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Convex upper bounds for last passage percolation (LPP) times in the Kardar-Parisi-Zhang (KPZ) universality class

Isaac Meilijson

Abstract

On the $Z^2$ lattice, vertices are assigned random weights $W(i,j)$. The point-to-point last passage percolation (LPP) time $S_{M,N+1-M}$ between $(1,1)$ and $(M,N+1-M)$ is the maximum total weight among all upward/right-directed paths connecting the two. Point-to-line LPP time $R_N$ is the maximum of these maximal total weights over $M$. Asymptotic distributions and fluctuations of these LPP times have been studied for i.i.d. weights. The current study deals with identically distributed but not necessarily independent weights, and maximizes LPP times in the sense of convex dominance. In particular, maximal expected LPP times are identified, in the class of all weight couplings with a given marginal distribution. For the case of mean-$1$ exponentially distributed weights, there is a coupling for which $R_N$ is the shifted exponential variable $R_N^* = N W(1,1) + \log(N!)$, such that $E[Ψ(R_N)] \le E[Ψ(R_N^*)]$ for all couplings and all convex non-decreasing functions $Ψ$ for which these expectations are well defined. In contrast to ${{R_N^*} \over N}= W(1,1)+{{\log(N!)} \over N}$, with variance $1$ and mean diverging to $\infty$ like $\log(N)$, ${{R_N} \over N}$ converges to $2$ for the commonly studied i.i.d. weights.

Convex upper bounds for last passage percolation (LPP) times in the Kardar-Parisi-Zhang (KPZ) universality class

Abstract

On the lattice, vertices are assigned random weights . The point-to-point last passage percolation (LPP) time between and is the maximum total weight among all upward/right-directed paths connecting the two. Point-to-line LPP time is the maximum of these maximal total weights over . Asymptotic distributions and fluctuations of these LPP times have been studied for i.i.d. weights. The current study deals with identically distributed but not necessarily independent weights, and maximizes LPP times in the sense of convex dominance. In particular, maximal expected LPP times are identified, in the class of all weight couplings with a given marginal distribution. For the case of mean- exponentially distributed weights, there is a coupling for which is the shifted exponential variable , such that for all couplings and all convex non-decreasing functions for which these expectations are well defined. In contrast to , with variance and mean diverging to like , converges to for the commonly studied i.i.d. weights.
Paper Structure (12 sections, 3 theorems, 20 equations)

This paper contains 12 sections, 3 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Convex majorization
  3. Characterizations of a distribution
  4. The concepts of stochastic and convex majorization
  5. Stochastic majorization of the maximum of r.v.'s with given marginals: anti-thetic coupling and the Fréchet bound
  6. Convex majorization of the sum of r.v.'s with given marginals: Co-monotone coupling
  7. Convex majorization of maxima of partial sums - The method of Meilijson & Nádas MeilijsonNadas
  8. LPP on the complete $N$-partite graph
  9. Distributions that are memoryless up to scaling
  10. On stochastically small last passage times
  11. Flat random walks on $C_N$
  12. A family of couplings generated by flat random walks

Key Result

Theorem 1

The convex bound model for the complete $N$-partite graph. In terms of a single observation $U \sim U(0,1)$, the point-to-line LPP time and the point-to-point LPP time (for $M \le {{N+1} \over 2}$) corresponding to the couplings with anti-thetic weights on anti-diagonals and sum of co-monotone maxima over the anti-diagonals, convexly dominate their counterparts for any other coupling.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3