  The properties of tissue interfaces -- between separate populations of cells,
or between a group of cells and its environment -- has attracted intense
theoretical, computational, and experimental study. Recent work on shape-based
models inspired by dense epithelia have suggested a possible ``topological
sharpening'' effect, by which four-fold vertices spatially coordinated along a
cellular interface lead to a cusp-like restoring force acting on cells at the
interface, which in turn greatly suppresses interfacial fluctuations. We
revisit these interfacial fluctuations, focusing on the distinction between
short length scale reduction of interfacial fluctuations and long length scale
renormalized surface tension. To do this, we implement a spectrally resolved
analysis of fluctuations over extremely long simulation times. This leads to
more quantitative information on the topological sharpening effect, in which
the degree of sharpening depends on the length scale over which it is measured.
We compare our findings with a Brownian bridge model of the interface, and
close by analyzing existing experimental data in support of the role of
short-length-scale topological sharpening effects in real biological systems.
