Fyzikální ústav Akademie věd ČR

Seminar Tuesday, 08/11/2011 15:00 - 16:00

Speakers: František Slanina (Division of Condensed Matter Physics, Institute of Physics ASCR, Prague)
Place: Na Slovance, main lecture hall
Presented in Czech
Organisers: Oddělení teorie kondenzovaných látek

Abstract: The theory of random matrices founds numerous applications in areas ranging from nuclear physics to quantum chaos to music-hall acoustics. We show some new results on the spectra of random sparse matrices.
The matrices we use encode the structure of a random graph. Two types of random graphs are considered, namely Erdös-Rényi (ER) graphs and random cubic graphs. In the former case, we investigate the influence of purely topological disorder, like in a glassy solid. In the latter one, topological disorder plays no role, but introducing diagonal disorder we come to a candidate model for a mean-field theory of Anderson localization.
For ER graph, we introduce a variational method. Minimizing the variation functional on a subspace of trial functions we obtain better approximations than those known previously. Especially, we are able to find corrections to the Coherent Potential Approximation.
Localization is studied by numerical diagonalization of large ensembles of matrices. We find the position of the localization edge both in the spectrum of ER graphs and random cubic graphs.