  A high-throughput investigation of local epitaxy (called combinatorial
substrate epitaxy) was carried out on Ca$_2$MnO$_4$ Ruddlesden-Popper thin
films of six thicknesses (from 20 to 400 nm), all deposited on isostructural
polycrystalline Sr$_2$TiO$_4$ substrates. Electron backscatter diffraction
revealed grain-over-grain local epitaxial growth for all films, resulting in a
single orientation relationship ($OR$) for each substrate-film grain pair. Two
preferred epitaxial $ORs$ accounted for more than 90 \% of all ORs on 300
different microcrystals, based on analyzing 50 grain pairs for each thickness.
The unit cell over unit cell $OR$ ([100][001]$_{film}$ $\parallel$
[100][001]$_{substrate}$, or $OR1$) accounted for approximately 30 \% of each
film. The $OR$ that accounted for 60 \% of each film ([100][001]$_{film}$
$\parallel$ [100][010]$_{substrate}$, or $OR2$) corresponds to a rotation from
$OR1$ by 90$^{\circ}$ about the a-axis. $OR2$ is strongly favored for substrate
orientations in the center of the stereographic triangle, and $OR1$ is observed
for orientations very close to (001) or to those near the edge connecting (100)
and (110). While $OR1$ should be lower in energy, the majority observation of
$OR2$ implies kinetic hindrances decrease the frequency of $OR1$. Persistent
grain over grain growth and the absence of variations of the $OR$ frequencies
with thickness implies that the growth competition is finished within the first
few \si{\nano\meter}, and local epitaxy persists thereafter during growth.
