In the original work of Esscher and Cramer Large Deviations arise through what is now known as Esscher or Cramer tilt, that changes the underlying distribution so that what was originally a rare event is now a very likely event. The exact tilt that leads to the large deviation allows one to calculate the precise rate of the large deviation. This provides a clue as to what the conditional distribution is, given that a particular rare event has occurred. While the original work is in the context of sums of independent random variables, subsequent work on large deviations has extended this idea considerably. It is the aim of this proposal to investigate this in the context of a random walk in a random environment. In particular if the walk travels with an unlikely velocity, what would it have experienced?
Large deviations deals with estimating probabilities of rare events. Rare events do occur and the theory deals with determining exactly how rare they are. When a rare event occurs it is not isolated. Other rare events happen as well. The same phenomenon that generated the rare event could very well have spawned other rare events. The theory deals with predicting such other rare events. One starts with a world of models as well as an assumption about a specific model. An event with very small probability under this model occurs. This changes one's belief in the particular model and a new model is chosen that is consistent with the rare event. If there are many such models an optimal one is chosen in some way. This model makes predictions, which were perhaps rare under the old model but not any more.
Large deviation theory estiamtes the probability of random events that arerare. Although they are rare, it is useful to understand the most likely cause for it, because that would tell us what other related rare events are likely to occur as well. This is a general theory that has been deceloped over the years with details that need to be woked out in specific contexts. During the grant period the theory was woked out in detail in several instances. One can, by using theory, estiamte the number of graphs with specified densities of a finite collection of finite subgraphs like edges, triangles etc. The occurrence of very large eigen values for a random matrix is another example. These results have been published in journal. Two graduate students, Lingjiong Zhu and Dmytro Karabash who were supported during the year worked on a class of point processes called Hawkws processes. They represent events that when they occur trigger other similar events in the future increasing the probability of their occurrence, such effect however waning over time. These processes have been used to model many natural phenomena. In their doctoral dissertation they established various asymptotic properties for such models. They are published in their theses submitted to New York University.
