  We derive stationary measures for certain zero-temperature random polymer
models, which we believe are new in the case of the zero-temperature limit of
the beta random polymer (that has been called the river delta model). To do
this, we apply techniques developed for understanding the stationary measures
of the corresponding positive-temperature random polymer models (and some
deterministic integrable systems). More specifically, the article starts with a
survey of results for the four basic beta-gamma models (i.e. the inverse-gamma,
gamma, inverse-beta and beta random polymers), highlighting how the maps
underlying the systems in question can each be reduced to one of two basic
bijections, and that through an `independence preservation' property, these
bijections characterise the associated stationary measures. We then derive
similar descriptions for the corresponding zero-temperature maps, whereby each
is written in terms of one of two bijections. One issue with this picture is
that, unlike in the positive-temperature case, the change of variables required
is degenerate in general, and so whilst the argument yields stationary
solutions, it does not provide a complete characterisation of them. On the
other hand, this degeneracy does allow us to explain the appearance of atoms in
the stationary measures of certain zero-temperature models. We also derive from
our viewpoint various links between random polymer models, some of which
recover known results, some of which are novel, and some of which lead to
further questions.
