Award Abstract #9801085
Limits and Deviations for Interacting Random Systems

NSF Org:	DMS Division of Mathematical Sciences
Initial Amendment Date:	July 8, 1998
Latest Amendment Date:	July 8, 1998
Award Number:	9801085
Award Instrument:	Standard Grant
Program Manager:	Keith N. Crank DMS Division of Mathematical Sciences MPS Directorate for Mathematical & Physical Sciences
Start Date:	July 15, 1998
Expires:	June 30, 2001 (Estimated)
Awarded Amount to Date:	$62076
Investigator(s):	Timo Seppalainen seppalai@math.wisc.edu (Principal Investigator)
Sponsor:	Iowa State University 1138 Pearson AMES, IA 50011 515/294-5225
NSF Program(s):	PROBABILITY
Field Application(s):	0000099 Other Applications NEC
Program Reference Code(s):	OTHR,0000
Program Element Code(s):	1263

ABSTRACT

9801085

Seppalainen

This project develops precise mathematical understanding of certain macroscopic phenomena that result from the interaction of a large number of microscopic components. There is randomness in the behavior of the microscopic components, so the models are stochastic in nature. The project covers several models of current interest: longest increasing subsequences, moving interfaces, Hammersley's process, the exclusion process, models with frozen disorder, and the random-cluster model. The investigation addresses two basic questions: (1) Is there predictable macroscopic behavior that does not depend on the particularities of the random microscopic motions? The answer sought can be the numerical value of an interesting quantity, a function that describes the shape of an interface, or a differential equation that describes the macroscopic evolution. (2) What are the chances that the microscopic evolution deviates noticeably from the macroscopic description? Answers to this question are large deviation theorems and central limit theorems, and may involve finding an entropy function that the system seeks to maximize or minimize.

In many situations macroscopic behavior of a system depends on interactions of microscopic components. The microscopic components can be of many types, depending on the model under study: vehicles on a freeway, customers in a sequence of queues, or fluid particles in a pipe or in a porous medium such as soil. The outcome of this project is a better understanding of the overall behavior of such a complex system. For example, in the traffic model the investigator seeks to describe two distinct situations, a low density phase and a high density phase, with qualitatively different patterns in the movement of vehicles. In a queueing model one seeks a description of the flow of customers through the network, to understand whether the system can become dangerously clogged and unstable. Such models form part of the theoretical underpinnings of modern communication technology. For a model of a moving interface the interesting properties are the speed, the eventual shape, and the roughness of the interface. These models and their relatives are intensely and concurrently studied by mathematicians, physical scientists, and engineers. In many cases simulations have not yielded conclusive pictures of the behavior. Consequently there is great need for basic mathematical work to aid the more applied scientists' understanding of such phenomena.

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