CAREER: Complexity of Disordered Systems

OVERVIEW: Research component. This proposal describes several projects at the interface of probability theory and statistical physics to be carried out by Antonio Auffnger, his students and post-docs. The main models considered here are first-passage percolation (FPP) and the Sherrington-Kirkpatrick (SK) spin glass model. These were introduced in the '50s and '70s, but have been the topics of renewed and vigorous research in the past decade. The PI proposes to (a) quantify the low-temperature structure of mean field spin glass models, with the long range goal of rigorously establish the Parisi description of full replica symmetry breaking, (b) to develop an intrinsic theory of geodesic rays and associated corays in FPP and (c) to establish further properties of limit shapes and uctuation exponents in these models. Educational Component. In years 2 and 4 of the grant, the PI will organize a two week graduate student / postdoc summer school in probability with 4 daily courses, and short talks given by young probabilists. The format of the school is modeled on the 2016 Summer school in probability held at Northwestern and organized by the PI. The school will be complemented by a one week pre-school "boot camp" aimed at undergraduate students and first year graduate students. Last, the PI will mentor REU students, two graduate students, and one postdoc. Computer projects for REU students in percolation are outlined in this document. INTELLECTUAL MERIT: Along with collaborators, the PI has recently made major contributions in (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, (c) the asymptotic number of critical points of random Morse functions on high-dimensional manifolds. 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 research problems considered in this proposal are not only relevant questions in pure probability but are connected to and motivated by problems in statistical physics, computer science and statistics. The educational component of this proposal, including the two summer schools in probability, the outlined numerical and partially theoretical project for undergraduates will substantially benefit many undergraduate and graduate US students. The PI will also collaborate with other mathematicians from the US, Europe and Asia, and disseminate the results obtained at international conferences and seminars (roughly 5-10 per year).