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Physical states in the canonical tensor model from the perspective of random tensor networks

Tensor models, generalization of matrix models, are studied aiming for quantum gravity in dimensions larger than two. Among them, the canonical tensor model is formulated as a totally constrained system with first-class constraints, the algebra of which resembles the Dirac algebra of general relativity. When quantized, the physical states are defined to be vanished by the quantized constraints. In explicit representations, the constraint equations are a set of partial differential equations for the physical wave-functions, which do not seem straightforward to be solved due to their non-linear character. In this paper, after providing some explicit solutions for $N=2,3$, we show that certain scale-free integration of partition functions of statistical systems on random networks (or random tensor networks more generally) provides a series of solutions for general $N$. Then, by generalizing this form, we also obtain various solutions for general $N$. Moreover, we show that the solutions for the cases with a cosmological constant can be obtained from those with no cosmological constant for increased $N$. This would imply the interesting possibility that a cosmological constant can always be absorbed into the dynamics and is not an input parameter in the canonical tensor model. We also observe the possibility of symmetry enhancement in $N=3$, and comment on an extension of Airy function related to the solutions.