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。特别是，该项目研究了建立在随机和伪随机行走基础上的聚集过程的统计物理学模型。例如，在前一种情况下，扩散受限的聚集过程通过与推动珊瑚形成的过程类似的过程产生复杂的分形图，但这些过程产生长而多刺的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一群粒子，最终导致复杂的分形图案，以及在自然界中重新出现的统计规律(例如，地震、雪崩等的频率分布)。除其他目标外，该项目旨在更好地了解阿贝尔沙堆的行为对底层晶格的依赖。一个特别有趣的情况是，周期格子受到随机边删除的影响；在这种情况下，阿贝尔沙堆是随机环境中的伪随机聚集过程，我们预计它的行为类似于周期环境中的随机聚集过程。其他主题包括欧几里得泛函方面的工作；特别是，通过典型(即随机)点集及其算法近似来完善我们对旅行者旅行等结构之间的渐近关系的理解。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.