Statistics of Extrema in Disordered Systems and Related Models

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A wide range of phenomena in nature, physics, and society can be viewed as the byproduct of many similar components or agents interacting in a seemingly random or disordered manner (e.g., stock markets, condensed matter in physics, prime numbers in mathematics). The focus of this project is the study of rare events, or extrema, that emerge in disordered systems. The vast number of agents and the disorder present in these complex systems make straightforward predictions impossible. The project goal is to develop tools of probability theory to improve our fundamental understanding of disordered systems and, ultimately, to lead to better statistical modeling of rare events, such as high volatility episodes in stock markets and certain kinds of phase transitions in physics.Specifically, the aim of this project is to extend the theory of extreme values in probability to random systems with strong and complex correlation structure. The first objective is to derive fine asymptotics for the maxima of characteristic polynomials of random matrices and for the local maxima of the Riemann zeta function (which controls the distribution of prime numbers). These systems are examples of stochastic processes with logarithmically decaying correlations. The second part is a study of extrema of spin glasses, an important class of physical systems with more complex correlations that includes disordered magnets. The main objective there is to find a new approach to investigate the structure of the Gibbs states of the models. In addition, the nature and the existence of phase transitions for spin glasses in finite dimension (which are realistic models of disordered magnets) will be studied by extending methods based on fluctuations of relevant thermodynamic quantities.