Date of Award

2014

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mathematics

First Advisor

Robin A. Pemantle

Abstract

The unifying objective of this thesis is to find the mixing time of the Markov chain on $S_n$ formed by applying a random $n$-cycle to a deck of $n$ cards and following with repeated random transpositions. This process can be viewed as a Markov chain on the partitions of $n$ that starts at $(n)$, making it a natural counterpart to the random transposition walk, which starts at $(1^n)$. By considering the Fourier transform of the increment distribution on the group representations of $S_n$ and then computing the characters of the representations, Diaconis and Shahshahani showed in \cite{DS81} that the order of mixing for the random transposition walks is $n\ln n$. We adapt this approach to find an upper bound for the mixing time of the $n$-cycle-to-transpositions shuffle. To obtain a lower bound, we derive the distribution of the number of fixed points for the chain using the method of moments. In the process, we give a nice closed-form formula for the irreducible representation decomposition of tensor powers of the defining representation of $S_n$. Along the way, we also look at the more general $m$-cycle-to-transpositions chain ($m \le n$) and give an upper bound for the mixing time of the $m=n-1$ case as well as characterize the expected number of fixed points in the general case where $m$ is an arbitrary function of $n$.

Included in

Mathematics Commons

Share

COinS