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Convergence of Half-Space Last Passage Percolation Away from the Boundary to the Directed Landscape

Xinyi Zhang

Abstract

In this note, we prove convergence of the half-space exponential last passage percolation (LPP) model, away from the boundary, to the directed landscape. Our approach couples the half-space and full-space LPP models and constructs two barrier events based on the monotonicity of last passage paths. Combining this coupling with moderate deviation estimates for both models and the known convergence of full-space LPP to the directed landscape, we establish the desired convergence.

Convergence of Half-Space Last Passage Percolation Away from the Boundary to the Directed Landscape

Abstract

In this note, we prove convergence of the half-space exponential last passage percolation (LPP) model, away from the boundary, to the directed landscape. Our approach couples the half-space and full-space LPP models and constructs two barrier events based on the monotonicity of last passage paths. Combining this coupling with moderate deviation estimates for both models and the known convergence of full-space LPP to the directed landscape, we establish the desired convergence.
Paper Structure (6 sections, 10 theorems, 52 equations)

This paper contains 6 sections, 10 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Model Definitions and Main Results
  3. Coupling Between Full-Space and Half-Space LPP
  4. Moderate Deviation Estimates
  5. Convergence Away from the Boundary
  6. Acknowledgments

Key Result

Theorem 2.3

Let $\mathbb{R}_+^4 = \{(x,s;r,t) \in \mathbb{R}^4: s<t\}$. Fix any $\delta>0$ and define the scaled half-space exponential last passage percolation away from the boundary $L^{\mathrm{half},\delta}_n$ as a random function on $\mathbb{R}_+^4$: Then $L^{\mathrm{half},\delta}_n$ converges to the directed landscape $\mathfrak{L}$ in distribution uniformly over compact subsets of $\mathbb{R}_+^4$.

Theorems & Definitions (17)

  • Definition 2.1: Full-space LPP
  • Definition 2.2: Half-space LPP
  • Theorem 2.3: Half-space exponential LPP converges to the directed landscape
  • Theorem 2.4
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Proposition 4.1: Full-space one-point moderate deviation
  • Proposition 4.2: Full-space constrained moderate deviation
  • Definition 4.3
  • ...and 7 more