The law of (1+1)D SOS with an area tilt in a wedge

2026/08/05 by Amir Dembo, Eyal Lubetzky, Ofer Zeitouni
Mathematics · Physics and Astronomy · #math.PR #math-ph #math.MP #msc:60F17 #msc:82B24 #msc:82B41

paper · pdf

34 pages, 4 figures

arxiv created 2026/08/05 · arxiv updated 2026/08/07

Abstract

Motivated by the study of the level lines of the (2+1)D Solid-On-Solid (SOS) model above a floor, near the corners of the box, we derive the limit law of an ensemble of K curves from a (1+1)D SOS model, with an area tilt, and above a wedge-shaped floor in \-N,…,N\. We show that there exist explicit critical points α₀=1>α₁>…>α_K>0 such that, for each r ≥ 1, along the intervals ±(α_r N,α_r-1 N), the bottom K+1-r curves, rescaled by (N^2/3,N^1/3), tend to the law of a Geometrically-Area-Tilted Ensemble of non-crossing Brownian paths (Brownian GATE), independently across those 2K intervals. All other curves, centered and rescaled by (N,√(N)), tend to a product of K suitable Brownian bridges.

Citations

• The limiting law of the Discrete Gaussian level-lines
• Scaling limit and tail bounds for a random walk model of SOS level lines
• Characterizing Gibbs states for area-tilted Brownian lines
• On the limiting law of line ensembles of Brownian polymers with geometric area tilts
• Confinement of Brownian Polymers under Geometric Area Tilts
• Tightness and Line Ensembles for Brownian Polymers under Geometric Area Tilts
• Dyson Ferrari--Spohn diffusions and ordered walks under area tilts
• An invariance principle to Ferrari-Spohn diffusions
• Scaling limit and cube-root fluctuations in SOS surfaces above a wall
• Dynamics of (2+1)-dimensional SOS surfaces above a wall: Slow mixing induced by entropic repulsion
• An invariance principle for random walk bridges conditioned to stay positive
• Constrained Brownian motion: Fluctuations away from circular and parabolic barriers
• The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas