9504513 Ioffe Abstract The research is devoted to a study of various limit phenomena which arise in a statistical mechanical context or are closely related to it. More specifically, the investigator works - partly in a collaboration with other researchers - on questions concerning the structure of limit theorems in the phase transition regime as well as on applications of ideas reminiscent of the concept of complete analyticity to a general probabilistic setting such as, for example, to the theory of Markov chains or to the theory of random fields. Principal mathematical issues addressed in this research include further investigation of the two dimensional Ising model in the whole of the phase transition region, analysis of Ornstein-Zernike behavior of two-and-higher dimensional correlations and related shape and fluctuation theorems, analysis of limit phenomena for almost Markovian random fields and justification of the Wulff construction in various two and three dimensional models (e.g. SOS, Ising and continuous spin Ising, percolation etc.), based on sharp subvolume order large deviation results. Long term objectives of this research are twofold. On the one hand, powerful ideas and methods developed over the past decade in mathematical statistical mechanics--such as, for example, the method of polymer expansions, the concept of complete analyticity and the method of induction in volume--may provide both useful insights and approaches to various limit problems in probability and stochastic processes. On the other hand, most of the problems addressed in this research emerge from physical and material science theories of a rigorous reconstruction of thermodynamical and kinetic properties of matter directly from microscopic considerations. Thus, a successful investigation is ultimately aimed at a better understanding of such issues as, for example, equilibrium structure and stability of crystals and dynamics of surface growth.
