Dimitri Lozeve
e4aab38353
https://javran.github.io/posts/2014-03-01-add-tags-to-your-hakyll-blog.html
61 lines
2.3 KiB
Org Mode
61 lines
2.3 KiB
Org Mode
---
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title: "Random matrices from the Ginibre ensemble"
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date: 2019-03-20
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tags: maths, linear algebra, statistics, julia, visualization
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---
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** Ginibre ensemble and its properties
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The /Ginibre ensemble/ is a set of random matrices with the entries
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chosen independently. Each entry of a $n \times n$ matrix is a complex
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number, with both the real and imaginary part sampled from a normal
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distribution of mean zero and variance $1/2n$.
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Random matrices distributions are very complex and are a very
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active subject of research. I stumbled on this example while
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reading an article in /Notices of the AMS/ by Brian C. Hall [[ref-1][(1)]].
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Now what is interesting about these random matrices is the
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distribution of their $n$ eigenvalues in the complex plane.
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The [[https://en.wikipedia.org/wiki/Circular_law][circular law]] (first established by Jean Ginibre in 1965 [[ref-2][(2)]])
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states that when $n$ is large, with high probability, almost all
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the eigenvalues lie in the unit disk. Moreover, they tend to be
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nearly uniformly distributed there.
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I find this mildly fascinating that such a straightforward definition
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of a random matrix can exhibit such non-random properties in their
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spectrum.
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** Simulation
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I ran a quick simulation, thanks to [[https://julialang.org/][Julia]]'s great ecosystem for linear
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algebra and statistical distributions:
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#+begin_src julia
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using LinearAlgebra
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using UnicodePlots
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function ginibre(n)
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return randn((n, n)) * sqrt(1/2n) + im * randn((n, n)) * sqrt(1/2n)
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end
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v = eigvals(ginibre(2000))
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scatterplot(real(v), imag(v), xlim=[-1.5,1.5], ylim=[-1.5,1.5])
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#+end_src
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I like using =UnicodePlots= for this kind of quick-and-dirty plots,
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directly in the terminal. Here is the output:
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[[../images/ginibre.png]]
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** References
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1. <<ref-1>>Hall, Brian C. 2019. "Eigenvalues of Random Matrices in
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the General Linear Group in the Large-$N$ Limit." /Notices of the
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American Mathematical Society/ 66, no. 4 (Spring):
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568-569. https://www.ams.org/journals/notices/201904/201904FullIssue.pdf
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2. <<ref-2>>Ginibre, Jean. "Statistical ensembles of complex,
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quaternion, and real matrices." Journal of Mathematical Physics 6.3
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(1965): 440-449. https://doi.org/10.1063/1.1704292
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