Colours of randomness
In the 1950s, Eugene Wigner introduced random matrices, whose elements are random variables, to model the energy-level structure of the nuclei of large atoms. At about the same time, Phillip Anderson used random Schrödinger operators to model disordered solids and explain the phase transition from insulator to metal. The concept of random graphs was put forward by Paul Erdös and Alfred Rényi to prove the existence of graphs satisfying various properties. In subsequent years, the study of these mathematical models of real-world networks transformed the field of combinatorics. Random matrices, random Schrödinger operators and random graphs have interesting spectral properties that were the focus of the project SPECTRA (Spectra of random matrices, graphs and groups). The researchers' aim was to tackle still open questions. The SPECTRA team looked into mathematical models of long wires approximating them with infinite discrete paths. The wave functions corresponding to such paths show that it could be a superconductor. But, thin wires do not conduct well and, therefore, a different model was proposed. Researchers worked on a randomly perturbed version and analysed the shape of wave functions as well as the regularity of the average spectrum and its behaviour at energy zero. In this case, wave functions were found to be localised, suggesting that the long wires are insulators. Next, the focus of SPECTRA research was set on sparse random regular graphs that can be used to theoretically test algorithms on real-world networks. The team studied random local algorithms in which a random value is put at every node and then nodes change state according to what they see in their neighbourhood. Such algorithms outperformed greedy algorithms developed in the past for this purpose when wave functions were used on the graphs. Nonetheless, random local algorithms could not function when the connectivity degree in the network was relatively large. The spectral analysis of matrices defined on random graphs has recently attracted growing attention because of its implications in computer science, mathematics and physics. SPECTRA work has revealed connections between the geometrical structure of matrices and graphs with their spectra. The findings are detailed in a series of publications, whose e-prints have been uploaded on arXiv.
Keywords
Random matrices, random Schrödinger operators, random graphs, real-world networks, SPECTRA
