Insights into Disordered Landscapes via Random Matrix Theory and Statistical Mechanics

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Random Matrix Theory (RMT) acquired in the last decade the status of a universal paradigm in mathematical description of phenomena in systems of extremely diverse nature. The theory deals with integrals over matrix measures defined on various sets of matrices of (typically) large dimensions. Behaviour of such matrix integrals provided invaluable insights into universal (i.e. insensitive tomicroscopic details) properties of a variety of physical systems. Among representative examples one could mention models ranging from those describing motion of electrons in nanoscale conductors or scattering of electromagnetic waves in a random environment, to those pertinent to understanding statistics of shapes of the growing aggregates, or to lattice simulations in Quantum Chromodynamics, or employed in continuing attempts to build theory of quantum gravity. Last, but not least RMT-inspired methods and ideas proved highly useful in predicting generic properties of the Riemann zeta-function.Similarly but somewhat independently the idea of complicated energy landscapes pervades the theoretical description of glasses, disordered systems, proteins, etc., and recently re-emerged in stringtheory and cosmology. Here the main goal is to describe behaviour of the whole system, or one of its subparts, by employing ideas of statistical mechanics of a single point particle (or sometimes higher dimensional objects like lines or membranes) moving in a random potential, which encodes the complexity of the original system. The hope then is to be able to classify the possible types ofrandom potential and to establish generic, universal properties, not unlike those emerging in the RMT. As the low-temperature behaviour in statistical mechanics is controlled by the lowest available energies, it is not surprising that a detailed description of the freezing phenomena in systems with disorder is intimately related to the so-called extreme-value statistics of random variables. This area of research is itself an active field at the intersection between Probability and Mathematical/Theoretical Physics. Again RMT ideas and results played an essential role in that development.The aim of the present project is to combine RMT tools and results with the ideas of statistical mechanics for exploring statistical properties of random landscapes, most importantly extreme and high values of landscapes with logarithmically growing correlations. This line of research proves to be intimately connected to questions of essential current interest in Probability and other areas of Mathematics, as e.g. the extreme value statistics of strongly correlated random variables, in particular, of the 2D Gaussian Free Field, branching random walks, as well as to the issue of constructing closed conformally invariant random curves in 2D and to the distribution of values of the Riemann zeta-function along the critical line. Similar questions emerged recently in studies of multifractal processes appearing in financial time series and turbulence. The unifying aspects are again provided by tools and ideas coming from the Random Matrix Theory.