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Convergence from the Log-Gamma Polymer to the Directed Landscape

Xinyi Zhang

TL;DR

The paper proves that the log-gamma polymer's two-parameter free-energy objects converge to the Airy sheet and, in turn, to the directed landscape, thereby placing a positive-temperature polymer model within the KPZ universality class. The authors exploit a geometric RSK invariance of free energy and develop off-diagonal fluctuation estimates to compensate for the discrete model’s lack of Brownian scaling, enabling tightness and full convergence. They connect the log-gamma free-energy to multi-layer Airy line ensembles, establish change-of-coordinates for scaling limits, and prove both Airy-sheet and directed-landscape convergence through rigorous probabilistic and variational arguments. This work advances the understanding of positive-temperature KPZ limits and provides a robust framework for analyzing similar polymer models via invariance, tail bounds, and multi-level convergence to the KPZ fixed point objects.

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.

Convergence from the Log-Gamma Polymer to the Directed Landscape

TL;DR

The paper proves that the log-gamma polymer's two-parameter free-energy objects converge to the Airy sheet and, in turn, to the directed landscape, thereby placing a positive-temperature polymer model within the KPZ universality class. The authors exploit a geometric RSK invariance of free energy and develop off-diagonal fluctuation estimates to compensate for the discrete model’s lack of Brownian scaling, enabling tightness and full convergence. They connect the log-gamma free-energy to multi-layer Airy line ensembles, establish change-of-coordinates for scaling limits, and prove both Airy-sheet and directed-landscape convergence through rigorous probabilistic and variational arguments. This work advances the understanding of positive-temperature KPZ limits and provides a robust framework for analyzing similar polymer models via invariance, tail bounds, and multi-level convergence to the KPZ fixed point objects.

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.
Paper Structure (24 sections, 35 theorems, 205 equations, 2 figures)

This paper contains 24 sections, 35 theorems, 205 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Log-gamma polymer
  3. Main Results
  4. Airy line ensemble, the Airy sheets, and the directed landscape
  5. Organization of the paper
  6. Notations
  7. Acknowledgments
  8. Geometric RSK correspondence: invariance of free energy
  9. Discrete free energy
  10. Invariance of discrete free energy
  11. Basic properties of discrete free energy
  12. Variational formula and monotonicity
  13. From free energy to last passage percolation
  14. Log-gamma line ensembles to parabolic Airy line ensemble
  15. Log-gamma sheet and monotonicity
  16. ...and 9 more sections

Key Result

Theorem 1.5

The continuous linear interpolation of $h^N(x,y)$ converges to the Airy sheet $S(x,y)$ in distribution as $C(\mathop{\mathrm{\mathbb{R}}}\nolimits^2,\mathop{\mathrm{\mathbb{R}}}\nolimits)$-random variables.

Figures (2)

  • Figure 1: An example of Theorem \ref{['thm_invariance']} when $n = 5$ and $k = 2$. The $i$-th row of the grid is associated with function $f_i$ on the left and $(Wf)_i$ on the right. The illustrated paths are possible non-intersecting paths from $U$ to $V$ on the left and $\Uparrow U$ to $V$ on the right.
  • Figure 2: An example of a multipath reorganization when $k = 2$, $y = 6$, and $x = 5$. The diagram on the right illustrates a reorganization of the multipath in the left diagram. The first $k$ rows are fully saturated by paths and the only remaining degree of freedom lies in the segment of $\pi_{k+1}$ connecting $(x, n \wedge x)$ to $(y, k+1)$. Importantly, the free energy is unchanged under this reorganization, as what matters is the collection of points visited by the multipath.

Theorems & Definitions (83)

  • Definition 1.1
  • Definition 1.2: Polymer partition function and polymer free energy
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5: Airy sheet convergence
  • Theorem 1.6: Directed landscape convergence
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 73 more