2022/11/25 by Piet Lammers, Lammers, Piet
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.2211.14365
openalex publication_date 2022/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
This text considers the discrete height functions associated with the Berezinskii--Kosterlitz--Thouless transition (BKT) at slope zero. Our main results are as follows. * Sharpness: If the model is localised, then the two-point function (covariance) decays exponentially fast in the distance between the points. * Effective temperature gap: If the model is delocalised, then the variance grows at least as clog n, where n is the distance to the boundary and c>0 a universal constant not depending on the temperature. Thus, the effective temperature must jump from 0 to at least c at the transition point; values in the interval (0,c) are forbidden. * Delocalisation at the transition point: The delocalised phase includes the transition point, in the sense that it is a closed set in the phase diagram in the appropriate topology. These results contribute to the understanding of the regime at and around the transition point which remained largely unexplored. In a follow-up paper, the sharpness derived here is used to establish that the localisation-delocalisation transition is equivalent to the BKT transition in the dual XY and Villain models.