Probabilistic approach to cellular automata and related models

其中最方便的物理过程模型是确定性和随机元胞自动机。这些数学对象描述了通过重复更新局部规则而进化的格子上的配置。它们被用来模拟自然现象，并提供了对许多科学领域的基本组织原则的见解，包括物理、生物学、计算机科学和社会科学。从更抽象的角度来看，元胞自动机是用于描述和分类产生规定的全球现象的局部动力学的合适工具。这个项目的第一个目标是开发和使用现代概率论的技术，以及计算工具，来研究细胞自动机的各个方面。然后，第二个目标是利用收集的见解来研究受晶体生长、遗传学和社会网络启发的扩散、非局部和高维模型的更复杂的动力学。三个主题将元胞自动机与概率论联系起来。第一个问题是研究自组织，涉及从随机初始状态到给定规则的演化。讨论的主题包括：成核，即协调接管可用空间的中心的随机形成；决定典型轨迹行为的高度相关的渗流结构；以及通过分支随机游动的大偏差进行的Lyapunov稳定性分析。第二个方向是随机规则，采用分支过程方法来形成稳定的周期结构。第三个经典主题是随机噪声对确定性更新的扰动。对于更一般的模型，该项目旨在研究非局部动力学，重点是成核理论、相变的性质、空间异质性的影响以及短程和长程相互作用之间的边界。计算是研究的重要组成部分，涉及模拟、算法分析、计算机辅助证明、数值和统计方法以及可视化技术。

Among the most convenient models of physical processes are deterministic and random cellular automata. These mathematical objects describe configurations on lattices which evolve by a repeated update of a local rule. They have been utilized to model natural phenomena and have offered insights into fundamental organizational principles in many scientific fields, including physics, biology, computer science, and social sciences. From a more abstract perspective, cellular automata are a suitable tool used to characterize and catalog local dynamics which generate a prescribed global phenomenon. The first aim of this project is to develop and use techniques from modern probability theory, as well as computational tools, to study diverse aspects of cellular automata. Then second aim is to use the gathered insights to investigate more complex dynamics of diffusive, nonlocal and high-dimensional models inspired by crystal growth, genetics and social networks.Three themes connect cellular automata with probability theory. The first issue, motivated by studying self-organization, involves evolution of a given rule from random initial states. The addressed topics include: nucleation, that is, random formation of centers that orchestrate a takeover of the available space; highly dependent percolation structures that determine the behavior of typical trajectories; and Lyapunov stability analysis through large deviations of branching random walks. The second direction are random rules, with a branching process approach to formation of stable periodic structures. The third, classical, theme is perturbation of deterministic updates by random noise. For more general models, the project aims to study non-local dynamics, focusing on nucleation theory, the nature of phase transitions, effects of space heterogeneity, and the boundary between short- and long-range interactions. Computation is a significant component of the research and involves simulations, analysis of algorithms, computer-aided proofs, numerical and statistical methods, and visualization techniques.