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My friends working on Thermalization of Black Holes explained solutions to their matrix-valued differential equations (from numerical implementation of the Berenstein-Maldacena-Nastase matrix model) result in chaotic solutions. They are literally getting random matrices. For the eigenvalue spectrum, would expect a semicircle distribution but for finite N get something slightly different.


The proof of the Wigner Semicircle Law comes from studying the GUE Kernel \[ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} \] The eigenvalue density comes from setting $\mu = \nu$. The Wigner semicircle identity is a Hermite polynomial identity \[ \rho(\lambda)=e^{-\mu^2} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)^2}{2^j j!} \approx \left\{\begin{array}{cc} \frac{\sqrt{2N}}{\pi} \sqrt{1 - \lambda^2/2N} & \text{if }|\lambda|< 2\sqrt{N} \\ 0 & \text{if }|\lambda| > 2 \sqrt{N} \end{array} \right. \] The asymptotics come from calculus identities like Christoffel-Darboux formula.
For finite size matrices the eigenvalue distribution is a semicircle yet.

Plotting the eigenvalues of a random $4 \times 4$ matrix, the deviations from semicircle law are noticeable with 100,000 trials and 0.05 bin size. GUE is in brown, GUE|trace=0 is in orange.

Axes not scaled, sorry!

Mathematica Code:

num[] := RandomReal[NormalDistribution[0, 1]]
herm[N_] := (h = 
   Table[(num[] + I num[])/Sqrt[2], {i, 1, N}, {j, 1, N}]; (h + 
     Conjugate[Transpose[h]])/2)

n = 4;
trials = 100000;

eigen = {};
Do[eigen = 
   Join[(mat = herm[n]; mat = mat - Tr[mat] IdentityMatrix[n]/n ; 
     Re[Eigenvalues[mat]]), eigen], {k, 1, trials}];
Histogram[eigen, {-5, 5, 0.05}]
BinCounts[eigen, {-5, 5, 0.05}];
a = ListPlot[%, Joined -> True, PlotStyle -> Orange]

eigen = {};
Do[eigen = 
   Join[(mat = herm[n]; mat = mat; Re[Eigenvalues[mat]]), eigen], {k, 
   1, trials}];
Histogram[eigen, {-5, 5, 0.05}]
BinCounts[eigen, {-5, 5, 0.05}];
b = ListPlot[%, Joined -> True, PlotStyle -> Brown]

Show[a, b]


My friend asks if traceless GUE ensemble $H - \frac{1}{N} \mathrm{tr}(H)$ can be analyzed. The charts suggest we should still get a semicircle in the large $N$ limit. For finite $N$, the oscillations (relative to semicircle) are very large. Maybe has something to do with the related harmonic oscillator eigenstates.
The trace is the average eigenvalue & The eigenvalues are being "recentered". We could imagine 4 perfectly centered fermions - they will repel each other. Joint distribution is: \[ e^{-\lambda_1^2 -\lambda_2^2 - \lambda_3^2 - \lambda_4^2} \prod_{1 \leq i,j \leq 4} |\lambda_i - \lambda_j|^2 \] On average, the fermions will sit where the humps are. Their locations should be more pronounced now that their "center of mass" is fixed.

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Interesting. Of course in the large dimension limit one expects no difference. However I am quite surprised to see such a big differences for N=4. Sorry I have no answer for the time being, but I will follow this post. –  user667 Jan 30 '12 at 14:46
    
Dont anybody close or migrate this just because some mathematica code is displayed ... ! –  Dilaton Dec 30 '12 at 17:37
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1 Answer 1

The Christoffel–Darboux formula is not an asymptotic (in the sense of $N$ going to infinity) result, whereas the semicircle is valid for random matrices of infinite size. For finite matrices you obtain the oscillations you've got. To see this check out and plot formula (97) in http://arxiv.org/abs/math-ph/0412017

As to the traceless GUE, I'm no expert but here is sth. I dug out, http://arxiv.org/abs/math/9909104 maybe start there.

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