The investigators will study a variety of systems all of which either directly concern classical spin systems in equilibrium or have their basis in some approximation to a statistical mechanics model. There are six individual projects, all of which they have already analyzed to some degree. The first project concerns an approach to a general theory of discontinuous phase transitions. For a certain class of nearest neighbor attractive systems on the usual d-dimensional cubic lattices they can show that whenever the associated mean field theory predicts a discontinuous phase transition, the actual system also undergoes a discontinuous transition whenever the dimension d is large. The second project is a proposal to study a model which emulates, through a deterministic evolution of a random environment, the celebrated SOC sandpile models. Third, they study systems at points of phase coexistence in a statistical ensemble fixing the excess of the minority phase in the system. It appears that, in a correct scaling, there is a sharp value of minority-phase volume where a droplet of the minority phase spontaneously emerges. Fourth, they will investigate certain percolation systems which locally appear similar to percolation on the complete graph but contain an underlying geometrical structure that is absent in the usual complete graph systems. In the fifth project, they will study a variety of interacting random walks with both attractive and repulsive (and mixed) interactions. The final project concerns the correct definition of thermodynamic limit of the free energy in polydisperse systems, e.g., systems with a continuum of particle species. The Principal Investigators will be examining a number of problems (six in the proposal with more on the horizon) in the area of Statistical Mechanics or closely related fields. The essence of these problems concerns systems consisting of simple constituents which interact in a simple manner. However, since there are a myriad of these constituents it turns out that such systems are capable of highly complex behavior. Of particular interest is the phenomena of phase transitions in which the entire collective character of the system undergoes a drastic change. This notion is either at the heart of or just behind the scenes in all of the systems that are proposed for study. One project concerns a demonstration that "most" systems undergo a more catastrophic sort of transition sometimes known as a phase change of the first kind. A second line of investigation concerns the formation of droplets (e.g. raindrops) which, as it turns out, takes place on a submacroscopic scale known as a mesoscopic scale. Third, under investigation is the infiltration of "disturbances" in a system which -- from one perspective -- models a social network. Fourth, there will be investigations of systems in which the above mentioned elementary constituents are all of different (but closely related) character. The remainder of the investigations concern the so called percolation and random walk systems. While these models have their origins in the study of chemical polymers, the applications cover a large range of topics, one of which, in modern terms, is the dissemination of information in communications networks.
