  In this article the geometry of quantum gravity is quantized in the sense of
being noncommutative (first quantization) but it is also quantized in the sense
of being emergent (second quantization). A new mechanism for quantum geometry
is proposed in which noncommutative geometry can emerge from "one-matrix
multitrace scalar matrix models" by probing the statistical physics of
commutative phases of matter. This is in contrast to the usual mechanism in
which noncommutative geometry emerges from "many-matrix singletrace Yang-Mills
matrix models" by probing the statistical physics of noncommutative phases of
gauge theory. In this novel scenario quantized geometry emerges in the form of
a transition between the two phase diagrams of the real quartic matrix model
and the noncommutative scalar phi-four field theory. More precisely, emergence
of the geometry is identified here with the emergence of the uniform-ordered
phase and the corresponding commutative (Ising) and noncommutative (stripe)
coexistence lines. The critical exponents and the Wigner's semicircle law are
used to determine the dimension and the metric respectively. Arguments from the
saddle point equation, from Monte Carlo simulation and from the matrix
renormalization group equation are provided in support of this scenario.
