  We analyze the high moments of the Stochastic Heat Equation (SHE) via a
transformation to the attractive Brownian Particles (BPs), which are Brownian
motions interacting via pairwise attractive drift. In those scaling regimes
where the particles tend to cluster, we prove a Large Deviation Principle (LDP)
for the empirical measure of the attractive BPs. Under what we call the
1-to-$n$ initial-terminal condition, we characterize the unique minimizer of
the rate function and relate the minimizer to the spacetime limit shapes of the
Kardar--Parisi--Zhang (KPZ) equation in the upper tails. The results of this
paper are used in the companion paper Lin and Tsai (2023) to prove an
$n$-point, upper-tail LDP for the KPZ equation and to characterize the
corresponding spacetime limit shape.
