Abstract
We introduce a new moment method in Random Matrix Theory specifically tailored to the spectral analysis of invariant ensembles. Our method produces a classification of invariant ensembles which exhibit a spectral Law of Large Numbers and yields an explicit description of the limiting eigenvalue distribution when it exists. We discuss the future development and applications of this new moment method.
Highlights
Random Matrix Theory (RMT) is one of the most active research topics in contemporary probability theory
We have outlined a new moment method in Random Matrix Theory tailored to the invariant ensembles
The method is based on the observation that, if the distribution of a random selfadjoint matrix is invariant under conjugation, it is completely determined by the joint distribution of its diagonal matrix elements, which form a family of real, identically distributed, exchangeable random variables
Summary
Random Matrix Theory (RMT) is one of the most active research topics in contemporary probability theory. Tr denotes the matrix trace, V is a sufficiently well-behaved real-valued function, and β is the Dyson index, which is equal to 1, 2, or 4 according to whether the ensemble is real, complex, or quaternionic The fixation on this special class of invariant ensembles stems from the fact that the joint density of eigenvalues is known explicitly, being proportional to. We suggest a new approach to the spectral analysis of general invariant ensembles, the implementation of which will both broaden and deepen current understanding of this important paradigm This new approach is based on a simple observation: the distribution of any conjuagation-invariant random selfadjoint matrix is completely determined by the joint distribution of its diagonal matrix elements. This reduces the spectral analysis of invariant ensembles to extracting eigenvalue statistics from the joint distribution of diagonal matrix elements
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
