Abstract
Let\(\bar H_V = \kappa \bar \Delta _V + \xi _V (x),x \in V \subset \mathbb{Z}^v \), be the mean-field Hamiltonian with\(\kappa > 0\) and random i.i.d. potential ξV. We prove limit theorems for the extreme eigenvalues of\(\bar H_V \) as |V|→∞. The limiting distributions are the same as for the corresponding extremes of ξV only if either (i) ξV is undbounded and\(\kappa > 0\), or (ii) ξV is bounded with “sharp” peaks and\(\kappa \ll 1\). Localization properties for the corresponding eigenfunctions are also studied.
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Additional information
Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 147–168, April–June, 1999.
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Astrauskas, A. Limit theorems for the maximal eigenvalues of the mean-field Hamiltonian with random potential. Lith Math J 39, 117–133 (1999). https://doi.org/10.1007/BF02469277
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DOI: https://doi.org/10.1007/BF02469277