Invariant Measures for Random Growth Processes

在这个计划中，我们将研究第一次通过渗流，最后一次通过渗流和理查森的增长模型。与把它们作为次可加过程来研究的典型方式不同，我们将把它们看作是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。对于这些系统中的每一个，我们建议分类不变的措施。我们认为，以这种方式研究第一通道渗流不仅是有趣的，在其本身的权利，它也将导致解决有关原始的第一通道渗流和理查森的增长模型的问题。这些包括表明，在第一通道渗流无限测地线的数量是无限的，并限制形状的边界没有尖锐的角落。对于两类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上的最后一次通过渗流与平面菱形平铺上某些动力学的不变测度（称为“完全不对称六边形翻转”）之间的联系。我们将分析完全非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中最后一次渗流的极限形状。在这个提议中，我们研究了几个模拟不同物体在空间和时间中生长的过程。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同一区域。 我们将研究在什么样的增长规则下，有可能不止一个对象生存任意长的时间，反之，在什么样的增长规则下，一个对象征服所有其他对象。 在这个建议将寻求分析的过程中的相互作用的粒子系统，这是在20世纪80年代和90年代开发的框架。

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.