Large deviations for random matrices with varying potential and sum rules

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In this project we investigate spectral measures of large random matrices. Random matrices are used to model complex physical systems and the spectral measure is a way to describe properties of the matrix. Since the matrix is random, the spectral measure is a random object as well. However, if the matrix is large, the spectral measure is typically close to a deterministic limit measure and deviations are exponentially unlikely. We study large deviation principles, which measure how likely such a fluctuation from this limit is precisely. The novel approach of this project is to show these concentration estimates when the mechanism generating the random matrix, determined by a potential, is not fixed, but changes with its size. In this case, several limit theorems are known for the spectral measure, but very few on the scale of large deviations. There are two ways to describe a random spectral measure: by its spectral information or by a series of coefficients. It is however not easy to relate between information given in spectral form and information given in terms of the coefficients. We utilize both encodings to study how a varying potential influences the concentration of the spectral measure, which allows to prove large deviation principles in two different ways. The great benefit of two independent proofs in both descriptions is a resulting sum rule. These sum rules are important equations, which allow to translate between spectral information and the coefficients. Besides the purely probabilistic large deviation estimate, the discovery of such identities is another main goal. By varying the mechanism generating the random spectral measure, we can derive new forms of sum rules.