High temperature behaviors of the directed polymer on a cylinder

In this paper, we study the free energy of the directed polymer on a cylinder of radius $L$ with the inverse temperature $\beta$. Assuming the random environment is given by a Gaussian process that is white in time and smooth in space, with an arbitrary compactly supported spatial covariance function, we obtain precise scaling behaviors of the limiting free energy for high temperatures $\beta\ll1$, followed by large $L\gg1$, in all dimensions. Our approach is based on a perturbative expansion of the PDE hierarchy satisfied by the multipoint correlation function of the polymer endpoint distribution. For the random environment given by the $1+1$ spacetime white noise, we derive an explicit expression of the limiting free energy, confirming the result obtained through the replica method in [12].