  We study unitary evolution of bipartite entanglement in a circuit with
nearest-neighbor random gates. Deriving a compact non-unitary description of
purity dynamics on qudits we find a sudden transition in the purity relaxation
rate the origin of which is in the underlying boundary localized eigenmodes --
the skin effect. We provide the full solution of the problem, being one of the
simplest iterations of two-site matrices, namely, that each is a sum of only
two projectors. This leads to rich dynamics influenced by the Jordan normal
form of the kernel and, most importantly, a spectrum that is completely
discontinuous in the thermodynamic limit. It provides a simple example of how a
seemingly innocuous many-body unitary evolution can harbor interesting
mathematical effects: an effective non-symmetric Toeplitz transfer matrix
description causes a phantom relaxation, such that the correct relaxation rate
is not given by the matrix spectrum, but rather by its pseudospectrum.
