Purity decay rate in random circuits with different configurations of gates

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 $\sim {\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}^t$, with ${\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ 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 ${\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ depends on the location of bipartition boundaries as well as on the geometry of applied gates.