Discrete Random and Pseudorandom Structures

这个项目旨在更好地理解随机和伪随机过程，以及它们之间的相互作用。特别地，该项目研究了建立在随机和伪随机行走上的聚合过程的统计物理模型。例如，在前一种情况下，扩散有限的聚集过程通过类似于驱动珊瑚形成的过程产生复杂的分形，但这些过程产生长刺状结构的倾向仍然没有得到严格的理解。在后一种情况下，阿贝尔沙堆过程通过粒子的伪随机分布建立了一个粒子群，最终产生了复杂的分形模式，以及在整个自然界中重现的统计规律（例如，地震、雪崩等的频率分布）。在其他目标中，该项目旨在更好地理解阿贝尔沙堆对底层晶格行为的依赖。一个特别有趣的情况是，当一个周期晶格遭受随机边删除时；在这种情况下，阿贝尔沙堆是一个随机环境中的伪随机聚集过程，我们期望它的行为类似于周期性环境中的随机聚集过程。其他课题包括欧几里得泛函的研究；特别是，通过典型的（即随机的）点集和它们的算法近似值，改进我们对诸如旅行推销员之旅等结构之间的渐近关系的理解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project aims to better understand random and pseudorandom processes, and the interplay between them. In particular, the project studies models from statistical physics of aggregation processes built on both random and pseudorandom walks. For example, in the former case, diffusion limited aggregation processes produce intricate fractals through processes analogous to those that drive the formation of corals, but the propensity of these processes to give rise to long, spiny, structures is still not rigorously understood. In the latter case, the Abelian sandpile process builds a cluster of particles through a pseudorandom distribution of particles that ends up giving rise to intricate fractal patterns, as well as statistical laws which reappear throughout nature (e.g., as the frequency distributions of earthquakes, avalanches, etc).Among other goals, this project aims to better understand the dependence of the behavior of the Abelian sandpile on the underlying lattice. One particularly interesting case is that where a periodic lattice is subjected to random edge-deletions; in this case the Abelian sandpile is a pseudorandom aggregation process on a random environment, which we expect to behave like a random aggregation process on a periodic environment. Other topics include work on Euclidean functionals; in particular, refining our understanding of asymptotic relationships between structures like Traveling Salesperson Tours through typical (i.e. random) point sets, and their algorithmic approximations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.