  We study purity decay -- a measure of bipartite entanglement -- in a chain of
$n$ qubits under the action of various geometries of nearest-neighbor random
two-site unitary gates. We use a Markov chain description of average purity
evolution, using further reduction to obtain a transfer matrix of only
polynomial dimension in $n$. In most circuits, an exception being the
brick-wall configuration, purity decays to its asymptotic value in two stages:
the initial thermodynamically relevant decay persisting up to extensive times
is $\sim \lambda_{\mathrm{eff}}^t$, with $\lambda_{\mathrm{eff}}$ not
necessarily being in the spectrum of the transfer matrix, while the ultimate
asymptotic decay is given by the second largest eigenvalue $\lambda_2$ of the
transfer matrix. The effective rate $\lambda_{\mathrm{eff}}$ depends on the
location of bipartition boundaries as well as on the geometry of applied gates.
