Probability and Statistical Mechanics

University Of Southern California, Los Angeles CA

Investigators

Abstract

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 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 discrepancy between the actual and ideal shape of this droplet. (2) Percolation models also conditioned to exhibit the formation of a large droplet, with emphasis on lower bounds for the local fluctuations of the boundary. (3) Mixing properties and decay of correlations and connectivities for spin systems and percolation models, particularly in finite volumes. (4) The spectral gap and rate of convergence to equilibrium, for the stochastic Ising model at low temperature. (5) Potts models in which the external field(s) and the boundary condition are opposed to each other. (6) The use of percolation ideas to create a model which mimics certain features of the freezing of water which contains impurities. The project aids human resource development through the support of one Ph.D. student working on similar problems. 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) composite materials with special properties; 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. The systems which Alexander will investigate--percolation, random cluster models, Ising and Potts models, and other spin systems--are examples of such abstract systems.