All Random with this Matrix

On our way from billiards to quantitative finance we stopped at quantum chaos in last "Physics Friday" blog post. Without going to deep into details we mentioned the term random matrix - a term some of the  readers may not be too familiar with - so perhaps, as they have a direct application in quant finance, we will go a bit more into detail today.
As the name implies, a random matrix is a matrix with random entries. For such a matrix several question can be asked:
  • What is the limiting shape of the histogram of the distribution of eigenvalues when the size of the matrix becomes large?
  • How does the distribution of spacings between eigenvalues look like and what is the limiting shape if the size of the matrix becomes large?
  • Is there universality of these shapes with respect to small changes in the distribution of the matrix entries?
  • Are the eigenvectors localized or are there no dominant components?
Different approaches to answer these questions exist - a few of them are mentioned in the following:
  • Method of traces (combinatorial)
  • Stieltjes transform method
  • Orthogonal polynomials 
  • Stochastic differential equations

The figure shows a histogram of eigenvalues from a random real symmetric matrix (N=3000). Each matrix element is a random number and these numbers are Gaussian distributed with a mean of zero and a standard deviation of one (except the diagonal elements which have a standard deviation of square root of 2). The distribution thins out toward the edges as it should as the theoretical density of eigenvalues is known as Wigner's semicircle law (because the curve takes the shape of a semicircle centered at zero with radius Sqrt[2N],where N is the dimension of the random matrix (and hence the number of eigenvalues).

Next week we will see how random matrices are connected to quant finance.