  In last passage percolation models lying in the KPZ universality class, long
maximizing paths have a typical deviation from the linear interpolation of
their endpoints governed by the two-thirds power of the interpolating distance.
This two-thirds power dictates a choice of scaled coordinates, in which these
maximizers, now called polymers, cross unit distances with unit-order
fluctuations. In this article, we consider Brownian last passage percolation in
these scaled coordinates, and prove that the probability of the presence of $k$
disjoint polymers crossing a unit-order region while beginning and ending
within a short distance $\epsilon$ of each other is bounded above by
$\epsilon^{(k^2 - 1)/2 \, + \, o(1)}$.
  This result, which we conjecture to be sharp, yields understanding of the
uniform nature of the coalescence structure of polymers, and plays a
foundational role in [Ham17c] in proving comparison on unit-order scales to
Brownian motion for polymer weight profiles from general initial data. The
present paper also contains an on-scale articulation of the two-thirds power
law for polymer geometry: polymers fluctuate by $\epsilon^{2/3}$ on short
scales $\epsilon$.
