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Coalescing random walks on n-block Markov chains

dc.contributor.author Lan, Kathleen
dc.date.accessioned 2014-01-11T23:01:30Z
dc.date.available 2014-01-11T23:01:30Z
dc.date.issued 2014-01-11
dc.identifier.uri http://hdl.handle.net/10161/8309
dc.description.abstract Fix a discrete-time Markov chain $(V,P)$ with finite state space $V$ and transition matrix $P$. Let $(V_n,P_n)$ be the Markov chain on n-blocks induced by $(V,P)$, which we call the n-block process associated with the base chain $(V,P)$. We study coalescing random walks on mixing n-block Markov chains in discrete time. In particular, we are interested in understanding the asymptotic behavior of $\mathbb{E} C_n$, the expected coalescence time for $(V_n,P_n)$, as $n\to\infty$. Define the quantity $L=-\log\lambda$, where $\lambda$ is the Perron eigenvalue of the matrix $Q$ that has entries $Q_{i,j}=P_{i,j}^2$. We prove the existence of four limits and show that all of them are equal to $L$: $\lim\limits_{n\to\infty}\frac{1}{n}\log\mathbb{E} C_n$, $\lim\limits_{n\to\infty}\frac{1}{n}\log m_n^*$, $\lim\limits_{n\to\infty} \frac{1}{n}\log \bar{m}_n$, and $\lim\limits_{n\to\infty} -\frac{1}{n}\log\Delta_n$, where $m_n^*$ and $\bar{m}_n$ are the maximum and average meeting times for $(V_n, P_n)$ respectively. We establish the inequalities $0<L\leq h$, where $h$ is the entropy of $P$, and show that $L=h$ iff $P$ is a measure of maximal entropy. The formulas and bounds for $L$ provide a complete characterization of $\mathbb{E} C_n$ on the exponential scale.
dc.language.iso en_US
dc.subject Markov chains
dc.subject coalescing random walk
dc.subject coalescence time
dc.subject entropy
dc.subject thermodynamic formalism
dc.title Coalescing random walks on n-block Markov chains
dc.type Honors thesis
dc.department Mathematics


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