2000/05/22 by Éric Brunet-Gouet, Eric Brunet, Bernard Derrida · 5 citations
Mathematics · Physics and Astronomy · #Combinatorics #Cumulant #Distribution (mathematics) #Energy (signal processing) #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Moment (physics) #Partition (number theory) #Partition function (quantum field theory) #Physics #Polymer #Quantum mechanics #Random Matrices and Applications #Scaling #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.61.6789
18 pages, no figure, tu appear in PRE 61 (2000)
arxiv created 2000/05/22 · openalex publication_date 2000/06/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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 nth 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 approximately (Lgamma)(-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.