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

TL;DR

This paper proves that the free energy of the half-space log-gamma polymer, when sufficiently far from the boundary, converges to the directed landscape, extending the universality class of these models.

Contribution

It establishes the convergence of the half-space log-gamma polymer free energy to the directed landscape away from the boundary, using coupling and tail bounds.

Findings

Convergence of half-space free energy to the directed landscape.

Paths contributing to free energy are confined within a $N^{2/3}$ transversal window.

Uniform exponential tail bounds for half-space free energies.

Abstract

We prove that the free energy of the half-space log-gamma polymer $N^{2/3+\delta}$ 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 $N^{2/3+\delta}$ 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.