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Interdisciplinary applications of Random Matrix Theory

Maestro project

The project is devoted to novel and interdisciplinary applications of random matrix theory, on the basis of recent results in this field, including the research performed by the members of the team.

Random matrix theory, and in particular its version known as Free Random variables (FRV) can be described as the probability calculus of the XXI century, since the role of the random variable is played by huge matrices stored in computers’ memory in practically all domains of science and life. As such, FRV allows for unprecedented successful analysis of gargantuan sets of data, identification of the hidden correlations and cleaning the signals from the noise.

The project outlines 10 strategic domains - 5 at the fundamental level and 5 at the interdisciplinary applied level. The tasks in the fundamental research cover: study of new matricial spectral probability distributions (of the Meixner type), generalization of the concepts of conditional probability and Bayesian inference for matrix-valued probability calculus, study of FRV analogues of extreme events central limit theorems, determinental processes and matrix valued stochastic differential equations, and finally, so-called multiplication laws for strictly non-hermitian random matrix ensembles. The applied research is based on using above concepts in information theory, at quantum level (2 tasks) and as well at classical level (3 tasks).

We plan the study of random matrix ensembles for the processing of quantum information in so-called composed ensembles. Then we plan to use similar techniques for the analysis of entanglement in random networks. Both projects constitute currently a challenge in quantum information. In the domain of classical information theory we have chosen three strategic domains:

Unification of the theory of random matrices, game theory and information and communication technologies (ICT) known as Multiple Input Multiple Output (MIMO) systems, for the purpose of better understanding the design of wireless telecommunication systems and peer-to-peer networks.

Analysis of spatiotemporal correlations of huge amounts of econometric data aimed at seeking correlations and identification of crucial economic indicators signaling the states of serious (global) crisises. Analysis based on random matrix techniques of large sets of electromagnetic data gathered from the brain. All applications are of crucial importance from the point of better understanding and designing info-, bio- and socio-economic technologies. On top of known expectations in quantum computing we mention here the possibility of Brain-Computer Interfaces (BCI) based on neuroinformatics, taming large extreme events in economy and maximizing capacity of information in various kinds of ICT networks. At the fundamental level, proposed goals fill the conceptual gap in matricial probability theory and will help to establish FRV as perhaps the most promising candidate for noncommutative probability theory.

The members of the team have performed successful pilot studies in all targeted areas, and have formed international scientific links with several leading centers interested in similar problems.