Insights into Disordered Landscapes via Random Matrix Theory and Statistical Mechanics

随机矩阵理论(RMT)在过去十年中获得了在极其不同性质的系统中的现象的数学描述中的普遍范式的地位。该理论处理定义在(通常)大维度的各种矩阵集合上的矩阵测度上的积分。这种矩阵积分的行为提供了对各种物理系统的普遍(即对微观细节不敏感)特性的宝贵见解。在具有代表性的例子中，人们可以提到各种模型，从描述纳米级导体中电子的运动或随机环境中电磁波的散射的模型，到与理解不断增长的聚集体形状的统计有关的模型，或者与量子色动力学中的晶格模拟有关的模型，或者用于继续尝试建立量子引力理论的模型。最后，但并非最不重要的是，RMT启发的方法和想法在预测Riemann Zeta函数的类属性质方面被证明非常有用。类似的但有点独立的复杂能量景观的想法渗透到玻璃、无序系统、蛋白质等的理论描述中，最近在弦理论和宇宙学中重新出现。这里的主要目标是通过使用统计力学的思想来描述整个系统或其一个子部分的行为，即单个点粒子(有时是高维物体，如线或膜)在随机势中运动的统计力学思想，随机势编码了原始系统的复杂性。因此，人们希望能够对随机势的可能类型进行分类，并建立通用的、普遍的性质，就像RMT中出现的那些性质一样。由于统计力学中的低温行为是由最低可用能量控制的，因此对无序系统中冻结现象的详细描述与所谓的随机变量的极值统计密切相关也就不足为奇了。这一研究领域本身就是概率和数学/理论物理学交叉点上的一个活跃领域。本项目的目标是将RMT工具和结果与统计力学的思想相结合，以探索随机景观的统计特性，最重要的是，具有对数增长相关性的景观的极值和高值。这条研究路线被证明与概率和其他数学领域中当前基本感兴趣的问题密切相关，例如，强相关随机变量的极值统计，特别是2D高斯自由场、分支随机游动的极值统计，以及构建2D中封闭的共形不变随机曲线的问题，以及Riemann Zeta函数的值沿临界线的分布。最近，在对金融时间序列和湍流中出现的多重分形过程的研究中，也出现了类似的问题。统一的方面再次由来自随机矩阵理论的工具和思想提供。

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.