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.
