2020/10/21 by Ofer Busani, Busani, Ofer, Timo Seppäläinen +1
Mathematics · Physics and Astronomy · #Combinatorics #Computer science #FOS: Mathematics #FOS: Physical sciences #Geometry #Independent and identically distributed random variables #Inverse #Lattice (music) #Mathematical Physics (math-ph) #Mathematics #Midpoint #Physics #Planar #Polymer #Probability (math.PR) #Random Matrices and Applications #Random variable #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.2010.11279
published in arXiv (Cornell University) (Cornell University) · 38 pages, 5 figures
openalex publication_date 2020/10/21 · arxiv created 2020/11/11 · arxiv updated 2020/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise technical result is that, except for measures supported on straight-line paths, such Gibbs measures do not exist in almost every environment when the weights are independent and identically distributed inverse-gamma random variables. The proof proceeds by showing that when two endpoints of a point-to-point polymer distribution are taken to infinity in opposite directions but not parallel to lattice directions, the midpoint of the polymer path escapes. The proof is based on couplings, planar comparison arguments, and a recently discovered joint distribution of Busemann functions.