Convex bounds for last passage percolation with dependent identically distributed weights

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 $E[Ψ(R_N)] \le E[Ψ(R_N^*)]$ 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 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.