The shape of emergent quantum geometry from an N = 4 SYM minisuperspace approximation David Berenstein $^{†}$, Yuichiro Nakada † , ‡ $^{†}$Department of Physics, University of California at Santa Barbara, CA 93106 $^{‡}$Institute for Cosmic Ray Research University of Tokyo, Kashiwa, Chiba 277-8582 JAPAN We study numerically various wave functions in a gauged matrix quantum me- chanics 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 de- termining 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 observ- ables 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 differ- ent 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. PACS numbers: 11.25.Tq, 11.25.Sq ,02.70.Uu, 11.15.Pg Keywords: Emergent geometry, AdS/CFT arXiv:1001.4509v2 [hep-th] 3 Mar 2010