The shape of emergent quantum geometry from an N=4 SYM minisuperspace approximation

We study numerically various wave functions in a gauged matrix quantum mechanics of six commuting hermitian $N\times N$ matrices. Our simulations span ranges of $N$ up to 10000. This system is a truncated and quenched version of N=4 SYM that serves as a minisuperspace approximation to the full ${\cal N}=4$ SYM system. This setup encodes aspects of the geometry of the AdS dual in terms of joint eigenvalue distributions for the matrices in the large $N$ limit. We analyze the problem of determining geometric measurements from these fluctuating distributions at finite $N$ and how fast they approach to the large N limit. We treat this eigenvalue geometry information as a proxy for geometric calculations in quantum gravity in a description where gravity is an emergent phenomenon. Our results show that care is needed in choosing the observables that measure the geometry: different choices of observables give different answers, have different size fluctuations at finite $N$ and they converge at different rates to the large $N$ limit. We find that some natural choices of observables are pathological at finite $N$ for $N$ sufficiently small. Finally, we note that the approach to the large $N$ limit does not seem to follow the expected convergence in powers of $1/N^2$ from planar diagram arguments. Our evidence suggests that different powers of $N$ appear, but convergence to large $N$ is rather slow so the values of $N$ we have explored might be too small to conclude this unambiguously.