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Last passage percolation in a product-type random environment

Yuri Bakhtin, Konstantin Khanin, András Mészáros, Jeremy Voltz

TL;DR

The paper analyzes last passage percolation in a 1+1 dimensional product-type random environment Φ(x,i)=F(x)B(i), with F supported on (-c,c) and near-edge density governed by κ>-1, and B(i) i.i.d. ±1. It proves the existence of a deterministic shape function Λ with Λ(0)=c, Λ(±1)=0, and a corner at 0, plus a nonlinear regime for small slopes, and shows that when κ>0 there is a linear segment of Λ near 0, indicating a distinct universality class. It further establishes a Poisson scaling limit for discrepancies and, under an extra assumption, a Tracy-Widom-type limiting distribution for suitably scaled optimal actions, with the scaling exponent and limit depending only on κ. The approach combines subadditive ergodic theory, a loop-decomposition, concentration inequalities, and a Poisson-approximation for edge discrepancies, yielding a new universality description tied to κ and highlighting qualitative departures from KPZ-type behavior. Overall, the work identifies a novel universality class for LPP in product environments and provides structural insights into the geometry of optimal paths and their fluctuations.

Abstract

We consider a last passage percolation model in dimension $1+1$ with potential given by the product of a spatial i.i.d. potential with symmetric bounded distribution and an independent i.i.d. in time sequence of signs. We assume that the density of the spatial potential near the edge of its support behaves as a power, with exponent $κ>-1$. We investigate the linear growth rate of the actions of optimal point-to-point lazy random walk paths as a function of the path slope and describe the structure of the resulting shape function. It has a corner at $0$ and, although its restriction to positive slopes cannot be linear, we prove that it has a flat edge near $0$ if $κ>0$. For optimal point-to-line paths, we study their actions and locations of favorable edges that the paths tends to reach and stay at. Under an additional assumption on the time it takes for the optimal path to reach the favorable location, we prove that appropriately normalized actions converge to a limiting distribution that can be viewed as a counterpart of the Tracy-Widom law. Since the scaling exponent and the limiting distribution depend only on the parameter $κ$, our results provide a description of a new universality class.

Last passage percolation in a product-type random environment

TL;DR

The paper analyzes last passage percolation in a 1+1 dimensional product-type random environment Φ(x,i)=F(x)B(i), with F supported on (-c,c) and near-edge density governed by κ>-1, and B(i) i.i.d. ±1. It proves the existence of a deterministic shape function Λ with Λ(0)=c, Λ(±1)=0, and a corner at 0, plus a nonlinear regime for small slopes, and shows that when κ>0 there is a linear segment of Λ near 0, indicating a distinct universality class. It further establishes a Poisson scaling limit for discrepancies and, under an extra assumption, a Tracy-Widom-type limiting distribution for suitably scaled optimal actions, with the scaling exponent and limit depending only on κ. The approach combines subadditive ergodic theory, a loop-decomposition, concentration inequalities, and a Poisson-approximation for edge discrepancies, yielding a new universality description tied to κ and highlighting qualitative departures from KPZ-type behavior. Overall, the work identifies a novel universality class for LPP in product environments and provides structural insights into the geometry of optimal paths and their fluctuations.

Abstract

We consider a last passage percolation model in dimension with potential given by the product of a spatial i.i.d. potential with symmetric bounded distribution and an independent i.i.d. in time sequence of signs. We assume that the density of the spatial potential near the edge of its support behaves as a power, with exponent . We investigate the linear growth rate of the actions of optimal point-to-point lazy random walk paths as a function of the path slope and describe the structure of the resulting shape function. It has a corner at and, although its restriction to positive slopes cannot be linear, we prove that it has a flat edge near if . For optimal point-to-line paths, we study their actions and locations of favorable edges that the paths tends to reach and stay at. Under an additional assumption on the time it takes for the optimal path to reach the favorable location, we prove that appropriately normalized actions converge to a limiting distribution that can be viewed as a counterpart of the Tracy-Widom law. Since the scaling exponent and the limiting distribution depend only on the parameter , our results provide a description of a new universality class.
Paper Structure (13 sections, 21 theorems, 99 equations, 1 figure)

This paper contains 13 sections, 21 theorems, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Formulation of the main results
  3. Notations
  4. Basic properties of the shape function
  5. The discrepancy of an edge
  6. Existence of the shape function
  7. Nonlinearity of $\Lambda$
  8. The existence of a linear segment of $\Lambda$ -- The proof of Theorem \ref{['shape function linear']}
  9. The main idea of the proof
  10. Choosing the weights $f$
  11. Concentration and choosing the signs $b_n$
  12. Loop decomposition
  13. Finishing the proof of Theorem \ref{['shape function linear']}

Key Result

Theorem 2.1

There is a continuous, even and concave deterministic function $\Lambda:[-1,1]\to\mathbb{R}_+$ and an event $\Omega_0\in\mathcal{F}$ with $\mathbb{P}\left(\Omega_0\right)=1$ such that for all $(B,F)\in\Omega_0$ and all $\alpha\in[-1,1]$, we have

Figures (1)

  • Figure 1: A simulation of the shape function $\Lambda$ for $\varrho=\frac{1}{2} \mathbbm{1}_{(-1,1)}$. Note that although this density is not covered by Theorem \ref{['shape function linear']}, $\Lambda$ seems to have a linear piece, see Section \ref{['secopenproblems']} for further discussion.

Theorems & Definitions (34)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 24 more