12 potential. The logarithmic repulsions in six dimensions are long range. Because of that, the associated statistical mechanical problem is not standard. If Q = 0 the background potential is confining towards the origin. If Q = 0, the background potential is confining in generic directions, but it is repulsive from the z 1 = 0 locus with a logarithmic repulsion (this is, particles are repelled from the locus x 1 = x 2 = 0). In the gravity theory, we usually take the large N limit to go to the classical limit. In this case, we can take the large N limit of the Boltzmann gas which is a thermodynamic limit of large number of particles. In the thermodynamic limit we should be able to replace individual eigenvalues by a density of eigenvalues on R $^{6}$, given by ρ ( ⃗ x ). The probability density for the eigenvalue distribution is then given by The density is constrained so that ρ ≥ 0 and ∫ d $^{6}$xρ ( x ) = N . As shown in [18], the size of the distribution for Q = 0 is of order √$_{N}$$_{. So we can rescale the coordinates}$ ⃗ x = √$_{N⃗a}$ and the density ˆ ρ ( a ) d $^{6}$a = ρ ( x ) N − $^{1}$d$^{6}$x . In this way, we have that the normalized distribution ˆ ρ satisfies If we do this substitution we find that where q = Q/N . Notice that the extra numerical factors that are obtained from the rescaling of x inside the logarithms give a constant, due to the constraint ∫ ˆ ρ = 1. This can be absorbed in the normalization of p . In this new setup, we see that a limit should be smooth at large N if q is held fixed in the process. Also, the factor of N 2 in front of the exponential term shows that when N → ∞ we should be able to use a saddle point configuration for ˆ ρ with confidence. It is also clear that 1 /N 2 plays a similar role to ℏ in semiclassical physics, where the saddle point equations to a Euclidean path integral involve finding the extrema of exp( − S/ ℏ ). 1 /N 2 can also be interpreted as a temperature, so large N cools the system and suppresses fluctuations. This is exactly what is expected from the match to supergravity [1] and to the planar diagram counting [42]. ∫ ˆ ρ ( a ) d $^{6}$a = 1 (9) p (ˆ ρ ) ∝ exp ( − N 2 ∫ d $^{6}$a ˆ ρ ( a ) [ − ⃗ a 2 + q log( a 2 1 + a 2 $_{2}$) ] + N 2 2 ∫ ∫ d $^{6}$ad$^{6}$b ˆ ρ ( a )ˆ ρ ( b ) log | ⃗ a − ⃗ b | 2 ) (10) p ( ρ ) ∝ exp ( − ∫ d $^{6}$xρ ( x ) [ ⃗ x 2 + Q log( x 2 1 + x 2 $_{2}$) ] + 1 2 ∫ ∫ d $^{6}$xd$^{6}$yρ ( x ) ρ ( y ) log | ⃗ x − ⃗ y | 2 ) (8)