We report a direct and robust calculation, free from ergodic problems, of the nonuniform-to-uniform (stripe) transition line of noncommutative Φ 4 2 by means of an exact Metropolis algorithm applied to the first non-trivial multitrace correction of this theory on the fuzzy sphere. In fact, we reconstruct the entire phase diagram including the Ising, matrix and stripe boundaries together with the triple point. We also report that the measured critical exponents of the Ising transition line agrees with the Onsager values in two dimensions. The triple point is identified as a termination point of the one-cut-totwo-cut transition line and is located at (b,c) = (−1.55, 0.4) which compares favorably with previous Monte Carlo estimate. *

A novel scenario for the emergence of geometry in random multitrace matrix models of a single hermitian matrix $M$ with unitary $U(N) $ invariance, i.e. without a kinetic term, is presented. In particular, the dimension of the emergent geometry is determined from the critical exponents of the disorder-to-uniform-ordered transition whereas the metric is determined from the Wigner semicircle law behavior of the eigenvalues distribution of the matrix $M$. If the uniform ordered phase is not sustained in the phase diagram then there is no emergent geometry in the multitrace matrix model.Comment: 18 pages, 7 figures (16 graphs), 1 tabl

A detailed Monte Carlo calculation of the phase diagram of bosonic IKKT Yang-Mills matrix models in three and six dimensions with quartic mass deformations is given. Background emergent fuzzy geometries in two and four dimensions are observed with a fluctuation given by a noncommutative U (1) gauge theory very weakly coupled to normal scalar fields. The geometry, which is determined dynamically, is given by the fuzzy spheres S 2 N and S 2 N ×S 2 N respectively. The three and six matrix models are in the same universality class with some differences. For example, in two dimensions the geometry is completely stable, whereas in four dimensions the geometry is stable only in the limit M −→ ∞, where M is the mass of the normal fluctuations. The behavior of the eigenvalue distribution in the two theories is also different. We also sketch how we can obtain a stable fuzzy foursphere S 2 N × S 2 N in the large N limit for all values of M as well as models of topology change in which the transition between spheres of different dimensions is observed. The stable fuzzy spheres in two and four dimensions act precisely as regulators which is the original goal of fuzzy geometry and fuzzy physics. Fuzzy physics and fuzzy field theory on these spaces are briefly discussed.

We study a six matrix model with global SO(3) × SO(3) symmetry containing at most quartic powers of the matrices. This theory exhibits a phase transition from a geometrical phase at low temperature to a Yang-Mills matrix phase with no background geometrical structure at high temperature. This is an exotic phase transition in the same universality class as the three matrix model but with important differences. The geometrical phase is determined dynamically, as the system cools, and is given by a fuzzy four-sphere background S 2 N × S 2 N , with an Abelian gauge field which is very weakly coupled to two normal scalar fields playing the role of dark energy.

We present a new model of quantum gravity as a theory of random geometries given explicitly in terms of a multitrace matrix model. This is a generalization of the usual discretized random surfaces of 2D quantum gravity which works away from two dimensions and captures a large class of spaces admiting a finite spectral triple. These multitrace matrix models sustain emergent geometry as well as growing dimensions and topology change.