On the Critical Capacity of the Hopfield Model

We estimate the critical capacity of the zero-temperature Hopfield model by using a novel and rigorous method. The probability of having a stable fixed point is one when $\alpha\le 0.113$ for a large number of neurons. This result is an advance on all rigorous results in the literature and the relationship between the capacity $\alpha$ and retrieval errors obtained here for small $\alpha$ coincides with replica calculation results.Comment: Latex 36 page macros: http://www.springer.de/author/tex/help-journals.htm

Central limit theorem for linear eigenvalue statistics of orthogonally invariant matrix models

We prove central limit theorem for linear eigenvalue statistics of orthogonally invariant ensembles of random matrices with one interval limiting spectrum. We consider ensembles with real analytic potentials and test functions with two bounded derivatives.Comment: 19 page

Photon correlations for colloidal nanocrystals and their clusters

Images of semiconductor `dot in rods' and their small clusters are studied by measuring the second-order correlation function with a spatially resolving ICCD camera. This measurement allows one to distinguish between a single dot and a cluster and, to a certain extent, to estimate the number of dots in a cluster. A more advanced measurement is proposed, based on higher-order correlations, enabling more accurate determination of the number of dots in a small cluster. Nonclassical features of the light emitted by such a cluster are analyzed.Comment: 4 pages, 4 figure