Complexity of Disordered Systems

Overview: Page A This proposal describes several projects at the interface of probability theory and statistical physics to be carried out by Antonio Auffinger. Some of the tasks are related to the geometry of random fields and percolation models and some involve the low-temperature states of mean field spin glasses. Both of these types of models originated with physical systems and are characterized by a collection of random variables with possibly long-range dependence structures. The main questions address how their extremes affect the overall macroscopic behavior of the system. The PI proposes to (a) continue work started with G. Ben Arous on the number of critical points of high dimensional random Morse functions, (b) develop further techniques to analyze and quantify the low-temperature structure of mean field spin glass models, (c) continue long collaboration with M. Damron on questions related to the limit shape and fluctuation exponents in first-passage percolation and polymer models, (d) further investigate the relationship between the number and location of extremes of smooth fields with the eigenvalues of random matrix theory. Intellectual Merit : Along with collaborators, the PI has recently made major contributions to (a) the low-temperature structure of mean field spin glasses, including the Sherrington-Kirkpatrick model, (b) scaling exponents and asymptotic shapes in first-passage percolation and directed polymers, (c) the asymptotic number of critical points of random Morse functions on high-dimensional manifolds, (d) the asymptotic behavior of eigenvalues of heavy tailed random matrices. Therefore he and his collaborators are in an ideal and opportune position to carry out the work in this proposal. The PI expects the projects described here to make important contributions to the study of disordered systems and these advances would affect other areas. A recurring goal of this proposal is the further understanding of the relationship between the number and location of extremes of these disordered systems and their macroscopic behavior. A long-range hope is to identify the underlying universal quantities and use this structure to understand dynamical phenomena such as aging and metastability on statistical physics models. Broader Impacts : The problems considered in this proposal are not only relevant questions in pure probability but are connected to and motivated by questions in statistical physics, telecommunications, medical research and statistics. For example, level sets of smooth random fields play a key role in detecting brain activity and improving quality and efficiency of MRIs. First passage percolation and spin glass ground state problems may be viewed as combinatorial optimization problems and have applications to computer science. Furthermore, the projects outlined in this document have the additional feature of being accessible to graduate researchers. The proposer is committed to organize seminars and workshops and disseminate the outcome of this work to young researchers.