The aim of this proposal is to investigate a class of problems on the borderline between probability and mathematical physics. Altogether ten specific projects are proposed which can be grouped into four categories. The first category deals with problems of first-order phase transitions. Here the specific objects of study include proofs of phase transitions by comparison to mean-field theory, influence of dilution on symmetry breaking in antiferromagnets, and various effects accompanying phase transitions in systems with continuous spins. The second category is focused on droplet and interface phenomena. The corresponding projects include a study of relaxation to equilibrium in solvent-solute systems and the mesoscopic concept of pressure in systems at phase coexistence. The third category of projects deals with problems related to conformal invariance in 2D critical models. Here the fractal properties of the so-called SLE boundaries will be investigated and a model of random fractal trees will be analyzed. The final category involves problems of small-world phenomena. The specific problem of interest here is the growth of the graph distance in a critical model of Euclidean-based random graphs.
On a broader level, the proposal hopes to address a series of questions that all have their origin in physics, chemistry, engineering and/or social sciences. A particular attention will be paid to the phenomena of phase transitions in which the overall character of the system undergoes a sudden, and often rather drastic, change while the external conditions vary smoothly through a particular, transitional, value. Examples of such transitions are ample in physical sciences (e.g., freezing or melting in physical chemistry) and engineering (e.g., jams in internet traffic); but they also have their natural counterparts in social sciences (e.g., opinion spreading in social networks). Much of this proposal is spent on studying very specific mathematical problems of these types. For instance, one of the proposed projects deals with the appearance of magnetism in transition-metal compounds, another offers to shed some light on the theoretical aspects of brine-pocket formation in the sea ice, yet another project is devoted to the "degree of separation" of two individuals (or computer servers) in a thinly connected (communication) network. The common ground of several of these problems is the need of proper mathematical tools for their successful resolution. One of the goals of the present proposal is to develop such tools and disseminate their main ideas to the scientific community at large.
