9706915 Wehr The proposal discusses three circles of problems arising in mathematical physics of random systems. While the problems concern very different physical situations, at the mathematical level they all deal with situations where presence of disorder can significantly change properties of the system. Part one studies ground states of random lattice models in statistical mechanics---a question related to understanding thermodynamics of disordered systems. In part two the focus is on effects of randomness on propagation of nonlinear waves described by partial differential equations---of the conservation law and Hamilton-Jacobi type. The content of part three is statistics of density and of pairwise velocities of galaxies in the universe. From the point of view of mathematics, each part of the proposal involves studying nonlinear functions of many independent random variables or of the realizations of a simple underlying process (often a Gaussian one). This relates the project thematically to the theory of large deviations and concentration of measure phenomena. The tools include functional limit theorems, ergodic theory, large deviation estimates and martingale methods in the first two parts and generalized Gaussian processes in the third part. The interplay of physical and mathematical ideas is essential in the whole proposal. Many interesting questions in mathematical physics are concerned with complicated inhomogeneous systems. For example, a realistic description of a solid often has to take into account presence of defects and impurities, which make its physical properties different from those of a homogeneous crystal. These special properties of disordered systems are of crucial importance in modern materials science and technology --- semiconductors and glasses are examples of disordered materials. Another example is the distribution of galaxies in the Universe --- their positions do not follow any simple regular pattern and us ing such a pattern in modeling their configuration would not account for observed phenomena. While detailed configuration (arrangement of atoms, galaxies or other constituents) of an inhomogeneous system varies from one region in space to another, there is often an underlying simple statistical regularity, which leads to modeling and studying such systems using concepts and methods of probability theory. Each of the three parts of the proposal addresses a different physical problem of the above type. Part one involves the study of models of disordered solids, focusing on their ground states --- the energetically preferred configurations. Part two is concerned with propagation of waves in disordered media --- several applications of mathematics to hydrology are of this type. The content of part three is fluctuations of various quantities related to the distribution of matter in the Universe and statistical questions arising in interpreting observational results. The general physical motivation behind the proposal is to understand possible effects resulting of the presence of randomness in different physical systems.
