随机图特征向量的研究。虽然这样的特征向量的理论和应用的兴趣，很少有结果是严格知道他们。PI建议使用随机矩阵理论和组合学的技术来继续研究稀疏随机图的谱特性。一个特定的目标将是研究图中的循环对特征向量的局部化/离域性的影响。另一个工作领域与数学金融中使用的模型有关。最近，统计物理学的粒子系统模型已成功地用于解释观察到的金融数据，并构建可以被认为是市场指数套利机会的投资组合。PI建议，一方面，进一步研究这些过程作为粒子系统的特性，另一方面，与行业团体合作，帮助他们建立更有效的投资组合。 我们学习的数学有以下更广泛的影响。随机图和网络在不同的领域都很受欢迎，例如社交网络、互联网模型、计算机视觉和数论。图上的几个自然优化问题（例如，计算出聚类，或排名算法，如Google PageRank）涉及所谓的图的特征向量。如果网络是随机增长的，那么它的特征向量也是随机的，它们的行为是很有趣的。拟议的金融工作是有用的，在建立投资组合，从金融市场的波动性中获益。因此，对于那些希望从市场波动中收获增长的从业者来说，这是非常有趣的。拟议的研究是一个领域，这是已经被应用的投资组合经理，其中一些人正在与PI和他的学生进行磋商的进一步研究。

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.