Abstracts Mathematics

Pictures, not merely photographs: authenticity, performance and the Hopi in Edward S. Curtis's The North American Indian

by Zachary David Voller

Excited random walks (ERW) or random walks in a cookie environment is a modification of the nearest neighbor simple random walk such that in several first visits to each site of the integer lattice, the walk’s jump kernel gives a preference to a certain direction and assigns equal probabilities to the remaining directions. If the current location of the random walk has been already visited more than a certain number of times, then the walk moves to one of its nearest neighbors with equal probabilities. The model was introduced by Benjamini and Wilson and extended by Martin Zerner. In the cookies jargon, upon first several visits to every site of the lattice, the walker consumes a cookie providing them a boost toward a distinguished direction in the next step. The excited random walk is a popular mainstream model of theoretical probability. An interesting application of this model to the motion of DNA molecular motors has been discovered by Antal and Krapivsky (Phys. Review E, 2007), see also the article of Mark Buchanan Attack of the cyberspider in Nature Physics, 2009. Many basic asymptotic properties of excited random walk have their counterparts for random walk in random environment (RWRE). The major difference between two processes is that while the random (cookie) environment is dynamic and rapidly changes with time the environments considered in the RWRE process are stationary both in space and in time. The similarity between the asymptotic behaviors of these two classes of random walks can be ex- plained using the fact that certain functionals (for instance, exit times and exit probabilities) of the local time (or occupation time, also referred to as the number of previous visits to a current location) process converge after a proper rescaling to diffusion processes with time- independent coefficients. Thus phenomenon, discovered by Kosygina and Mountford, can be exploited for a heuristic explanation of the analogy between the role of the local drift of ERW (bias created by the cookie environment) and a random potential which governs the behavior of RWRE. In this thesis we consider an excited random walk on Z with the jump kernel that depends not only on the number of cookies present at the current location of the walker, but also on direction from which the current location is entered. Random walks with the jump kernel that depends not only on the current location and possibly the history of the random walk at this location but also on the direction where the current location is visited from are usually referred to as persistent random walks. We therefore refer to our model as an persistent random walk in a cookie environment (PRWCE). We prove recurrence and transience criteria and derive a necessary and sufficient condition for the asymptotic speed of the walk to be strictly positive. The law of large number in the transient case is complement by a central limit theorem for the position of the random walk. Surprisingly, it turns out that a transient PRWCE even in one dimension does not necessarily satisfy…