  We study paths of time-length $t$ of a continuous-time random walk on
$\mathbb Z^2$ subject to self-interaction that depends on the geometry of the
walk range and a collection of random, uniformly positive and finite edge
weights. The interaction enters through a Gibbs weight at inverse temperature
$\beta$; the "energy" is the total sum of the edge weights for edges on the
outer boundary of the range. For edge weights sampled from a
translation-invariant, ergodic law, we prove that the range boundary
condensates around an asymptotic shape in the limit $t\to\infty$ followed by
$\beta\to\infty$. The limit shape is a minimizer (unique, modulo translates) of
the sum of the principal harmonic frequency of the domain and the perimeter
with respect to the first-passage percolation norm derived from (the law of)
the edge weights. A dense subset of all norms in $\mathbb R^2$, and thus a
large variety of shapes, arise from the class of weight distributions to which
our proofs apply.
