  Random matrix theory has proven very successful in the understanding of the
spectra of chaotic systems. Depending on symmetry with respect to time reversal
and the presence or absence of a spin 1/2 there are three ensembles, the
Gaussian orthogonal (GOE), Gaussian unitary (GUE), and Gaussian symplectic
(GSE) one. With a further particle-antiparticle symmetry the chiral variants of
these ensembles, the chiral orthogonal, unitary, and symplectic ensembles (the
BDI, AIII, and CII in Cartan's notation) appear which are the main point of
interest in this paper. Following a recently published work on chiral random
matrix ensembles and their experimental realizations (Rehemanjiang et al., PRL
124, 116801 (2020)), this is achieved by using dielectric cylinders placed
between two parallel aluminium plates. These cylinders act as microwave
resonators which are used to create tight-binding chains of finite length up to
N=5. The different ensembles are achieved by using different types of
couplings: For the orthogonal case spatial proximity is used, for the unitary
case microwave circulators are used, and for the symplectic case a combination
of circulators and cables is used to create the necessary symmetry. In all
cases the predicted repulsion behavior between positive and negative
eigenvalues for energies close to zero are verified by a comparison with theory
taking the finite size of the systems into account. We will show that the
difference to the expected universal behavior is given by logarithmic
corrections only. These corrections stem from the Hamiltonians having zero
entries in their off-diagonal blocks.
