Convex bounds for last passage percolation with dependent identically distributed weights

TL;DR

This paper extends last passage percolation analysis to dependent weights, identifying maximal expected LPP times and their asymptotic behavior, especially for exponential weights, under various coupling strategies.

Contribution

It introduces a framework for analyzing dependent weights in LPP, establishing bounds on expected passage times and their fluctuations, and characterizing optimal couplings.

Findings

Maximal expected LPP times are identified within all couplings with a fixed marginal.

For exponential weights, a coupling exists where the LPP time is a shifted exponential variable with specific properties.

Expected LPP time for small scales is at least proportional to the number of vertices, with minimal variance asymptotically zero.

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-oriented 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 increasing 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 $E[\Psi(R_N)] \le E[\Psi(R_N^*)]$ for all couplings and all convex non-decreasing functions $\Psi$ 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 a.s. to $2$ for the commonly studied i.i.d. weights. As for {\em small} LPP, expected LPP time is at least $NE[W(1,1)]$, attained by assigning to each anti-diagonal identical weights. The minimal possible variance of $R_N$ is asymptotically zero for exponential weights.