Brief summary and outlook

Abstract

Besides giving a short summary of the contents of Chaps. 2 - 15, some of the important open problems in the subject of embedded ensembles are listed. These are: (i) developing binary correlation method (or some other methods) to derive asymptotic results for EGOE(1+2)-s and also for bosonic ensembles; (ii) analysis of embedded symplectic ensembles (EGSE) [so far no EGSE has bee analyzed]; (iii) deriving formulas for the Racah coefficients needed for analyzing higher moments of EGUEs and this calls for new advances in the general SU(N) Wigner-Racah algebra; (iv) more higher level "computational experiments" than those that are carried out till now; (v) derivation of the two-point correlation function. Results for (v) even for simplest spinless EGOE(2) and BEGOE(2) are not available. Finally, to strengthen the numerically established result that EGOE fluctuations follow GOE, results of the analysis using the new measure, statistics of quotients of successive spacings, that is independent of unfolding procedure and the unfolding functions, are presented. © 2014 Springer International Publishing Switzerland.