Asymmetric particle systems and last-passage percolation in one and two dimensions - Schmidt, P (2012)
This thesis studies three models: Multi-type TASEP in discrete time, long-range lastpassage percolation on the line and convoy formation in a travelling servers model. All three models are relatively easy to state but they show a very rich and interesting behaviour. The TASEP is a basic model for a one-dimensional interacting particle system with non-reversible dynamics. We study some aspects of the TASEP in discrete time and compare the results to recently obtained results for the TASEP in continuous time. In particular we focus on stationary distributions for multi-type models, speeds of secondclass particles, collision probabilities and the speed process. We consider various natural update rules. The second model we study is directed last-passage percolation on the random graph G = (V,E) where V = ℤ and each edge (i,j), for i <: j ∈ , is present in E independently with some probability p ∈ (0,1]. To every (i,j) ∈ E we attach i.i.d. random weights vi,j > 0. We are interested in the behaviour of w0,n, which is the maximum weight of all directed paths from 0 to n, as n tends to infinity. We see two very different types of behaviour, depending on whether 𝔼[vi,j2] is finite or infinite. In the case where 𝔼[vi,j2] is finite we show that the process has a certain regenerative structure, and prove a strong law of large numbers and, under an extra assumption, a functional central limit theorem. In the situation where 𝔼[vi,j2] is infinite we obtain scaling laws and asymptotic distributions expressed in terms of a continuous last-passage percolation model on [0,1]. In the last model customers arrive on the non-negative half-line as a Poisson process of rate λ ∈ (0,∞) and n servers start at the origin at time 0. After completing some initial holding times, each server jumps to the first free customer it sees to its right and serves this customer. All service times are i.i.d. exponentials with parameter ν. After completion of a service the customer leaves the system and the server jumps to the next customer to the right, ignoring customers that are currently being served. We study the formation of convoys, groups of servers that travel together at the same speed, and the asymptotic behaviour of Xt(j), see position of server j at time t. ;