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

Understand

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.