The random walk among Bernoulli obstacles model describes a system in which particles move randomly in a space containing random traps. More precisely, obstacles are placed independently at sites in $\Z^d$ ($d \geq 2$) with probability $p \in (0,1)$, and the random walk is killed if it hits one of these obstacles. Of interest to this study is the random walk's behavior, conditional on the event that it survives for a long time. The most prominent feature of the model is a strong localization effect: the conditioned random walk will be localized in a tiny region. This is closely related to the so-called Anderson localization studied in condensed matter physics. The past 40 years have seen many important contributions to explicating the localization phenomenon, notably Donsker and Varadhan's large deviation results and Sznitman's method of enlargement of obstacles. However, the understanding of the random walk's path behavior remains incomplete. This is because, in part, the required analysis usually has to reach a level far beyond the large deviation results. This thesis investigates the behavior of the random walk path in the obstacle model. The contents are based on joint works with Jian Ding, Ryoki Fukushima, and Rongfeng Sun. Under the quenched law, we will show that the random walk conditional on survival up to time $N$ will first rush to a small ball that is free of obstacles and of volume asymptotically $d\log_{1/p}N$, and then be localized there until time $N$. Under the annealed law, it was known that for any $d \geq 2$, the random walk range is contained in a ball of radius $CN^{\frac{1}{d+2}}$, and for $d = 2$ it also contains a ball of asymptotically the same radius. We will show that the latter is also true for $d \geq 3$, and we give a bound for the boundary size of the random walk range. Under the annealed law with bias, the model undergoes a phase transition from the sub-ballistic regime to the ballistic regime, depending on the size of the bias. We show the following description of the behavior of the random walk in the sub-ballistic regime: the random walk is contained in a ball of radius $CN^{\frac{1}{d+2}}$, and the endpoint lies near a fixed point on the boundary of this ball.




Downloads Statistics

Download Full History