  Recently a matrix model with non-pairwise index contractions has been studied
in the context of the canonical tensor model, a tensor model for quantum
gravity in the canonical formalism. This matrix model also appears in the same
form with different ranges of parameters and variables, when the replica trick
is applied to the spherical $p$-spin model ($p=3$) in spin glass theory.
Previous studies of this matrix model suggested the presence of a continuous
phase transition around $R\sim N^2/2$, where $N$ and $R$ designate its matrix
size $N\times R$. This relation between $N$ and $R$ intriguingly agrees with a
consistency condition of the tensor model in the leading order of $N$,
suggesting that the tensor model is located near or on the continuous phase
transition point and therefore its continuum limit is automatically taken in
the $N\rightarrow \infty$ limit. In the previous work, however, the evidence
for the phase transition was not satisfactory due to the slowdown of the Monte
Carlo simulations. In this work, we provide a new setup for Monte Carlo
simulations by integrating out the radial direction of the matrix. This new
strategy considerably improves the efficiency, and allows us to clearly show
the existence of the phase transition. We also present various characteristics
of the phases, such as dynamically generated dimensions of configurations,
cascade symmetry breaking, and a parameter zero limit, to discuss some
implications to the canonical tensor model.
