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We introduce an exact mapping of Clifford (stabilizer) random tensor networks(RTNs) and monitored quantum circuits, onto a statistical mechanics model. WithHaar unitaries, the fundamental degrees of freedom ('spins') are permutationsbecause all operators commuting with the action of the unitaries on a tensorproduct arise from permutations of the tensor factors ('Schur-Weyl duality').For unitaries restricted to the smaller Clifford group, the set of commutingoperators, the 'commutant', forming the new 'spin' degrees of freedom, will belarger. We use the recent full characterization of this commutant by Gross etal., Comm. Math. Phys. 385, 1325 (2021), to construct the Clifford statisticalmechanics models for on-site Hilbert space dimensions which are powers of aprime number $p$. We show that the Boltzmann weights are invariant under asymmetry group involving orthogonal matrices with entries in the finite numberfield ${\bf F}_p$. This implies that the symmetry group, and consequently alluniversal properties of entanglement transitions in Clifford circuits and RTNswill in general depend on, and only on the prime $p$. We show that Cliffordmonitored circuits with on-site Hilbert space dimension $d=p^M$ are describedby percolation in the limits $d \to \infty$ at (a) $p=$ fixed but $M\to\infty$, and at (b) $M= 1$ but $p \to \infty$. In the limit (a) we calculatethe effective central charge, and in the limit (b) we derive the followinguniversal minimal cut entanglement entropy $S_A =(\sqrt{3}/\pi)\ln p \ln L_A$for $d=p$ large at the transition. We verify those predictions numerically, andpresent extensive numerical results for critical exponents at the transition inmonitored Clifford circuits for prime number on-site Hilbert space dimension$d=p$ for a variety of different values of $p$, and find that they approachpercolation values at large $p$.

Statistical Mechanics Model for Clifford Random Tensor Networks and Monitored Quantum Circuits