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  random matrix ensembles from BMN model

+ 3 like - 0 dislike

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 + 

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. This post has been migrated from (A51.SE)
asked Jan 29, 2012 in Theoretical Physics by john mangual (310 points) [ no revision ]
retagged Apr 19, 2014 by dimension10
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.

This post has been migrated from (A51.SE)

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