Displacement of biased random walk in a one-dimensional percolation model

• Main Author: Lübbers, Jan-Erik
• Format: Others
• Language: en
• Published: 2019
• Online Access: https://tuprints.ulb.tu-darmstadt.de/8537/1/Displacement%20of%20biased%20random%20walk%20in%20a%20one-dimensional%20percolation%20model.pdf; Lübbers, Jan-Erik <http://tuprints.ulb.tu-darmstadt.de/view/person/L=FCbbers=3AJan-Erik=3A=3A.html> (2019): Displacement of biased random walk in a one-dimensional percolation model.Darmstadt, Technische Universität, [Ph.D. Thesis]

Suppose an ant is placed in a randomly generated, infinite maze. Having no orientation whatsoever, it starts to move along according to a nearest neighbour random walk. Now furthermore, suppose the maze is slightly tilted, such that the ant makes a step along the slope with higher probability than in the opposite direction. Tracking the ant's position, we are interested in the long-term behaviour of the corresponding random walk. We study this model in the context that the maze is given by a one-dimensional percolation cluster. Depending on the bias parameter of the walk, its linear speed converges almost surely towards a deterministic value. This limit exhibits a phase transition from positive value to zero at a critical value of the bias. We investigate the typical order of fluctuations of the walk around its linear speed in the ballistic speed regime, and the order of displacement from the origin in the critical and subballistic speed regimes. Additionally, we show a law of iterated logarithm in the subdiffusive speed regime.