Statistical Mechanics and the Probability Theory

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0405915Alexander Probability models from statistical mechanics are a framework for studying how small-scale randomness produces global-scale phenomena, such as phase transitions, which are essentially nonrandom. Alexander proposes to investigate the following aspects of the subject. (1) Pinning and depinning of lattice polymers and interfaces due to potentials existing in lower-dimensional subspaces of the lattice. Particular emphasis will be placed on the disordered case, in which the potential varies randomly with location. Competition between a potential in a subspace and random potentials in the bulk will also be considered. (2) Models in which certain features of the freezing of solutions can be rigorously derived from a local interaction. (3) The Ising model, conditioned, by fixing an excess of the minority spin, so as to exhibit the formation of a large droplet, with emphasis on the critical size of this excess and the width of the droplet-formation "transition" as this critical size threshold is crossed. (4) Moving interfaces, e.g. between phases of the Ising model, which may "hang up" at locations where a weakened interaction reduces the energetic cost of the interface, resulting in an energy barrier which must be crossed. (5) Eigenvalues of the covariance matrix of the two-dimensional Potts model and their relation to decay rates of certain probabilities in the FK model. (6) Potts models in which the external field(s) and the boundary condition are opposed to each other. With regard to intellectual merit, this work is part of an ongoing effort by mathematicians and physicists to understand various systems in the natural world in which nonrandom global-scale phenomena reflect aspects of small-scale randomness. Examples include (i) magnetic properties of materials; (ii) waves traveling through irregular materials, such as seismic waves through the earth's crust; (iii) impurities in semiconductors; (iv) denaturation of DNA; and (v) percolation of liquid through a porous material, such as water or oil through underground rock. It has long been understood that many qualitative aspects of the relation between small-scale randomness and macroscopic properties, including critical phenomena, do not depend too closely on the particular system being studied. One can therefore gain insight into real-world phenomena by studying abstract systems not intended to model specifically magnets, or porous rock, or any other particular part of the physical world. The systems need only exhibit parallel features, such as clustering and critical phenomena. Some of the systems, which Alexander will investigate--percolation, random cluster models, Ising and Potts models, and other spin systems--are examples of such abstract systems. Other systems, which Alexander will investigate, are somewhat more closely based on specific physical systems, such as polymers, DNA molecules, and solutions.