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

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