Universality in RMT

Meet a random matrix...

n=3;MatrixForm @ RandomVariate[GaussianOrthogonalMatrixDistribution[n]]

...and its eigenvalues

Sort[Eigenvalues[%]]

Histograms of eigenvalues as N increases

Table[Histogram[RandomVariate[MatrixPropertyDistribution[Eigenvalues[x],x\[Distributed]GaussianOrthogonalMatrixDistribution[n]]],Floor[Sqrt[n]]],{n,{50,300,2000}}]

Histograms of rescaled eigenvalues as N increases

Table[Histogram[RandomVariate[MatrixPropertyDistribution[Eigenvalues[x],x\[Distributed]GaussianOrthogonalMatrixDistribution[1/Sqrt[n],n]]],Floor[Sqrt[n]],"ProbabilityDensity"],{n,{50,300,2000}}]

Appearence of Wigner’s semicircle law at large N

Show[Histogram[RandomVariate[MatrixPropertyDistribution[Eigenvalues[x],x\[Distributed]GaussianOrthogonalMatrixDistribution[1/Sqrt[2000],2000]]],Floor[Sqrt[2000]],"ProbabilityDensity"],Plot[1/Pi Sqrt[2-x^2],{x,-2,2}]]