  Using numerical diagonalization we study the crossover among different random
matrix ensembles [Poissonian, Gaussian Orthogonal Ensemble (GOE), Gaussian
Unitary Ensemble (GUE) and Gaussian Symplectic Ensemble (GSE)] realized in two
different microscopic models. The specific diagnostic tool used to study the
crossovers is the level spacing distribution. The first model is a one
dimensional lattice model of interacting hard core bosons (or equivalently spin
1/2 objects) and the other a higher dimensional model of non-interacting
particles with disorder and spin orbit coupling. We find that the perturbation
causing the crossover among the different ensembles scales to zero with system
size as a power law with an exponent that depends on the ensembles between
which the crossover takes place. This exponent is independent of microscopic
details of the perturbation. We also find that the crossover from the
Poissonian ensemble to the other three is dominated by the Poissonian to GOE
crossover which introduces level repulsion while the crossover from GOE to GUE
or GOE to GSE associated with symmetry breaking introduces a subdominant
contribution. We also conjecture that the exponent is dependent on whether the
system contains interactions among the elementary degrees of freedom or not and
is independent of the dimensionality of the system.
