  We consider Mabuchi rays of toric K\"ahler structures on symplectic toric
manifolds which are associated to toric test configurations and that are
generated by convex functions on themoment polytope, $P$, whose second
derivative has support given by a compact subset $K<P$. Associated to the test
configuration there is a polyhedral decomposition of $P$ whose components are
approximated by the components of $P \setminus K$. Along such Mabuchi rays, the
toric complex structure remains unchanged on the inverse image under the moment
map of $(P \setminus \check {K})$, where $\check {K}$ denotes the interior of
$K$. At infinite geodesic time, the K\"ahler polarizations along the ray
converge to interesting new toric mixed polarizations. The quantization in
these limit polarizations is given by restrictions of the monomial holomorphic
sections of the K\"ahler quantization, for monomials corresponding to integral
points in $P \setminus \check {K}$, and by sections on the fibers of the moment
map over the integral points contained in $\check {K}$, which, along the
directions parallel to $K$ are holomorphic and which along the directions
transverse to $K$ are distributional. These quantizations correspond to
quantizations of the central fiber of the test family, in the symplectic
picture. We present the case of $S2$ in detail and then generalize to higher
dimensional symplectic toric manifolds. Metrically, at infinite Mabuchi
geodesic time, the sphere decomposes into two discs and a collection of
cylinders, separated by infinitely long lines. Correspondingly, the
quantization in the limit polarization decomposes into a direct sum of the
contributions from the quantizations of each of these components.
