Invariant Measures for Random Growth Processes

在这个提案中，我们将研究第一通道渗流、最后通道渗流和Richardson增长模型。与将它们作为次加性过程研究的典型方式不同，我们将它们视为相互作用的粒子系统。我们将考虑几种将首次通过渗流转化为马尔可夫过程的方法。这些过程是相互作用的粒子系统的新例子。我们将在许多图上研究这些过程，包括R^d中的Z^d三角剖分和Delaunay三角剖分。我们建议对每个系统的不变度量进行分类。我们认为，以这种方式研究首通道渗流不仅本身很有趣，而且还将导致关于原始首通道渗流和Richardson增长模型的猜想的解决。其中包括证明了第一通道渗流中无限多条测地线的数目是无限的，并且极限形状的边界没有尖锐的角点。对于这两种类型的Richardson模型，我们认为这种方法将表明当两种感染具有不同的速度时，共存是不可能的。我们还将研究Z^3上的最后一次渗流与平面菱形瓦片上某些动力学不变测度(称为“完全不对称六边形翻转”)之间的联系。我们将分析完全不对称六边形翻转过程的不变度量，并利用不变度量研究Z^3中最后一次通过的渗流的极限形状。在这个方案中，我们研究了几个模拟不同对象在空间和时间上增长的过程。这些物体可能是不同的物种或种群。在模型中，我们看到不同的对象正在扩张和争夺领土。我们的模型还假设两个不同的对象不能同时占据相同的区域。我们将研究在什么增长规则下，不止一个物体可能在任意长时间内存活，反之，在什么增长规则下，一个物体可以征服所有其他物体。在这项提案中，将寻求在1980年代和1990年代开发的相互作用粒子系统的框架内分析过程。

In this proposal we will study first-passage percolation, last-passage percolation and Richardson's growth model. In contrast to the typical manner of studying these as subadditive processes we will view them as interacting particle systems. We will consider several ways of turning first-passage percolation into a Markov process. These processes are new examples of interacting particle systems. We will study these processes on many graphs including Z^d and Delaunay triangulations in R^d. For each of these systems we propose to classify the invariant measures. We feel that studying first passage percolation in this manner is not only this is interesting in its own right, it will also lead to the resolution of conjectures about the original first-passage percolation and Richardson's growth model. These include showing that the number of infinite geodesics in first-passage percolation is infinite and that the boundary of the limiting shape has no sharp corners. For the two type Richardson's model we feel this method will show that coexistence is impossible when the two infections have different speeds. We will also study the connection between last-passage percolation on Z^3 and invariant measures for certain dynamics (called "totally asymmetric hexagon flipping") on lozenge tilings of the plane. We will analyze the invariant measures for the totally asymmetric hexagon flipping process and use the invariant measures to study the limiting shape of last-passage percolation in Z^3.In this proposal we study several processes which model the growth of different objects in space and time. These objects may be different species or populations. In the models we look at the different objects are expanding and competing for territory. Our models also assume that two different objects cannot occupy the same area at the same time. We will study under what rules of growth it is possible that more than one of the objects survive for an arbitrary large time and conversely under which rules of growth does one object conquer all of the others. In this proposal will seek to analyze the processes in the framework of interacting particle systems which was developed in the 1980s and 1990s.