Last Passage Percolation with a Defect Line and the Solution of the Slow Bond Problem

Abstract

We address the question of how a localized microscopic defect, especially if it is small with respect to certain dynamic parameters, affects the macroscopic behavior of a system. In particular we consider two classical exactly solvable models: Ulam's problem of the maximal increasing sequence and the totally asymmetric simple exclusion process. For the first model, using its representation as a Poissonian version of directed last passage percolation on $\mathbb R^2$, we introduce the defect by placing a positive density of extra points along the diagonal line. For the latter, the defect is produced by decreasing the jump rate of each particle when it crosses the origin. The powerful algebraic tools for studying these processes break down in the perturbed versions of the models. Taking a more geometric approach we show that in both cases the presence of an arbitrarily small defect affects the macroscopic behavior of the system: in Ulam's problem the time constant increases, and for the exclusion process the flux of particles decreases. This, in particular, settles the longstanding Slow Bond Problem.