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Directed landscape convergence for the half-space log-gamma polymer $N^{2/3+δ}$ away from the boundary

Xinyi Zhang

Abstract

We prove that the free energy of the half-space log-gamma polymer $N^{2/3+δ}$ away from the boundary in the non-attractive regime converges to the directed landscape. Based on the convergence of the full-space log-gamma free energy to the directed landscape, we couple the full-space and the half-space model and prove that the dominant contributions to free energy in both cases come from paths that remain confined to a transversal window of order $N^{2/3}$. The result follows from three main inputs: a deterministic leading-order gap between paths that deviate from the transversal window on the $N^{2/3+δ}$ scale and those within the typical $N^{2/3}$ scale; uniform exponential upper-tail bounds for half-space free energies with general slope; and existing full-space estimates on constrained and exiting free energies.

Directed landscape convergence for the half-space log-gamma polymer $N^{2/3+δ}$ away from the boundary

Abstract

We prove that the free energy of the half-space log-gamma polymer away from the boundary in the non-attractive regime converges to the directed landscape. Based on the convergence of the full-space log-gamma free energy to the directed landscape, we couple the full-space and the half-space model and prove that the dominant contributions to free energy in both cases come from paths that remain confined to a transversal window of order . The result follows from three main inputs: a deterministic leading-order gap between paths that deviate from the transversal window on the scale and those within the typical scale; uniform exponential upper-tail bounds for half-space free energies with general slope; and existing full-space estimates on constrained and exiting free energies.
Paper Structure (14 sections, 16 theorems, 102 equations, 1 figure)

This paper contains 14 sections, 16 theorems, 102 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model Definition
  3. Main Results
  4. Coupling Between Full-Space and Half-Space Log-Gamma Polymers
  5. Proof of Proposition \ref{['prop:diff_loggamma']}
  6. Analysis of the shape function
  7. Proof of Proposition \ref{['prop:uniform_upper_tail']}
  8. An identity in distribution between full-space and half-space log-gamma polymer
  9. Full-space point-to-point model.
  10. Half-space point-to-line model.
  11. Homogeneous full-space estimates
  12. Inequalities of shape function
  13. Proof of Proposition \ref{['prop:inhomo_full_space']}
  14. Acknowledgments

Key Result

Theorem 1.3

We define the following scaling operators $\overline{x}_N = \lfloor N^{2/3}xq^{-2} \rfloor + 1$, $t_N = \lfloor 2Nt \rfloor$. Fix any $\delta>0$ and define the scaled half-space log-gamma polymer free energy away from the boundary, $h_{\mathrm{half}}^{N, \delta}$, as a random function on $\mathbb{R} Then the continuous linear interpolation of $h_{\mathrm{half}}^{N,\delta}$ converges to the directe

Figures (1)

  • Figure 1: Parameter configuration for the Barraquand--Wang log-gamma polymers: (A) full-space point-to-point geometry, (B) octant/trapezoidal geometry.

Theorems & Definitions (25)

  • Definition 1.1: Full-space log-gamma polymer
  • Definition 1.2: Half-space log-gamma polymer
  • Theorem 1.3
  • Theorem 2.1
  • Proposition 2.2
  • proof : Proof of Theorem \ref{['thm_half_loggamma_to_DL']}
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 15 more