File:Wigner Semi Circle Law (3000x3000 Gaussian Matrix).JPG

The code is normalized to :${\displaystyle f(x)={\frac {1}{2\pi }}\mu (x)={\sqrt {4-x^{2}}}}$
The real graph is very flat semi-circle due to 2π factor without this normalization.

Octave code

sc(3000,64);

function sc(n,d)
# Wigner semicircle law
# p(x) = 1/(2*pi) x sqrt(4-x^2)
# 
X=randn(n,n);
M=(X+transpose(X))/sqrt(2);
E=eig(full(M));
#--------------------------------
# number of histgram = d
# histgram: delta    = 4/d
#--------------------------------
[y,x]=hist(E,d);
#----------------------
# Normalization
# x : 1/sqrt(n)
# y : 2*pi/(n*4/d)
#----------------------
nx=1/sqrt(n);
ny=2*3.1415*d/(4*n);
bar(x*nx,y*ny,"facecolor","b");
axis([-2.5,2.5,0,2.5],"equal");
end