  We develop the study of a particular class of biorthogonal measures,
encompassing at the same time several random matrix models and partition
functions of polymers. This general framework allows us to characterize the
partition functions of the Log Gamma polymer and the mixed polymer in terms of
explicit biorthogonal measures, as it was previously done for the homogeneous
O'Connell-Yor polymer by Imamura and Sasamoto. In addition, we show that the
biorthogonal measures associated to these three polymer models (Log Gamma,
O'Connell-Yor, and mixed polymer) converge to random matrix eigenvalue
distributions in small temperature limits. We also clarify the connection
between different Fredholm determinant representations and explain how our
results might be useful for asymptotic analysis and large deviation estimates.
