  This paper quantifies the asymptotic order of the largest singular value of a
centered random matrix built from the path of a Block Markov Chain (BMC). In a
BMC there are $n$ labeled states, each state is associated to one of $K$
clusters, and the probability of a jump depends only on the clusters of the
origin and destination. Given a path $X_0, X_1, \ldots, X_{T_n}$ started from
equilibrium, we construct a random matrix $\hat{N}$ that records the number of
transitions between each pair of states. We prove that if $\omega(n) = T_n =
o(n^2)$, then $\| \hat{N} - \mathbb{E}[\hat{N}] \| =
\Omega_{\mathbb{P}}(\sqrt{T_n/n})$. We also prove that if $T_n = \Omega(n
\ln{n})$, then $\| \hat{N} - \mathbb{E}[\hat{N}] \| =
O_{\mathbb{P}}(\sqrt{T_n/n})$ as $n \to \infty$; and if $T_n = \omega(n)$, a
sparser regime, then $\| \hat{N}_\Gamma - \mathbb{E}[\hat{N}] \| =
O_{\mathbb{P}}(\sqrt{T_n/n})$. Here, $\hat{N}_{\Gamma}$ is a regularization
that zeroes out entries corresponding to jumps to and from most-often visited
states. Together this establishes that the order is
$\Theta_{\mathbb{P}}(\sqrt{T_n/n})$ for BMCs.
