The flat edge in last passage percolation

最后通道渗透的严格数学模型研究了一个随机簇的演化，它遵循一些精确的、预定的规则在二维空间中生长。生长的速度取决于附着在空间上的潜在“随机环境”，有效地创造了一种“随机介质”。理论上，人们可以设计各种机制来解释增长是如何发生的，在最后通道渗透中，最常见的一种可以用队列串联系统（称为“完全不对称简单排除过程”）来解释，在该系统中，许多客户在一系列服务器前排成一行排队。为了便于说明，请考虑一个具体但噩梦般的场景，在一个小机场，乘客排成一行排队办理登机手续，然后是行李托运，接着是护照检查，安检，最后是在登机口排队。在不妨碍其他顾客的情况下，所有的服务器都需要每个人清理干净。当第一个客户离开第一个服务器时，她会立即前往第二个服务器。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，系统继续发展。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）加入随机集群的时间。每当客户通过服务器时，相应的点就会添加到集群中。随机服务时间对应随机环境。这是最近近邻最后通道渗透的粒子系统解释。还有一种几何解释：在任何给定的时间，我们都可以把这看作是一个随机曲面的演变，因此，我们关注一些定义明确的几何概念，如形状、随机测地线和曲率。毫不奇怪，随机环境的分布对表面的形状有很大的影响。在一些例子中，数学家可以明确地计算形状并检查，当适当缩放时，它可以被视为一个“严格凹的，可微的”表面；换句话说，如果我们“缩小”并从远处观察随机集群的边界，我们将看到一个漂亮的曲线形状，没有平坦的部分（严格的凹性），没有尖锐的点（可微性）。该提案中的项目研究了这种形状，当它呈现出一种称为“平边”的东西时，这意味着形状不再是严格的凹形。这是一个理论上的数学建议。我们将研究“非均匀”环境下的最后一段渗流形状。“不均匀”这个词的意思是每个服务人员在一段时间后会改变服务的方式，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。我们不仅将这些结果扩展到最近邻居定向最后通道渗透，而且将其扩展到更一般的模型类别，例如哈默斯利过程的离散版本，它作为更难的“最长公共子序列”问题的独立版本。在这个过程中，我们还将开发平面边缘微观效应的数学基础，特别是它如何影响模型的测地线（可以告诉我们延迟发生在粒子系统中的特殊路径）。此外，我们将建立平面边缘的产生与测地线宏观波动之间的联系。理解这一点至关重要，原因有很多，其中最重要的是，这将提供进一步的证据，证明具有平边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.