We propose a working strategy to describe the eigenvalue statistics of random spin systems along whole phase diagram with thermal many-body localization (MBL) transition. Our relies on two matrix (RM) models well-defined construction, namely mixed (Brownian) ensemble and Gaussian β ensemble. show both RM are capable capturing lowest-order level correlations during transition, while deviations b...

Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We include the important mathematics which is a very modern development, as well as the computational softwa...

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-...

This paper discusses random matrix models that exhibit the unusual phenomena of having multiple solutions at the same point in phase space. These matrix models have gaps in their spectrum or density of eigenvalues. The free energy and certain correlation functions of these models show differences for the different solutions. This study presents evidence for the presence of multiple solutions bo...