Particle systems, growth models and their probabilistic structures

概率论中的许多经典结果考虑独立（或弱相关）且通常是同分布的随机变量。在这种情况下，关于样本平均值的收敛性（大数定律）及其波动（中心极限定理）的最基本问题都得到了很好的理解。当失去独立性时，情况就大不相同了。人们发明了几个模型来帮助我们理解各种现象中的观察结果，如凝聚态物质和统计物理学、分子、细胞水平和种群生物学、地理学、社会学和工程学。这里考虑的流程的一个共同特征是独立性的丧失（或弱依赖性）。物理系统中的原子运输过程相互作用，分子在小细胞通道或身体狭窄血管中的血细胞中移动时也会相互作用。工程学看到了发生在我们街道上的交通流参与者之间的类似互动。趋化性和其他生物自主运动依赖于移动细胞的局部环境，使其行为与看到相同环境的路径的任何早期片段相关。种群中的感染或树木之间的森林火灾作为一个随机生长的表面推进，其生长速度取决于当地社区中相同表面的形状。雪崩被建模为雪块的自我强化相互作用的随机运动，而河床一段的变化会强烈影响其他部分的变化。投票意见与谣言传播显然是一个相互关联的社会学过程，具有大量的空间交互作用。数据传输系统使用的队列通过路由算法以复杂的方式相互作用。在以上所有的例子中，任何试图建立随机观察模型的尝试都会很快导致随机变量的随机依赖序列。在许多这些过程中，时间和/或空间的依赖性使得不可能应用经典方法。在某些情况下，可以发明新的想法来证明大数定律和中心极限定理的行为。在其他一些情况下，中心极限定理的缩放和正态分布作为普遍极限将不再有效。新的，仍然非常普遍的尺度顺序和极限分布出现了，它们是相互作用过程的一般类别的特征，但与通常的独立图像本质上不同。近年来取得突破性成果的一个例子是所谓的kardar - paris - zhang方程，其特征是时间^{1/3}缩放和Tracy-Widom极限分布。因此，这些领域的数学研究本质上需要与数学（泛函分析、代数、组合学、动力系统等）和物理学等各个领域密切互动的新思想。我们的研究目标是在一些上述模型的结构，平稳行为，波动和缩放限制的基本问题。更具体地说，我们研究：-相互作用的粒子系统，其中许多，否则简单的运动相互作用，-在模拟多孔介质，道路网络，雷击和其他现象的随机环境中的最佳路径。我们的每一个研究问题都与经典的既定情景相去甚远。为了得出结果，我们应用了来自其他数学领域的原始概率思想和工具。我们可以在这些系统中展示的行为是新的，并且极大地提高了我们对使用这些模型的其他科学的一般理解。我们的工作也代表了对数学的宝贵贡献，因为与其他领域的相互作用，因为在缺乏传统工具的情况下，人们必然会发明各种各样的想法。

Many classical results in probability theory consider independent (or weakly dependent) and often identically distributed random variables. The most fundamental questions regarding convergence of the sample average (Law of Large Numbers) and the fluctuations thereof (Central Limit Theorem) are well understood in these cases. The picture becomes very different when independence is lost.Several models have been invented to help our understanding of observations in diverse phenomena such as condensed matter and statistical physics, molecular, cell-level and population biology, geography, sociology and also engineering. A common feature of the processes considered here is the loss of independence (or weak dependence). Atomic transport processes in physical systems interact with each other, as well as molecules do when progressing in small cellular channels, or blood cells in narrow vessels of the body. Engineering sees similar interactions between participants of traffic flow that is happening on our streets. Chemotaxis and other autonomous motion of biology depend on the local environment of the moving cell, making its behaviour correlated to any earlier segment of the path that saw the same environment. Infections in a population or forest fires among trees advance as a randomly growing surface where the speed of growth depends on the shape of the very same surface in a local neighbourhood. Avalanches are modeled as self-reinforcing interacting random motion of blocks of snow, while changes in one stretch of a riverbed strongly influence those in other bits. Voting opinions and rumour spreading is obviously a very cross-correlated process of sociology with lots of spatial interactions. Data-transmission systems use queues that interact in complicated ways via routing algorithms. In all of the above examples any attempts to build a stochastic model of observations quickly lead to stochastically dependent sequences of random variables.The temporal and/or spatial dependence in many of these processes makes it impossible to apply the classical methods. In some instances new ideas can be invented to prove the Law of Large Numbers and Central Limit Theorem behaviour. In some other cases the scaling of the Central Limit Theorem and the Normal distribution being the universal limit will simply not be valid anymore. New, still very universal scaling orders and limit distributions emerge, characteristic to general classes of interacting processes but essentially different from the usual independent picture. An example where several groundbreaking results have been achieved in recent years is the so-called Kardar-Parisi-Zhang equation with its characteristic time^{1/3} scaling and Tracy-Widom limit distributions.Mathematical research in these areas thus require essentially new ideas that often strongly interact with various fields of mathematics (functional analysis, algebra, combinatorics, dynamical systems among others) and physics. Our research aims at fundamental questions of constructions, stationary behaviour, fluctuations and scaling limits in some of the above models. More specifically, we investigate:- interacting particle systems, where many, otherwise simple motions interact with each other,- optimal paths in a random environment that models porous media, road networks, lightning strikes and other phenomena.Each of our research questions concerns cases that are far from the classical well-established scenarios. To come up with results we apply original probabilistic ideas and tools from other fields of mathematics. The behaviour we can demonstrate in these systems is new, and greatly improves our general understanding in other sciences which use these models. Our work also represents valuable contributions to mathematics because of interactions with other areas and because of the variety of ideas that one necessarily invents in the lack of traditional tools.