  Understanding thermalisation in quantum many-body systems is among the most
enduring problems in modern physics. A particularly interesting question
concerns the role played by quantum mechanics in this process, i.e. whether
thermalisation in quantum many-body systems is fundamentally different from
that in classical many-body systems and, if so, which of its features are
genuinely quantum. Here we study this question in minimally structured
many-body systems which are only constrained to have local interactions, i.e.
local random circuits. We introduce a class of random permutation circuits
(RPCs), where the gates locally permute basis states modelling generic
microscopic classical dynamics, and compare them to random unitary circuits
(RUCs), a standard toy model for generic quantum dynamics. We show that, like
RUCs, RPCs permit the analytical computation of several key quantities such as
out-of-time order correlators (OTOCs), or entanglement entropies. RPCs can be
interpreted both as quantum or classical dynamics, which we use to find
similarities and differences between the two. Performing the average over all
random circuits, we discover a series of exact relations, connecting quantities
in RUC and (quantum) RPCs. In the classical setting, we obtain similar exact
results relating (quantum) purity to (classical) growth of mutual information
and (quantum) OTOCs to (classical) decorrelators. Our results indicate that
despite of the fundamental differences between quantum and classical systems,
their dynamics exhibits qualitatively similar behaviours.
