We calculate exactly the first cumulants of the free energy of a directed polymer in a random medium for the geometry of a cylinder. By using the fact that the $n\mathrm{th}$ moment $〈Z^n〉$ of the partition function is given by the ground-state energy of a quantum problem of n interacting particles on a ring of length L, we write an integral equation allowing to expand these moments in powers of the strength of the disorder $\gamma$ or in powers of n. For n small and $n\sim (L\gamma {)}^{-1/2},$ the moments $〈Z^n〉$ take a scaling form which allows us to describe all the fluctuations of order $1/L$ of the free energy per unit length of the directed polymer. The distribution of these fluctuations is the same as the one found recently in the asymmetric exclusion process, indicating that it is characteristic of all the systems described by the Kardar-Parisi-Zhang equation in $1+1$ dimensions.

• Received 10 December 1999

DOI:https://doi.org/10.1103/PhysRevE.61.6789