An excited random walk is a non-Markovian extension of the simple random walk, in which the walk’s behavior at time n is impacted by the path it has taken up to time n. The properties of an excited random walk are more difficult to investigate than those of a simple random walk. For example, the limiting speed of an excited random walk is either zero or unknown depending on its initial conditions. While its limiting speed is unknown in most cases, the qualitative behavior of an excited random walk is largely determined by a parameter δ which can be computed explicitly. Despite this, it is known that the limiting speed cannot be written as a function of δ. We offer a new proof of this fact, and use techniques from this proof to further investigate the relationship between δ and limiting speed. We also generalize the standard excited random walk by introducing a “bias” to the right, and call this generalization an excited asymmetric random walk. Under certain initial conditions we are able to compute an explicit formula for the limiting speed of an excited asymmetric random walk.

Author Bio

Mike Cinkoske is a sophomore at Purdue University majoring in Mathematics and Computer Science. In his free time, he enjoys biking and playing piano.

Joseph Jackson is a senior at Swarthmore College studying mathematics and economics. He enjoys playing sports, especially baseball, basketball, and Ultimate Frisbee. He is planning to pursue graduate school in mathematics after graduation.

Claire Plunkett is a fourth year Applied Mathematics major at Case Western Reserve University, where she expects to earn a B.S. in May 2018. Following her undergraduate degree, she hopes to pursue a PhD. in mathematics. Her research interests include mathematical biology and probability, and she is a member of the CWRU varsity Cross Country and Track & Field teams.