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Coupling derivation of optimal-order central moment bounds in exponential last-passage percolation

Elnur Emrah, Nicos Georgiou, Janosch Ortmann

TL;DR

This work develops a soft probabilistic framework for optimal-order central-moment bounds in planar directed exponential LPP and its one-sided-boundary variants. Central to the approach is a coupling with increment-stationary LPP and the use of MGF identities of Rains type, enabling sharp left-tail and right-tail control across bulk, increments, and boundary settings, including a KPZ-to-Gaussian transition. The authors obtain uniform moment bounds for bulk LPP for all $p\ge1$, prove a left-tail bound with exponent $3/2$ (improvable to the optimal $3$ in some directions), and extend the analysis to LPP with boundary, obtaining optimal-order bounds in both KPZ and Gaussian regimes. The results demonstrate the power of the coupling method, offering a robust probabilistic path to sharp fluctuation bounds in KPZ-class models and informing potential extensions to related polymers and random matrix settings.

Abstract

We introduce new probabilistic arguments to derive optimal-order central moment bounds in planar directed last-passage percolation. Our technique is based on couplings with the increment-stationary variants of the model, and is presented in the context of i.i.d. exponential weights for both zero and near-stationary boundary conditions. A main technical novelty in our approach is a new proof of the left-tail fluctuation upper bound with exponent 3/2 for the last-passage times.

Coupling derivation of optimal-order central moment bounds in exponential last-passage percolation

TL;DR

This work develops a soft probabilistic framework for optimal-order central-moment bounds in planar directed exponential LPP and its one-sided-boundary variants. Central to the approach is a coupling with increment-stationary LPP and the use of MGF identities of Rains type, enabling sharp left-tail and right-tail control across bulk, increments, and boundary settings, including a KPZ-to-Gaussian transition. The authors obtain uniform moment bounds for bulk LPP for all , prove a left-tail bound with exponent (improvable to the optimal in some directions), and extend the analysis to LPP with boundary, obtaining optimal-order bounds in both KPZ and Gaussian regimes. The results demonstrate the power of the coupling method, offering a robust probabilistic path to sharp fluctuation bounds in KPZ-class models and informing potential extensions to related polymers and random matrix settings.

Abstract

We introduce new probabilistic arguments to derive optimal-order central moment bounds in planar directed last-passage percolation. Our technique is based on couplings with the increment-stationary variants of the model, and is presented in the context of i.i.d. exponential weights for both zero and near-stationary boundary conditions. A main technical novelty in our approach is a new proof of the left-tail fluctuation upper bound with exponent 3/2 for the last-passage times.
Paper Structure (30 sections, 34 theorems, 181 equations, 1 figure)

This paper contains 30 sections, 34 theorems, 181 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Exponential LPP
  3. Stochastic growth and KPZ universality
  4. Coupling approach to KPZ fluctuation exponents
  5. Recent developments in the coupling approach
  6. Contribution of the present work
  7. Central moment bounds for exponential LPP
  8. Bounds for the bulk LPP
  9. Bounds for LPP increments
  10. Bounds for LPP with boundary
  11. Proofs of the tail bounds for LPP increments
  12. Proof of Theorem \ref{['TIncRTUB']}
  13. Proof of Theorem \ref{['TIncRTLB']}
  14. Proofs of the bounds for bulk LPP
  15. Proof of Theorem \ref{['T:BlkLT']}
  16. ...and 15 more sections

Key Result

Theorem 2.1

Let $\delta > 0$. There exists a constant $C_0 = C_0(\delta) > 0$ such that for $\mathbf{v} \in S_\delta \cap \mathbb{Z}_{>0}^2$ and $p \ge 1$.

Figures (1)

  • Figure 3.1: Illustrates the justification for equation \ref{['E:103']} with $\mathbf{v} = (m, n)$. When the geodesic $\pi = \widetilde{\pi}_{(1, 1), \mathbf{v} + (1, 1)}^{\mathbf{v}, u}$ (black) visits $\mathbf{v}+(1, 0) = (m+1, n)$, for any $i \in [m+1]$, the geodesic $\pi' = \widetilde{\pi}_{(i, 1), \mathbf{v} + (1, 1)}^{\mathbf{v}, u}$ (any of the dashed gray) must intersect $\pi$ strictly before the endpoint $\mathbf{v} + (1, 1) = (m+1, n+1)$. Once they intersect, $\pi$ and $\pi'$ a.s. coalesce by the a.s. uniqueness of geodesics. In particular, $\pi'$ also visits $(m+1, n)$ a.s., which implies \ref{['E:103']}.

Theorems & Definitions (61)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 51 more