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I think there are still some more mistakes in this post and in the corresponding chapter of “topics in RMT”:

1): Using (34), I found (35) should be

when n is odd. Comparing with (35), the sign is different (the power of -1 should be (n-1)/2).

2) The rescaled functions should be defined as which are orthonormal. The scale outside of the functions do not make them orthonormal.

Then the following 3), 4), 5) are related to this.

3) In the book, the harmonic oscillator operator on should be instead of .

4) In eq.(36), the coefficient of should be .

5) In the integral formula of (in the sketch of steepest descent), the factor is missing.

6) In Exercise 12, the density of should be

*[Corrected, thanks – T.]*

In Exercise 4 should there be a factor of 2 before the logarithmic potential? Otherwise taking the Laplacian we would get $\rho(z)=2/\pi$ on its support.

*[Corrected, thanks – T.]*

Oops, there was a factor of missing in front of . One can indeed proceed using BCH: the fact that Haar measure is invariant under rotations of the form for some skew-adjoint with zero diagonal means that the distribution of S is invariant under a transformation which looks like translation by plus errors of size (and whose derivatives also have size o(1)), so after taking Jacobians we see that the density of this distribution is constant up to factors of .

]]>Right after making the ansatz {U=\exp(\epsilon S)R} you say

As {U} is distributed using Haar measure on {U(n)}, {S} is (locally) distributed using {\epsilon^{n^2-n}} times a constant multiple of Lebesgue measure on the space {W} of skew-adjoint matrices with zero diagonal, which has dimension {n^2-n}.

Is there any refference to this step in details? Do i need to use the Baker–Campbell–Hausdorff formula?

]]>I very much like this post as always.

Minor typos perhaps:

The first paragraph below Eq. 11 the I think instead of 1/|\lambda_i – \lambda_j|, it should say \log 1/|\lambda_i – \lambda_j|.

Also three lines below “\lambda_i should be have…” to “\lambda_i should have…”

*[Corrected, thanks – T.]*

On the 5th line, in the paragraph above Eq. 2 starting with ” Let D be…”, M should probably be M_n.

*[Corrected, thanks -T.]*

The points that maximise the pdf are known as Fekete points; asymptotics for these points are known (by using a more rigorous version of the mean field approximation), but I doubt that there is any closed form for them.

]]>Does the fact that you used a heuristic mean field approximation imply that in general, you can’t find the actual eigenvalues that maximize the pdf for n > 4?

Thanks ]]>