Probability Models from Statistical Mechanics

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9802368 Alexander 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) The use of random-cluster representations to study Gibbs uniqueness, dilution and its effect on critical temperatures, disordered models, lattice gases with weak interactions, bounds on critical points, and monotonicity of various quantities as a function of system parameters. (2) Two-dimensional systems (percolation and the Ising model) conditioned so as to exhibit the formation of a large droplet, that is, a macroscopic region enclosed by a long contour, with emphasis on the discrepancy between the actual and ideal shape of this droplet. (3) Confetti percolation, a model which is both isotropic and, in two dimensions, self-dual, and hence is a natural setting in which to study conformal invariance and related phenomena. (4) The use of percolation ideas to create a model which mimics certain features of the freezing of water which contains impurities. (5) The geometry of finite clusters, and its relation to the cluster size distribution, for certain continuum models of percolation. 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; and (iv) 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. The systems which Alexander will investigate--percolation, random cluster models, Ising and Potts models, and other spin systems--are examples of such abstract systems.