dc.description.abstract 
Fix a discretetime Markov chain $(V,P)$ with finite state space $V$ and transition
matrix $P$. Let $(V_n,P_n)$ be the Markov chain on nblocks induced by $(V,P)$, which
we call the nblock process associated with the base chain $(V,P)$. We study coalescing
random walks on mixing nblock 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.

