  In last passage percolation models lying in the KPZ universality class, the
energy of long energy-maximizing paths may be studied as a function of the
paths' pair of endpoint locations. Scaled coordinates may be introduced, so
that these maximizing paths, or polymers, now cross unit distances with
unit-order fluctuations, and have scaled energy, or weight, of unit order. In
this article, we consider Brownian last passage percolation in these scaled
coordinates. In the narrow wedge case, when one endpoint of such polymers is
fixed, say at $(0,0) \in \mathbb{R}^2$, and the other is varied horizontally,
over $(z,1)$, $z \in \mathbb{R}$, the polymer weight profile as a function of
$z \in \mathbb{R}$ is locally Brownian; indeed, by Theorem $2.11$ and
Proposition $2.5$ of [Ham16], the law of the profile is known to enjoy a very
strong comparison to Brownian bridge on a given compact interval, with a
Radon-Nikodym derivative in every $L^p$ space for $p \in (1,\infty)$, uniformly
in the scaling parameter, provided that an affine adjustment is made to the
weight profile before the comparison is made. In this article, we generalize
this narrow wedge case and study polymer weight profiles begun from a very
general initial condition. We prove that the profiles on a compact interval
resemble Brownian bridge in a uniform sense: splitting the compact interval
into a random but controlled number of patches, the profile in each patch after
affine adjustment has a Radon-Nikodym derivative that lies in every $L^p$ space
for $p \in (1,3)$. This result is proved by harnessing an understanding of the
uniform coalescence structure in the field of polymers developed in [Ham17a]
using techniques from [Ham16] and [Ham17b].
