  Random Matrix Theory (RMT) has successfully modeled diverse systems, from
energy levels of heavy nuclei to zeros of $L$-functions; this correspondence
has allowed RMT to successfully predict many number theoretic behaviors.
However there are some operations which to date have no RMT analogue. Our
motivation is to find an RMT analogue of Rankin-Selberg convolution, which
constructs a new $L$-functions from an input pair. We report one such attempt;
while it does not appear to model convolution, it does create new ensembles
with properties hybridizing those of its constituents.
  For definiteness we concentrate on the ensemble of palindromic real symmetric
Toeplitz (PST) matrices and the ensemble of real symmetric matrices, whose
limiting spectral measures are the Gaussian and semi-circular distributions,
respectively; these were chosen as they are the two extreme cases in terms of
moment calculations. For a PST matrix $A$ and a real symmetric matrix $B$, we
construct an ensemble of random real symmetric block matrices whose first row
is $\lbrace A, B \rbrace$ and whose second row is $\lbrace B, A \rbrace$. By
Markov's Method of Moments and the use of free probability, we show this
ensemble converges weakly and almost surely to a new, universal distribution
with a hybrid of Gaussian and semi-circular behaviors. We extend this
construction by considering an iterated concatenation of matrices from an
arbitrary pair of random real symmetric sub-ensembles with different limiting
spectral measures. We prove that finite iterations converge to new, universal
distributions with hybrid behavior, and that infinite iterations converge to
the limiting spectral measure of the dominant component matrix.
