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

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We study numerically various wave functions in a gauged matrix quantum mechanics of six commuting hermitian N× 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 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/N² 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.