  The dynamics of quantum information in many-body systems with large onsite
Hilbert space dimension admits an enlightening description in terms of
effective statistical mechanics models. Motivated by this fact, we reveal a
connection between three separate models: the classically chaotic $d$-adic
R\'{e}nyi map with stochastic control, a quantum analog of this map for qudits,
and a Potts model on a random graph. The classical model and its quantum analog
share a transition between chaotic and controlled phases, driven by a randomly
applied control map that attempts to order the system. In the quantum model,
the control map necessitates measurements that concurrently drive a phase
transition in the entanglement content of the late-time steady state. To
explore the interplay of the control and entanglement transitions, we derive an
effective Potts model from the quantum model and use it to probe
information-theoretic quantities that witness both transitions. The
entanglement transition is found to be in the bond-percolation universality
class, consistent with other measurement-induced phase transitions, while the
control transition is governed by a classical random walk. These two phase
transitions merge as a function of model parameters, consistent with behavior
observed in previous small-size numerical studies of the quantum model.
