Purity decay rate in random circuits with different configurations of gates

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