Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Large Deviation Principle for Last Passage Percolation Models

Pranay Agarwal

TL;DR

The paper develops a metric-level large deviation principle for last passage percolation models in the KPZ class, using hypo-convergence of upper semicontinuous functionals and a network-based rate function to go beyond exactly solvable cases. It introduces a preliminary rate $\Theta$ and upgrades to a true rate function $I$ via $e$-complete (planted) networks, proving both upper and lower bounds and establishing a geodesic LDP through a contraction principle and variational framework. The results provide a robust method to study large transversal fluctuations of geodesics and to quantify deviations of metric objects in LPP, with explicit extensions to Poisson LPP and directed polymers. The framework unifies and extends known solvable-case results, offering insights into the geometry of optimal paths and the universality of KPZ-type fluctuations in non-solvable settings.

Abstract

Study of the KPZ universality class has seen the emergence of universal objects over the past decade which arise as the scaling limit of the member models. One such object is the directed landscape, and it is known that exactly solvable last passage percolation (LPP) models converge to the directed landscape under the KPZ scaling (see \cite{DV21}). Large deviations of the directed landscape on the metric level were recently studied in \cite{DDV24}, which also provides a general framework for establishing such large deviation principle (LDP). The main goal of the article is to apply and refine that framework to establish a LDP for LPP models at the metric level without relying on exact solvability. We then use the LDP on the metric level to establish a LDP for geodesics in these models, thus providing a streamlined way to study large transversal fluctuations of geodesics in these models. We briefly touch on how the theory extends to other planar models like directed polymers and Poisson LPP.

Large Deviation Principle for Last Passage Percolation Models

TL;DR

The paper develops a metric-level large deviation principle for last passage percolation models in the KPZ class, using hypo-convergence of upper semicontinuous functionals and a network-based rate function to go beyond exactly solvable cases. It introduces a preliminary rate and upgrades to a true rate function via -complete (planted) networks, proving both upper and lower bounds and establishing a geodesic LDP through a contraction principle and variational framework. The results provide a robust method to study large transversal fluctuations of geodesics and to quantify deviations of metric objects in LPP, with explicit extensions to Poisson LPP and directed polymers. The framework unifies and extends known solvable-case results, offering insights into the geometry of optimal paths and the universality of KPZ-type fluctuations in non-solvable settings.

Abstract

Study of the KPZ universality class has seen the emergence of universal objects over the past decade which arise as the scaling limit of the member models. One such object is the directed landscape, and it is known that exactly solvable last passage percolation (LPP) models converge to the directed landscape under the KPZ scaling (see \cite{DV21}). Large deviations of the directed landscape on the metric level were recently studied in \cite{DDV24}, which also provides a general framework for establishing such large deviation principle (LDP). The main goal of the article is to apply and refine that framework to establish a LDP for LPP models at the metric level without relying on exact solvability. We then use the LDP on the metric level to establish a LDP for geodesics in these models, thus providing a streamlined way to study large transversal fluctuations of geodesics in these models. We briefly touch on how the theory extends to other planar models like directed polymers and Poisson LPP.

Paper Structure

This paper contains 12 sections, 36 theorems, 175 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminary Rate Function
  4. Finite Rate Metrics
  5. Networks and Metrics
  6. Proof of Upper Bound
  7. Proof of Lower Bound
  8. Large Deviations of Geodesics
  9. Discussion on Properties
  10. Poisson Last Passage and Polymer Models
  11. Poisson Last Passage
  12. Directed Polymers

Key Result

Theorem 1.1

There exists a lower semi-continuous function $I : {\mathcal{E}} \to [0, \infty]$, such that for every Borel measurable set ${\mathcal{A}} \subseteq {\mathcal{E}}$, as $n \to \infty$ we have Furthermore, $I$ is a good rate function i.e., $I^{-1}([0,a])$ is compact for all $a < \infty$. We also show that $I^{-1}(0) =d$ and $I$ is strictly increasing on $I^{-1}([0, \infty))$.

Figures (1)

  • Figure 1: Illustration for the construction of Proposition \ref{['prop: A3']}. The blue line denotes the path corresponding to $X_k$ and the red line illustrates the geodesic.

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1: Att84, DSW83
  • Remark 2.2
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Remark 4.3
  • Lemma 4.4
  • ...and 61 more