The project addresses problems covering a rather broad spectrum of discrete mathematics (DM), some drawn from related areas such as statistical mechanics (SM), probability, and theoretical computer science (TCS). A common thread in many of these problems is:one is given a probability distribution on some large discrete space (e.g. independent sets in, or proper colorings of, a graph), and wants to know whether behavior in one part of the system can significantly affect behavior in other, distant parts. The loose term ``long range effects" (LRE's) is used for such interactions. Both strong and weak LRE's are of interest: For SM what's wanted is, more often than not, strong LRE's, usually under the name ``phase transition" (PT). In very recent work, the investigator and his student David Galvin used mainly combinatorial ideas to settle one much-studied question in this area (concerning PT's in the ``hard-core lattice gas model"), and a particular hope at present is that ideas from this work, and, more generally, the investigator's combinatorial perspective, will lead to progress on various more or less related questions. From the perspectives of DM and TCS, on the other hand, weak LRE's are often desirable, e.g. because of their connections with rapid mixing of Markov chains (a topic of major interest in TCS), and because in combinatorial applications of probability, systems with sufficiently weak LRE's can sometimes substitute for probability spaces based on purely independent choices. (This again involves the hard-core model, which arises in surprisingly diverse mathematical contexts.)
Speaking very generally, the (main part of the) project aims at understanding the degree of order or disorder in various large random systems, for instance the extent to which the global structure of a physical system can be predicted from knowledge about interactions between neighboring particles. In recent years, random systems similar to those used to model physical reality have played increasingly important roles in TCS and DM, partly because they can be helpful in analyzing (algorithmically and/or theoretically) even non-random problems in these areas; and there has been an increasing realization of the degree to which the issues arising in these three disciplines are related. One major theme of the present project is an interest in applying ideas and methods across mathematical boundaries: especially in using ideas from the investigator's core area (DM) to attack problems in SM and TCS, but also in bringing an increasing familiarity with techniques from the latter areas to bear on several problems of longstanding interest in DM.
