Convergence from the Log-Gamma Polymer to the Directed Landscape

2025/05/08 by Xinyi Zhang, Zhang, Xinyi · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2505.05685

openalex publication_date 2025/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define the log-gamma sheet and the log-gamma landscape in terms of the 2-parameter and 4-parameter free energy of the log-gamma polymer model and prove that they converge to the Airy sheet and the directed landscape, which are central objects in the Kardar-Parisi-Zhang (KPZ) universality class. Our proof of convergence to the Airy sheet relies on the invariance of free energy through the geometric RSK correspondence and the monotonicity of the free energy. To upgrade the convergence to the directed landscape, tail bounds in both spatial and temporal directions are required. However, due to the lack of scaling invariance in the discrete log-gamma polymer--unlike the Brownian setting of the O'Connell-Yor model--existing on-diagonal fluctuation bounds are insufficient. We therefore develop new off-diagonal local fluctuation estimates for the log-gamma polymer.

Citations

• Characterization of the directed landscape from the KPZ fixed point
• Scaling limit of the colored ASEP and stochastic six-vertex models
• Strong Characterization for the Airy Line Ensemble
• Temporal correlation in the inverse-gamma polymer
• The KPZ equation and the directed landscape
• Tightness of (H, H^RW)-Gibbsian line ensembles
• The scaling limit of the longest increasing subsequence
• Spatial tightness at the edge of Gibbsian line ensembles
• Fluctuations of the log-gamma polymer free energy with general parameters and slopes
• Invariance of polymer partition functions under the geometric RSK correspondence
• The directed landscape
• Brownian Gibbs property for Airy line ensembles
• Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions
• Scale Invariance of the PNG Droplet and the Airy Process

Cited by

• The discrete periodic Pitman transform: invariances, braid relations, and Burke properties

Related

• Fluctuations of the log-gamma polymer free energy with general parameters and slopes
• Convergence to stationary measures for the half-space log-gamma polymer
• An identity in distribution between full-space and half-space log-gamma polymers
• Stationary measures for the log-gamma polymer and KPZ equation in half-space
• Tracy-Widom fluctuations for perturbations of the log-gamma polymer in intermediate disorder