LAUSR

Let $$Z^1$$ Z 1 and $$Z^2$$ 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 $$Z^1$$ Z 1 and $$Z^2$$ 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 Lévy 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 $$Z^1\preceq _{cv}Z^2$$ Z 1 ⪯ cv Z 2 if the lattice is replaced by a regular tree.