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Large deviations of geodesic midpoint fluctuations in last-passage percolation with general i.i.d. weights

Tom Alberts, Riddhipratim Basu, Sean Groathouse, Xiao Shen

TL;DR

The paper addresses large deviations of transversal (midpoint) fluctuations for geodesics in general i.i.d. last-passage percolation, introducing the right-tail rate function ${\mathcal J}_t(r)$ and the shape function ${\mu}_t$ to formulate a large deviation principle for the midpoint location. The authors develop a probabilistic framework that leverages subadditivity (via ${\mathcal J}_t$), left/right tail estimates, and combinatorial tools (BK inequality, FKG) to derive matching upper and lower bounds for the midpoint displacement, valid whenever ${\mu_0}>{\mu}_t$. In the special case of exponential weights, they verify a conjecture of Liu by showing the corner-path probability decays as $({4}/{e^2})^{n+o(n)}$, aligning with exact-solver predictions and highlighting interplay with KPZ universality. Overall, the work extends large deviation analysis to non-solvable LPP models, connects to directed landscape results, and provides a robust framework for understanding geodesic geometry in the KPZ class.$

Abstract

The study of transversal fluctuations of the optimal path is a crucial aspect of the Kardar-Parisi-Zhang (KPZ) universality class. In this work, we establish the large deviation limit for the midpoint transversal fluctuations in a general last-passage percolation (LPP) model with mild assumption on the i.i.d. weights. The rate function is expressed in terms of the right tail large deviation rate function of the last-passage value and the shape function. When the weights are chosen to be i.i.d. exponential random variables, our result verifies a conjecture communicated to us by Liu [Liu'22], showing the asymptotic probability of the geodesic from $(0,0)$ to $(n,n)$ following the corner path $(0,0) \to (n,0) \to (n,n)$ is $({4}/{e^2})^{n+o(n)}$.

Large deviations of geodesic midpoint fluctuations in last-passage percolation with general i.i.d. weights

TL;DR

The paper addresses large deviations of transversal (midpoint) fluctuations for geodesics in general i.i.d. last-passage percolation, introducing the right-tail rate function and the shape function to formulate a large deviation principle for the midpoint location. The authors develop a probabilistic framework that leverages subadditivity (via ), left/right tail estimates, and combinatorial tools (BK inequality, FKG) to derive matching upper and lower bounds for the midpoint displacement, valid whenever . In the special case of exponential weights, they verify a conjecture of Liu by showing the corner-path probability decays as , aligning with exact-solver predictions and highlighting interplay with KPZ universality. Overall, the work extends large deviation analysis to non-solvable LPP models, connects to directed landscape results, and provides a robust framework for understanding geodesic geometry in the KPZ class.$

Abstract

The study of transversal fluctuations of the optimal path is a crucial aspect of the Kardar-Parisi-Zhang (KPZ) universality class. In this work, we establish the large deviation limit for the midpoint transversal fluctuations in a general last-passage percolation (LPP) model with mild assumption on the i.i.d. weights. The rate function is expressed in terms of the right tail large deviation rate function of the last-passage value and the shape function. When the weights are chosen to be i.i.d. exponential random variables, our result verifies a conjecture communicated to us by Liu [Liu'22], showing the asymptotic probability of the geodesic from to following the corner path is .

Paper Structure

This paper contains 11 sections, 9 theorems, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Relations to the existing literature
  4. The $t=1/2$ Case
  5. Preliminaries
  6. Right tail large deviation rate function for the last-passage value
  7. Large deviation estimates of the last-passage value
  8. Proof of the upper bound in Theorem \ref{['main_thm']}
  9. Proof of the lower bound in Theorem \ref{['main_thm']}
  10. The case $0<t<1/2$
  11. The case $t=1/2$

Key Result

Theorem 1.1

Fix $0< t \leq 1/2$ such that $\mu_0 > \mu_t$. For each $\epsilon>0$, there exists a constant $n_0$ such that for each $n \geq n_0$, it holds that Thus, in the limit, we have

Figures (3)

  • Figure 1.1: Looking within the supercritical oriented percolation cone at $(0,0)$ in the northeast direction and the cone at $(n,n)$ in the southwest direction, if we fix a direction (shown in orange) that lies strictly inside the cone at $(0,0)$, as $n$ becomes large, with high probability, there will exist a geodesic formed by concatenating open paths from within the two cones, whose midpoint is to the right of the chosen direction.
  • Figure 4.1: Planting of the vertices $\{\mathbf{h}_j\}_{j=0}^{2J_0}$. The starting point $\mathbf{h}_0$, the midpoint $\mathbf{h}_{J_0}$, and the endpoint $\mathbf{h}_{2J_0}$ are labeled in the figure above.
  • Figure 4.2: An illustration of the definitions: The concatenation of geodesics connecting the points $\mathbf{h}_i$ is depicted in red and is denoted by $\Theta$. The geodesic from $(0,0)$ to $(n,n)$ is represented in blue. From these two paths, we construct a new path, $\gamma^*$, starting at $\mathbf{h}_{m^*-1}$ and ending at $\mathbf{h}_{k^*+1}$, shown as a yellow dotted line. Subsequently, the path $\gamma^*$ is further divided into three disjoint segments.

Theorems & Definitions (12)

  • Theorem 1.1
  • Proposition 1.2
  • Corollary 1.3
  • Proposition 2.1: LDP_poly Kes-86-stflour
  • Proposition 2.2: LDP_poly
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • proof : Sketch
  • ...and 2 more