(* nxn GOE matrix, with a semicircular eigenvalue density *)

n=1000;Histogram[RandomVariate[MatrixPropertyDistribution[Eigenvalues[x],x\[Distributed]GaussianOrthogonalMatrixDistribution[((Sqrt[2])/(Sqrt[n])),n]]]]

(* generate a random GOE matrix, compute eigenvalues lambda and eigenvectors h, and find the estimated value of tn=1/(4*(lambda2-lambda1) *)

n=1000;m=RandomVariate[GaussianOrthogonalMatrixDistribution[((Sqrt[2])/(Sqrt[n])),n]];{eval,evec}=Eigensystem[m];h=evec[[All,1]];imin=Ordering[eval,1][[1]];lambda2minlambda1=Sort[eval][[2]]-Sort[eval][[1]];tn=1/(4*lambda2minlambda1);

(* plot the function H1(t); the thin vertical line at tn is the time scale for the approach to the asymptote at 1 *)

Plot[Abs[h[[imin]]]*Exp[-2*eval[[imin]]*t]/Sqrt[Sum[h[[i]]^2*Exp[-4*eval[[i]]*t],{i,1,n}]],{t,0,10*tn},GridLines->{{tn},{}},PlotRange->Full]