Stochastic interacting systems: connections, fluctuations and applications

概率论中的许多经典结果考虑独立（或弱相依）且通常为同分布的随机变量。在这些情况下，关于样本平均值的收敛性（大数定律）及其波动性（中心极限定理）的最基本问题得到了很好的理解。当失去独立性时，情况就大不相同了，人们发明了几种模型来帮助我们理解各种现象的观测，如凝聚态和统计物理学、分子、细胞水平和种群生物学、地理学、社会学以及工程学。这里考虑的过程的一个共同特征是失去独立性（或弱依赖性）。物理系统中的原子运输过程相互作用，分子在小细胞通道中或身体狭窄血管中的血细胞中前进时也是如此。工程师们看到了我们街道上发生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部分相关。种群中的感染或树木间的森林火灾都是作为随机生长的表面推进的，其中生长速度取决于局部邻域中同一表面的形状。雪崩被模拟为自我加强相互作用的随机运动的雪块，而在一个延伸的河床的变化强烈影响那些在其他位。舆论投票与谣言传播是一个交叉关联的社会学过程，存在大量的空间互动。数据传输系统使用通过路由算法以复杂方式交互的队列。在上述所有例子中，任何试图建立随机观测模型的尝试都会很快导致随机变量序列的随机依赖性，其中许多过程的时间和/或空间依赖性使得经典方法无法应用。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，中心极限定理和正态分布作为普遍极限的比例将不再有效。新的，仍然是非常普遍的标度阶和极限分布出现，特点是一般类的相互作用过程，但本质上不同于通常的独立图片。最近几年取得了一些突破性成果的例子是所谓的Kardar-Parisi-Zhang方程及其特征时间^{1/3}标度和Tracy-Widom极限分布。因此，这些领域的数学研究需要本质上的新思想，这些思想往往与数学（泛函分析，代数，组合学，动力系统等）和物理学的各个领域密切相关。我们的研究目的是在一些上述模型的建设，平稳的行为，波动和标度限制的基本问题。更具体地说，我们调查：-随机游走在固定和动态变化的随机环境中，在那里一个简单的随机运动改变其行为取决于其位置;-相互作用的粒子系统，其中许多，否则简单的运动相互作用。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，并且大大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宝贵的贡献，因为数学与其他领域的互动，因为各种各样的想法，一个必然发明的缺乏传统的工具。

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:- random walks both in fixed and dynamically changing random environments, where an otherwise simple random motion changes its behaviour depending on its position;- interacting particle systems, where many, otherwise simple motions interact with each other.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.