  We introduce the strict-weak polymer model, and show the KPZ universality of
the free energy fluctuation of this model for a certain range of parameters.
Our proof relies on the observation that the discrete time geometric q-TASEP
model, studied earlier by A. Borodin and I. Corwin, scales to this polymer
model in the limit q->1. This allows us to exploit the exact results for
geometric q-TASEP to derive a Fredholm determinant formula for the strict-weak
polymer, and in turn perform rigorous asymptotic analysis to show KPZ scaling
and GUE Tracy-Widom limit for the free energy fluctuations. We also derive
moments formulae for the polymer partition function directly by Bethe ansatz,
and identify the limit of the free energy using a stationary version of the
polymer model.
