Last Passage Percolation in Macroscopically Inhomogeneous Media

In this note we investigate the last passage percolation model in the presence of macroscopic inhomogeneity. We analyze how this affects the scaling limit of the passage time, leading to a variational problem that provides an ODE for the deterministic limiting shape of the maximal path. We obtain a sufficient analytical condition for uniqueness of the solution for the variational problem. Consequences for the totally asymmetric simple exclusion process are discussed. This is a reprint from Electr. Comm. Probab., v. 13, p. 131-139. Published on: March 5, 2008. 1 The model and results The last passage percolation process has been widely studied over the last few years [1, 2, 3, 7, 8, 9, 10]. There are several equivalent physical interpretations for the model. Examples include zerotemperature directed polymer in a random environment, a certain growth process, queuing theory, a randomly increasing Young diagram and random partitions [6, 12, 17]. By a simple coupling argument, results obtained for last passage percolation have their duals for the totally asymmetric simple exclusion process and thus the former model is often useful for the study of the later one [4, 5, 11, 13, 14, 16]. It is for the two-dimensional case that the most explicit results and estimates are known. In particular for geometric or exponential distributions, to which an exact solution was given by [8]. In this note we study last passage percolation with exponentially distributed passage times in the presence of macroscopic inhomogeneity. We begin by restating known results with a different point of view: instead of taking the limit of a large rectangle on the usual lattice we consider the limit of a fine lattice on some fixed rectangle, which is equivalent but resembles hydrodynamics. A continuous function α is defined on the macroscopic rectangle and locally modifies the parameter of the process. The problem of studying the random microscopic path of maximal passage time leads to the variational problem of finding a deterministic macroscopic curve maximizing a certain functional. We shall see that the rescaled passage time indeed converges to that given by the variational problem, and give sufficient analytical conditions for convergence of the maximal path’s shape to a deterministic curve. The viewpoint adopted here has immediate implications for the totally asymmetric simple exclusion process. On the scaling limit we can describe the behavior of the total current through the origin up to a given time when the jump rate has macroscopic fluctuations in space (as well as in blocks of particles, or even both). In particular for spatial inhomogeneity it is easy to see that the instantaneous current is AMS Subject Classifications: 60K35, 82D60, 60F10.