Abstract
For a mean field operator with a random potential, asymptotic properties of the eigenvalues and eigenfunctions are studied and applied to investigate the longerm behavior of the solutions of a corresponding large system of differential equations. The total mass of the system is approximately concentrated in the record point of the random potential (complete localization). A more detailed inspection of the peaks shows that there is a phase transition: Only in the case of a moderate increase of time relatively to the growth of the space size the model behaves similarly to the system without “diffusion”. But also in the non-moderate case the asymptotic height of peaks can exactly be described.
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