  We consider universal aspects of two problems: (i) the slow purification of a
large number of qubits by repeated quantum measurements, and (ii) the singular
value structure of a product ${m_t m_{t-1}\ldots m_1}$ of many large random
matrices. Each kind of process is associated with the decay of natural measures
of entropy as a function of time or of the number of matrices in the product.
We argue that, for a broad class of models, each process is described by
universal scaling forms for purification, and that (i) and (ii) represent
distinct ``universality classes'' with distinct scaling functions. Using the
replica trick, these universality classes correspond to one-dimensional
effective statistical mechanics models for a gas of ``kinks'', representing
domain walls between elements of the permutation group. (This is an instructive
low-dimensional limit of the effective statistical mechanics models for random
circuits and tensor networks.) These results apply to long-time purification in
spatially local monitored circuit models on the entangled side of the
measurement phase transition.
