  Directed last passage percolation models on the plane, where one studies the
weight as well as the geometry of optimizing paths (called polymers) in a field
of i.i.d. weights, are paradigm examples of models in the KPZ universality
class. In this article, we consider the large deviation regime, i.e., when the
polymer has a much smaller (lower tail) or larger (upper tail) weight than
typical. Precise asymptotics of large deviation probabilities have been
obtained in a handful of the so-called exactly solvable scenarios, including
the Exponential (Johansson, '00) and Poissonian (Sepp\"al\"ainen, '98 and
Deuschel, Zeitouni, '99) cases. How the geometry of the optimizing paths change
under such a large deviation event was considered in (Deuschel, Zeitouni, '99),
where it was shown that the paths (from $(0,0)$ to $(n,n)$, say) remain
concentrated around the straight line joining the end points in the upper tail
large deviation regime, but the corresponding question in the lower tail was
left open. We establish a contrasting behavior in the lower tail large
deviation regime, showing that conditioned on the latter, in both the models,
the optimizing paths are not concentrated around any deterministic curve. Our
argument does not use any ingredient from integrable probability, and hence can
be extended to other planar last passage percolation models under fairly mild
conditions; and also to other non-integrable settings such as high dimensions.
