---
title: "Random matrices from the Ginibre ensemble"
date: 2019-03-20
---

** Ginibre ensemble and its properties

   The /Ginibre ensemble/ is a set of random matrices with the entries
   chosen independently. Each entry of a $n \times n$ matrix is a complex
   number, with both the real and imaginary part sampled from a normal
   distribution of mean zero and variance $1/2n$.

   Random matrices distributions are very complex and are a very
   active subject of research. I stumbled on this example while
   reading an article in /Notices of the AMS/ by Brian C. Hall [[ref-1][(1)]].

   Now what is interesting about these random matrices is the
   distribution of their $n$ eigenvalues in the complex plane.

   The [[https://en.wikipedia.org/wiki/Circular_law][circular law]] (first established by Jean Ginibre in 1965 [[ref-2][(2)]])
   states that when $n$ is large, with high probability, almost all
   the eigenvalues lie in the unit disk. Moreover, they tend to be
   nearly uniformly distributed there.

   I find this mildly fascinating that such a straightforward definition
   of a random matrix can exhibit such non-random properties in their
   spectrum.

** Simulation

   I ran a quick simulation, thanks to [[https://julialang.org/][Julia]]'s great ecosystem for linear
   algebra and statistical distributions:

   #+begin_src julia
     using LinearAlgebra
     using UnicodePlots

     function ginibre(n)
         return randn((n, n)) * sqrt(1/2n) + im * randn((n, n)) * sqrt(1/2n)
     end

     v = eigvals(ginibre(2000))

     scatterplot(real(v), imag(v), xlim=[-1.5,1.5], ylim=[-1.5,1.5])
   #+end_src

   I like using =UnicodePlots= for this kind of quick-and-dirty plots,
   directly in the terminal. Here is the output:

   [[../images/ginibre.png]]

** References

   1. <<ref-1>>Hall, Brian C. 2019. "Eigenvalues of Random Matrices in
      the General Linear Group in the Large-$N$ Limit." /Notices of the
      American Mathematical Society/ 66, no. 4 (Spring):
      568-569. https://www.ams.org/journals/notices/201904/201904FullIssue.pdf
   2. <<ref-2>>Ginibre, Jean. "Statistical ensembles of complex,
      quaternion, and real matrices." Journal of Mathematical Physics 6.3
      (1965): 440-449. https://doi.org/10.1063/1.1704292