86 lines
5.5 KiB
HTML
86 lines
5.5 KiB
HTML
<!doctype html>
|
||
<html lang="en">
|
||
<head>
|
||
<meta charset="utf-8">
|
||
<meta http-equiv="x-ua-compatible" content="ie=edge">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1, user-scalable=yes">
|
||
<meta name="description" content="Dimitri Lozeve's blog: Random matrices from the Ginibre ensemble">
|
||
|
||
<title>Dimitri Lozeve - Random matrices from the Ginibre ensemble</title>
|
||
<link rel="stylesheet" href="../css/tufte.css" />
|
||
<link rel="stylesheet" href="../css/pandoc.css" />
|
||
<link rel="stylesheet" href="../css/default.css" />
|
||
<link rel="stylesheet" href="../css/syntax.css" />
|
||
|
||
<!-- KaTeX CSS styles -->
|
||
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.11.0/dist/katex.min.css" integrity="sha384-BdGj8xC2eZkQaxoQ8nSLefg4AV4/AwB3Fj+8SUSo7pnKP6Eoy18liIKTPn9oBYNG" crossorigin="anonymous">
|
||
|
||
<!-- The loading of KaTeX is deferred to speed up page rendering -->
|
||
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.11.0/dist/katex.min.js" integrity="sha384-JiKN5O8x9Hhs/UE5cT5AAJqieYlOZbGT3CHws/y97o3ty4R7/O5poG9F3JoiOYw1" crossorigin="anonymous"></script>
|
||
|
||
<!-- To automatically render math in text elements, include the auto-render extension: -->
|
||
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.11.0/dist/contrib/auto-render.min.js" integrity="sha384-kWPLUVMOks5AQFrykwIup5lo0m3iMkkHrD0uJ4H5cjeGihAutqP0yW0J6dpFiVkI" crossorigin="anonymous" onload="renderMathInElement(document.body);"></script>
|
||
|
||
</head>
|
||
<body>
|
||
<article>
|
||
|
||
<header>
|
||
<nav>
|
||
<a href="../">Home</a>
|
||
<a href="../projects.html">Projects</a>
|
||
<a href="../archive.html">Archive</a>
|
||
<a href="../contact.html">Contact</a>
|
||
</nav>
|
||
|
||
<h1 class="title">Random matrices from the Ginibre ensemble</h1>
|
||
|
||
|
||
<p class="byline">March 20, 2019</p>
|
||
|
||
</header>
|
||
|
||
|
||
|
||
</article>
|
||
|
||
<article>
|
||
<section class="header">
|
||
|
||
</section>
|
||
<section>
|
||
<h3 id="ginibre-ensemble-and-its-properties">Ginibre ensemble and its properties</h3>
|
||
<p>The <em>Ginibre ensemble</em> is a set of random matrices with the entries chosen independently. Each entry of a <span class="math inline">\(n \times n\)</span> matrix is a complex number, with both the real and imaginary part sampled from a normal distribution of mean zero and variance <span class="math inline">\(1/2n\)</span>.</p>
|
||
<p>Random matrices distributions are very complex and are a very active subject of research. I stumbled on this example while reading an article in <em>Notices of the AMS</em> by Brian C. Hall <a href="#ref-1">(1)</a>.</p>
|
||
<p>Now what is interesting about these random matrices is the distribution of their <span class="math inline">\(n\)</span> eigenvalues in the complex plane.</p>
|
||
<p>The <a href="https://en.wikipedia.org/wiki/Circular_law">circular law</a> (first established by Jean Ginibre in 1965 <a href="#ref-2">(2)</a>) states that when <span class="math inline">\(n\)</span> is large, with high probability, almost all the eigenvalues lie in the unit disk. Moreover, they tend to be nearly uniformly distributed there.</p>
|
||
<p>I find this mildly fascinating that such a straightforward definition of a random matrix can exhibit such non-random properties in their spectrum.</p>
|
||
<h3 id="simulation">Simulation</h3>
|
||
<p>I ran a quick simulation, thanks to <a href="https://julialang.org/">Julia</a>’s great ecosystem for linear algebra and statistical distributions:</p>
|
||
<div class="sourceCode" id="cb1"><pre class="sourceCode julia"><code class="sourceCode julia"><a class="sourceLine" id="cb1-1" title="1">using LinearAlgebra</a>
|
||
<a class="sourceLine" id="cb1-2" title="2">using UnicodePlots</a>
|
||
<a class="sourceLine" id="cb1-3" title="3"></a>
|
||
<a class="sourceLine" id="cb1-4" title="4"><span class="kw">function</span> ginibre(n)</a>
|
||
<a class="sourceLine" id="cb1-5" title="5"> <span class="kw">return</span> randn((n, n)) * sqrt(<span class="fl">1</span>/<span class="fl">2</span>n) + im * randn((n, n)) * sqrt(<span class="fl">1</span>/<span class="fl">2</span>n)</a>
|
||
<a class="sourceLine" id="cb1-6" title="6"><span class="kw">end</span></a>
|
||
<a class="sourceLine" id="cb1-7" title="7"></a>
|
||
<a class="sourceLine" id="cb1-8" title="8">v = eigvals(ginibre(<span class="fl">2000</span>))</a>
|
||
<a class="sourceLine" id="cb1-9" title="9"></a>
|
||
<a class="sourceLine" id="cb1-10" title="10">scatterplot(real(v), imag(v), xlim=[-<span class="fl">1.5</span>,<span class="fl">1.5</span>], ylim=[-<span class="fl">1.5</span>,<span class="fl">1.5</span>])</a></code></pre></div>
|
||
<p>I like using <code>UnicodePlots</code> for this kind of quick-and-dirty plots, directly in the terminal. Here is the output:</p>
|
||
<p><img src="../images/ginibre.png" /></p>
|
||
<h3 id="references">References</h3>
|
||
<ol>
|
||
<li><span id="ref-1"></span>Hall, Brian C. 2019. “Eigenvalues of Random Matrices in the General Linear Group in the Large-<span class="math inline">\(N\)</span> Limit.” <em>Notices of the American Mathematical Society</em> 66, no. 4 (Spring): 568-569. <a href="https://www.ams.org/journals/notices/201904/201904FullIssue.pdf" class="uri">https://www.ams.org/journals/notices/201904/201904FullIssue.pdf</a></li>
|
||
<li><span id="ref-2"></span>Ginibre, Jean. “Statistical ensembles of complex, quaternion, and real matrices.” Journal of Mathematical Physics 6.3 (1965): 440-449. <a href="https://doi.org/10.1063/1.1704292" class="uri">https://doi.org/10.1063/1.1704292</a></li>
|
||
</ol>
|
||
</section>
|
||
</article>
|
||
|
||
|
||
<footer>
|
||
Site proudly generated by
|
||
<a href="http://jaspervdj.be/hakyll">Hakyll</a>
|
||
</footer>
|
||
</body>
|
||
</html>
|