Deterministic and Stochastic Growth Models

Gravner教授将在细胞自动机方面进行进一步的研究。这项研究将继续朝着几个新的方向发展。该项目中展示的增长模型范围从简单的门槛规则和竞争动力学，到低维和高维空间中的随机漫步和渗流模型的交互系统。其中主要的主题有：环境和规则中的随机扰动的影响，随机界面的波动，分形图案的产生，大邻域系统的行为，随机行走系统的大范围限制，以及基因空间结构与遗传多样性的关系。这个项目的目的是了解各种物理系统将扰动传播到远离平衡的地方的微观和宏观原理。这种生长动态的例子包括晶体生长、流行病的传播、波浪的传播、物种之间的竞争和遗传多样性。这类研究有助于理解以下自然现象：生存和生长的阈值密度与物种在不友好环境中的相互作用范围、尖锐几何晶体形状和粗糙随机晶体形状之间的转换、噪声环境对感染传播的影响、基于大范围相互作用或快速扩散的近似的有效性以及中性突变在遗传漂移中的作用。

0204367Gravner Professor Gravner will pursue further research in cellular automata. The research will continue in several new directions. The growth models featured in this project range from simple threshold rules and competition dynamics, to interacting systems of random walks and percolation models in low- and high-dimensional spaces. Among the principle topics are: effects of random perturbations in the environment and in the rule, fluctuations of random interfaces, generation of fractal patterns, behavior of large--neighborhood systems, large scale limits of systems of random walks, and relationship between structure of genotype space and genetic diversity. The aim of this project is to understand microscopic and macroscopic principles by which various physical systems propagate disturbances far from the equilibrium. Among instances of such growth dynamics are crystal growth, spread of epidemics, propagation of waves, competition between species, and genetic diversification. Such research can contribute to understanding of such natural phenomena as: threshold density for survival and growth vs. the range of interaction for species in an unfriendly environment, transition between sharp geometric crystal shapes and rough random ones, effects of noisy environments on spread of infections, validity of approximation based on large range of interaction or fast diffusion, and role of neutral mutations in genetic drift.