Resonant delocalisation in random operators

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The goal of this project is to address fundamental issues in the analysis of random quantum many-particle operators. This in particular requires to develop mathematical tools for the analysis. Of particular interest is the possibility of a localization-delocalization transition in random systems of interacting particles. This phase transition, which in physical models causes vanishing or non-vanishing of conduction, is rather well understood in random single-particle models. However, the effects of interaction on this phenomenon still lie rather in the dark. The present work will build on recent progress together with Michael Aizenman which showed that in certain situations extended states emerge through tunnelling which is facilitated in systems with an exponentially growing configuration space by resonances between well separated localization centers. This is the phenomenon of resonant delocalisation which shall be further explored here. In relation to the above mentioned localization-delocalization transition it works on the side of delocalisation. We propose to study this phenomenon in effective mean-field models such as the hierarchical Anderson model. From the mathematical perspective, the proposed research concerns topics which present challenges and opportunities for probability theory and analysis. In particular, a method for analysing random matrix-valued Herglotz-Nevanlinna-Pick functions will have to be developed.