Mathematical Sciences: Percolation and Related Problems

9504462亚历山大抽象渗流可能是人们可以研究小规模随机性如何产生大规模现象的最基本的模型，例如相变，这些现象本质上是非随机的。亚历山大建议对这一主题的以下几个方面进行研究。(1)利用渗流思想分析具有随机势的二维不可压缩流动的行为。(2)与标准的FK随机簇模型一样，与伊辛模型密切相关的一种新的随机簇模型的性质。(3)使用渗流思想为含有杂质的水结冰时发生的大规模现象创建一个小范围定义的模型。(4)某些渗流连续介质模型中有限团簇的几何形状及其与团簇大小分布的关系。(5)图的渗流行为与平稳随机标号图的“最小生成林”的其他性质之间的相互关系。(6)首次通过渗流在一定时间内可达到的区域边界起伏的大小。此外，Alexander还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一项努力的一部分，目的是了解小规模的随机性如何反映在自然界各种系统的大范围或“宏观”属性中。一个典型的例子是一块铁--每个原子都有一个沿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在大致相同的方向上排列，特别是在温度较低的时候。本质上，随机性的倾向随着温度的升高而增加，这与趋同的倾向相竞争。具有相似排列的原子团簇--都是向上的，都是向下的，等等--形成了，这些原子团的随机几何形状--以不同的概率出现的团簇的大小--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，铁的宏观行为就会突然发生变化--排列的倾向占上风，因此形成了一个非常大的排列的原子团，铁可以成为一块磁铁。这种“临界现象”--当一些测量值超过一个临界值时，宏观行为的突然变化--在各种情况下都会发生；例如，最近有人担心，围绕地球轨道运行的人造垃圾的密度正在接近一个临界值，超过这个临界值，碰撞的频率将急剧增加。在小尺度随机性决定宏观性质的其他系统中，可能会出现临界现象，其中包括：(1)波在不规则物质中传播，如地震波穿过地壳；(2)在有湍流存在的情况下，洋流的热量输送，这会影响全球气候；(3)液体通过多孔材料的渗流，如水或石油穿过地下岩石。数学家和物理学家早就认识到，小规模随机性和宏观性质之间的关系的许多方面，包括临界现象，并不取决于正在研究的特定系统。因此，人们可以通过研究抽象系统来洞察现实世界的现象，这些抽象系统并不是专门为磁体、多孔岩石或物理世界的任何其他特定部分建模的。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.