  We introduce a statistical system on random networks of trivalent vertices
for the purpose of studying the canonical tensor model, which is a rank-three
tensor model in the canonical formalism. The partition function of the
statistical system has a concise expression in terms of integrals, and has the
same symmetries as the kinematical ones of the canonical tensor model. We
consider the simplest non-trivial case of the statistical system corresponding
to the Ising model on random networks, and find that its phase diagram agrees
with what is implied by regrading the Hamiltonian vector field of the canonical
tensor model with N = 2 as a renormalization group flow. Along the way, we
obtain an explicit exact expression of the free energy of the Ising model on
random networks in the thermodynamic limit by the Laplace method. This paper
provides a new example connecting a model of quantum gravity and a random
statistical system.
