Title:

Asymmetric particle systems and lastpassage percolation in one and two dimensions

This thesis studies three models: Multitype TASEP in discrete time, longrange 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 onedimensional interacting particle system with nonreversible 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 multitype models, speeds of secondclass particles, collision probabilities and the speed process. We consider various natural update rules. The second model we study is directed lastpassage 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 v_{i,j} > 0. We are interested in the behaviour of w_{0,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[v_{i,j}^{2}] is finite or infinite. In the case where 𝔼[v_{i,j}^{2}] 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[v_{i,j}^{2}] is infinite we obtain scaling laws and asymptotic distributions expressed in terms of a continuous lastpassage percolation model on [0,1]. In the last model customers arrive on the nonnegative halfline 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 X_{t}^{(j)}, see position of server j at time t.
