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
KN(μ,ν)=e−12(μ2+ν2)⋅1√πN−1∑j=0Hj(λ)Hj(μ)2jj!
The eigenvalue density comes from setting
μ=ν. The Wigner semicircle identity is a Hermite polynomial identity
ρ(λ)=e−μ2⋅1√πN−1∑j=0Hj(λ)22jj!≈{√2Nπ√1−λ2/2Nif |λ|<2√N0if |λ|>2√N
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 4×4 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−1Ntr(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−λ21−λ22−λ23−λ24∏1≤i,j≤4|λi−λj|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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