Limit theorems for the maximal eigenvalues of the mean-field Hamiltonian with random potential (Q1568062)

Let \(\overline H_V:=\kappa\overline\Delta_V+ \xi(x)\), \(x\in V\subset\mathbb{Z}^\nu\), denote the mean field (Curie-Weiss) Hamiltonian with i.i.d. potential \(\xi\). Limit theorems are derived for the extreme eigenvalue of \(\overline H_V\) as \(|V|\to \infty\). The limit laws are the same as for the corresponding extremes of \(\xi\) only if either \(\xi\) is unbounded, or if \(\xi\) is bounded with ``sharp'' peaks and \(\kappa\ll 1\). Localization properties of the corresponding eigenfunctions are also studied.