Invariant Measures for Random Growth Processes

在本论文中，我们将研究首道渗流、末道渗流和理查森的增长模型。与将这些过程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。我们将考虑几种将第一遍渗透转化为马尔可夫过程的方法。这些过程是相互作用粒子系统的新例子。我们将在许多图上研究这些过程，包括Z^d和R^d中的Delaunay三角剖分。对于每一个系统，我们建议对不变测度进行分类。我们认为，以这种方式研究第一通道渗透不仅本身就很有趣，而且还将有助于解决有关原始第一通道渗透和理查森增长模型的猜测。这包括证明第一通道渗流中无限测地线的数目是无限的，以及极限形状的边界没有尖角。对于两种类型的Richardson模型，我们认为这种方法将表明，当两种感染的速度不同时，共存是不可能的。我们还将研究Z^3上的最后通道渗透与平面菱形平铺上某些动力学（称为“完全不对称六边形翻转”）的不变度量之间的联系。我们将分析完全不对称六边形翻转过程的不变测度，并利用不变测度研究Z^3中最后通道渗流的极限形状。在这个建议中，我们研究了几个过程，这些过程模拟了不同物体在空间和时间中的生长。这些对象可能是不同的物种或种群。在模型中，我们看到不同的物体在扩张和争夺领土。我们的模型还假设两个不同的物体不能同时占据同一区域。我们将研究在什么样的生长规律下，一个以上的物体有可能存活任意长的时间，反过来，在什么样的生长规律下，一个物体会征服所有其他的物体。在本提案中，将试图分析在20世纪80年代和90年代发展起来的相互作用粒子系统框架内的过程。

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.