Mathematical Sciences: Percolation and Related Problems

摘要渗透可能是最基本的模型，人们可以用它来研究小尺度随机性如何产生大尺度现象，如相变，而相变本质上是非随机的。亚历山大建议调查这个问题的以下几个方面。(1)利用渗流思想分析具有随机势的二维不可压缩流的行为。(2)一种新的随机聚类模型的性质，与标准FK随机聚类模型相似，与Ising模型密切相关。(3)利用渗透思想创建一个模型，在小尺度上定义，用于包含杂质的水结冰时发生的大尺度现象。(4)对于某些连续渗流模型，有限簇的几何形状及其与簇大小分布的关系。(5)平稳随机标记图的“最小生成森林”的图的渗透行为与其他性质之间的相互关系。(6)首道渗流在一定时间内所能达到的区域边界波动大小。此外，Alexander将研究与从许多短重叠片段重建DNA序列相关的字符串匹配问题。这项工作是数学家和物理学家正在进行的努力的一部分，目的是了解自然世界中各种系统的大规模或“宏观”特性如何反映小规模随机性。一个典型的例子是一块铁——每个原子都有一个按特定方向排列的磁场。这些方向是随机的，但附近的原子倾向于以大致相同的方向排列，特别是在温度较低的时候。从本质上讲，随温度升高而增加的随机性趋势与排列趋势是相互竞争的。类似排列的原子簇——全部“向上”、全部“向下”等——形成，这些簇的随机几何形状——簇的大小和概率——有助于确定铁的宏观性质。当温度低于某个精确的“临界点”时，铁的宏观行为会突然发生变化——排列的倾向胜出，因此形成了一个非常大的排列原子簇，铁可以变成磁铁。这种“临界现象”——当某些测量值超过临界值时宏观行为的突然变化——发生在各种情况下；例如，最近有人担心绕地球运行的人造垃圾的密度正在接近一个临界水平，超过这个水平，碰撞的频率将急剧增加。在其他系统中，小尺度随机性决定宏观特性，并可能发生临界现象，包括(i)穿过不规则物质的波，例如穿过地壳的地震波；（ii）在湍流存在的情况下由洋流输送热量，从而影响全球气候；（三）液体通过多孔物质渗透，如水或油通过地下岩石。数学家和物理学家早就认识到，小尺度随机性和宏观性质（包括临界现象）之间关系的许多方面并不取决于所研究的特定系统。因此，人们可以通过研究抽象系统来洞察现实世界的现象，而不是专门为磁铁、多孔岩石或物理世界的任何其他特定部分建模。亚历山大将要研究的系统就是这种抽象系统的例子。

9504462 Alexander Abstract Percolation is perhaps the most fundamental model in which one can study how small-scale randomness produces large-scale phenomena, such as phase transitions, which are essentially nonrandom. Alexander proposes to investigate the following aspects of the subject. (1) The use of percolation ideas to analyze the behavior of two-dimensional incompressible flows with random potentials. (2) The properties of a new random cluster model which, like the standard FK random cluster model, is closely related to the Ising model. (3) The use of percolation ideas to create a model, defined on a small scale, for large-scale phenomena which occur when water containing impurities freezes. (4) The geometry of finite clusters, and its relation to the cluster size distribution, for certain continuum models of percolation. (5) The interrelation between percolation behavior of the graph and other properties for the "minimal spanning forest" of a stationary random labeled graph. (6) The size of the fluctuations in the boundary of the region which can be reached by a fixed time in first-passage percolation. Additionally, Alexander will investigate a string-matching problem related to the reconstruction of DNA sequences from many short overlapping segments. This work is part of an ongoing effort by mathematicians and physicists to understand how small-scale randomness is reflected in large-scale, or "macroscopic," properties of various systems in the natural world. A typical example is a piece of iron--each atom has a magnetic field aligned in a particular direction. These directions are random, but nearby atoms tend to align in approximately the same direction, particularly when the temperature is low. In essence, the tendency toward randomness, which increases with temperature, competes with the tendency to align. Clusters of atoms with similar alignments--all "up,", all "down," etc.--are formed, and the random geometry of these clusters--what sizes of clusters occur with w hat probabilities--helps determine macroscopic properties of the iron. When the temperature goes below a certain precise "critical point," there is a sudden change in the macroscopic behavior of the iron--the tendency to align wins out, so that a very large cluster of aligned atoms is formed, and the iron can become a magnet. Such "critical phenomena"--sudden changes in macroscopic behavior when some measurement crosses a critical value--occur in a variety of contexts; recently, for example, there has been concern that the density of manmade junk orbiting the earth is approaching a critical level, above which the frequency of collisions will dramatically increase. Other systems in which small-scale randomness determines macroscopic properties, and critical phenomena may occur, include (i) waves traveling through irregular materials, such as seismic waves through the earth's crust; (ii) transport of heat by ocean currents in the presence of turbulence, which affects global climate; and (iii) percolation of liquid through a porous material, such as water or oil through underground rock. Mathematicians and physicists have long understood that many aspects of the relation between small-scale randomness and macroscopic properties, including critical phenomena, do not depend on the particular system being studied. One can therefore gain insight into real-world phenomena by studying abstract systems not intended to model specifically magnets, or porous rock, or any other particular part of the physical world. The systems which Alexander will investigate are examples of such abstract systems.