Abstract
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 ∼ λ_eff^t, with λ_eff not necessarily being in the spectrum of the transfer matrix, while the ultimate asymptotic decay is given by the second largest eigenvalue λ₂ of the transfer matrix. The effective rate λ_eff depends on the location of bipartition boundaries as well as on the geometry of applied gates.