Comparison of partition functions in a space-time random environment

Let $Z^1$ and $Z^2$ be partition functions in the random polymer model in the same environment but driven by different underlying random walks. We give a comparison in concave stochastic order between $Z^1$ and $Z^2$ if one of the random walks has "more randomness" than the other. We also treat some related models: The parabolic Anderson model with space-time L\'evy noise; Brownian motion among space-time obstacles; and branching random walks in space-time random environments. We also obtain a necessary and sufficient criterion for $Z^1\preceq_{cv}Z^2$ if the lattice is replaced by a regular tree.