  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.
