q-RANDOM MATRIX ENSEMBLES

With a few notable exceptions, the interaction between the community of mathematicians who work in special functions, in particular, those that are in the area of q-series and basic Hypergeometric functions and the physics community has so far been minimal. In this review article, we will describe some developments in one area in physics, namely the Theory of Random Matrix Ensembles, where a q-generalization has been proposed to be relevant for multifractal states near a critical regime. The term multifractal states refers to the novel fractal characteristics of the quantum mechanical wave functions of a disordered electronic conductor. It is well known in the physics literature that depending on the strength of the disorder, a quantum conductor may undergo a phase-transition from a metallic state which describes a good conductor to those of an insulating state where the (static) conductivity in a finite sample is very small. In a region near the transition such multifractal wave functions are observed at least in numerical experiments.