The flat edge in last passage percolation

最后一次通过渗流的严格数学模型研究随机簇的演化，该簇通过遵循某些精确的预定规则在二维中生长。增长的速度取决于一个潜在的“随机环境”，它附着在空间上，有效地创造了一个“随机介质”。从理论上讲，人们可以设计各种机制来解释增长是如何发生的，在最后一次通过渗透中，最常见的一种可以用串联系统中的队列来解释（称为“完全不对称的简单排除过程”），其中许多客户在一系列服务器前排成一行。为了便于说明，请考虑一个小机场的具体但噩梦般的情况，乘客排队办理登机手续，然后是行李托运，然后是护照检查，安全检查，最后是在登机口排队。在不影响其他客户的情况下，所有服务器都需要由每个人按顺序清除。当第一个客户离开第一个服务器时，她会立即转到第二个服务器。然后，第二个客户开始由服务器1服务，而第二个客户为客户1服务，系统不断发展。服务时间是随机的，问题是某个客户（例如第n个）何时会清除给定的服务器（例如第m个）。这个随机时间（其中n清除m）被称为最后通过时间（对于点（m，n）），它正是点（m，n）加入随机簇的时间。每当客户通过服务器时，相应的点被添加到集群。随机服务时间对应于随机环境。这是最近邻末次渗流的粒子系统解释。还有几何解释：在任何给定的时间，我们可以将其视为随机曲面的演化，因此，我们关注一些定义良好的几何概念，如形状，随机测地线和曲率。毫不奇怪，随机环境的分布对曲面的形状有很大的影响。在一些例子中，数学家可以显式地计算形状，并检查，当适当缩放时，它可以被视为“严格凹的，可微的”曲面;换句话说，如果我们“缩小”并从远处观察随机聚类的边界，我们将看到一个漂亮的曲线形状，没有平坦的部分（严格的曲线），没有尖锐的点（可微性）。本提案中的项目研究这种形状时，它表现出所谓的“平边”，这意味着形状不再是严格的凹。这是一个理论上的数学命题。我们将研究在一个“非均匀”的环境中的最后一个通道的渗流形状。“非同质”这个词意味着每个服务器在一段时间后改变它为人们服务的方式，例如，他可能会累，所以他会放慢速度，或者变得更有效率，所以他会加快速度。在一个研究的例子中，极限形状函数表现出平坦的边缘和至少一个不可微点。我们将扩展这些结果不仅最近的邻居直接最后通过渗透，但更一般的一类模型，如离散版本的Hammersley过程，它作为一个独立的版本更难的“最长的共同子序列”的问题。在这个过程中，我们还将为平坦边缘的微观效应奠定数学基础，特别是它如何影响模型的测地线（可以告诉我们粒子系统中延迟发生的特殊路径）。此外，我们将建立一个连接之间的创建平坦的边缘和宏观波动测地线。理解这一点是至关重要的，原因有很多，其中最重要的一个是，这将提供进一步的证据，证明具有平坦边缘的模型不属于著名的KPZ普适类。

Rigorous mathematical models of last-passage percolation study the evolution of a random cluster which grows in two dimensions by following some precise, predetermined rules. The speed of the growth depends on an underlying 'random environment' that is attached to the space, effectively creating a 'random medium'. Theoretically, one can devise various mechanisms that explain how growth occurs, and in last-passage percolation the most common one can be explained using a queues in series system (called a 'totally asymmetric simple exclusion process') in which many customers queue up in a single line in front of a series of servers. For the benefit of exposition, consider a concrete yet nightmarish situation in a small airport, where passengers queue in a single file for check-in, then luggage-drop, followed by passport control, security and a finally queue at the gate. Without cutting in front of other customers, all of the servers need to be cleared by everyone, in order. When the first customer leaves the first server, she goes immediately to the second one. The second customer then starts being serviced by server 1, while the second one serves customer 1 and the system keeps evolving. Service times are random and the question is when a certain customer (e.g. the n-th) will clear a given server (e.g. the m-th). This random time (where n clears m) is called the last-passage time (for point (m,n)) and it is precisely the time that point (m,n) joins the random cluster. Every time a customer passes through a server, a corresponding point is added to the cluster. The random service times correspond to the random environment. This is the particle system interpretation of nearest neighbor last-passage percolation. There is also the geometric interpretation: At any given time we can be view this as the evolution of a random surface and, as such, we are concerned with some well-defined geometric notions such as shape, random geodesics and curvature. Not surprisingly, the distribution of the random environment has an effect on the shape the surface has for large times.In some examples, mathematicians can compute the shape explicitly and check that, when suitably scaled, it can be viewed as a 'strictly concave, differentiable' surface; in other words, if we 'zoom out' and look at the boundary of the random cluster from afar, we will see a nice curved shape with no flat segments (strict concavity) with no sharp points (differentiability). Projects in this proposal study this shape when it exhibits something called a `flat edge' , which says that the shape is not strictly concave anymore. This is a theoretical mathematical proposal. We will study the last-passage percolation shape in an 'inhomogeneous' environment. The word 'inhomogeneous' means that each server changes the way it serves people after some time, e.g. he can get tired, so he slows down, or becomes more efficient, so he speeds up. In one studied example the limiting shape function exhibits a flat edge and at least one point of non-differentiability. We will extend these results not only to nearest neighbour directed last-passage percolation but to a more general class of models, like a discrete version of the Hammersley process, which acts as an independent version of the harder 'longest common subsequence' problem. In the process, we will also develop the mathematical groundwork for the microscopic effects of the flat edge, in particular how it affects the geodesics' (special paths that can tell us where the delays happened in the particle system) of the model. Furthemore, we will establish a connection between the creation of the flat edge and macroscopic fluctuations of geodesics. Understanding this is crucial for many reasons, one of the most important being that this will provide further evidence that models with flat edges do not belong in the famous KPZ universality class.