The Random Matrix Technique of Ghosts and Shadows

A. Edelman

2010, v.16, №4, 783-790

ABSTRACT

We propose to abandon the notion that a random matrix exists only if it can be sampled. Much of today's applied finite random matrix theory concerns real or complex random matrices ($\beta=1,2$). The "threefold way" so named by Dyson in 1962 [F.J. Dyson, The threefold way. Algebraic structures of symmetry groups and ensembles in Quantum Mechanics. J. Math. Phys., 1963, v. 3, pp. 1199-1215] adds quaternions ($\beta=4$). While it is true there are only three real division algebras ($\beta$="dimension over the reals"), this mathematical fact while critical in some ways, in other ways is irrelevant and perhaps has been over interpreted over the decades. We introduce the notion of a "ghost" random matrix quantity that exists for every beta, and a shadow quantity which may be real or complex which allows for computation. Any number of computations have successfully given reasonable answers to date though difficulties remain in some cases. Though it may seem absurd to have a "three and a quarter" dimensional or "pi" dimensional algebra, that is exactly what we propose and what we compute with. In the end $\beta$ becomes a noisiness parameter rather than a dimension.

Keywords: random matrix theory,ghost random variables