Eigenvectors of random graphs, random matrices and triple collisions

PI提出了未来三年的两大研究主题。一个是研究稀疏但大的随机图的特征向量。尽管这些特征向量在理论和应用上都很有意义，但很少有关于它们的严谨的结果。PI建议使用随机矩阵理论和组合学的技术继续他对稀疏随机图的谱性质的研究。一个特定的目标将是研究图中循环对特征向量的局部/非局部性质的影响。另一项工作与数学金融中使用的模型有关。来自统计物理学的粒子系统模型最近被成功地用于解释观察到的金融数据和构建投资组合，这些投资组合可以被认为是相对于市场指数的套利机会。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，与工业团体合作，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。随机图和网络在社交网络、互联网模型、计算机视觉和数论等各个领域都很流行。图上的一些自然优化问题（例如，计算聚类，或排序算法，如b谷歌PageRank）涉及图的特征向量。如果网络是随机增长的，那么它的特征向量就是随机的，它们的行为是很有趣的。在金融领域提出的工作有助于建立从金融市场波动中获益的投资组合。因此，希望从市场波动中获得增长的从业者对此非常感兴趣。拟议的研究是在一个已经被投资组合经理应用的领域的进一步研究，其中一些人正在与PI及其学生协商。

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.