Lee Gibson

Lee Gibson
Ph.D. (2005) Cornell University

First Position

Assistant professor at University of Louisville

Dissertation

The Number of Sites Visited By a Random Walk on an Infinite Graph

Advisor:

Laurent Saloff-Coste

Research Area:
Probability Theory, Random Walks on Groups

Abstract: The research contained in this dissertation develops the most specific knowledge to date regarding the logarithmic asymptotic behavior of the negative exponential moments of the number of sites visited by a random walk on an infinite graph.  Due to the relationship between this quantity and the survival time of a random walk among i.i.d. Bernoulli traps at the vertices of the graph, the accomplishment is achieved by first adapting and then applying Sznitman’s method of enlargement of obstacles to the graph context.  For Cayley graphs of finitely generated groups with polynomial volume growth, it is shown that the upper and lower limits of the logarithmic quantity are bounded above and below, respectively, by the upper and lower limits of a variational formula on sets in the graph which balances the volume of the set and the least non-zero Dirichlet eigenvalue of the generator of the random walk restricted to the set.  It is known that for the simple random walk on \mathbb{Z}^d, these upper and lower limits match to yield a precise limit, and progress is reported here regarding the possibility that these limits will also match for the Heisenberg group and others of similar nature.