Eigenvectors of random graphs, random matrices and triple collisions

国际和平研究所提出了未来三年的两个主要研究主题。一个是研究稀疏的大型随机图的特征向量。虽然这种特征向量在理论上和应用上都很有意义，但很少有关于它们的结果是严格已知的。PI建议使用随机矩阵理论和组合学的技术来继续他对稀疏随机图的谱性质的研究。一个特别的目标将是研究图中的圈对特征向量的局部化/非局部化属性的影响。另一项工作与数学金融中使用的模型有关。最近，统计物理学中的粒子系统模型被成功地用于解释观察到的金融数据，并构建可被视为相对于市场指数的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性质，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很流行，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，找出簇或诸如Google PageRank的排名算法)涉及所谓的图的特征向量。如果网络随机增长，那么它的特征向量就是随机的，它们的行为是令人感兴趣的。拟议的金融工作对于建立从金融市场波动中获益的投资组合是有用的。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，他们中的一些人正在与PI和他的学生进行磋商。

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.