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<h1>Random matrices from the Ginibre ensemble</h1>
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<h1 class="title">Random matrices from the Ginibre ensemble</h1>
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<p class="byline">March 20, 2019</p>
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Posted on March 20, 2019
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Posted on March 20, 2019
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<h2 id="ginibre-ensemble-and-its-properties">Ginibre ensemble and its properties</h2>
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<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>
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<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>
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<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>
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<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>
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<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>
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<h2 id="simulation">Simulation</h2>
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<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>
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<div class="sourceCode" id="cb1"><pre class="sourceCode julia"><code class="sourceCode julia"><a class="sourceLine" id="cb1-1" title="1">using LinearAlgebra</a>
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<a class="sourceLine" id="cb1-2" title="2">using UnicodePlots</a>
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<a class="sourceLine" id="cb1-3" title="3"></a>
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<a class="sourceLine" id="cb1-4" title="4"><span class="kw">function</span> ginibre(n)</a>
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<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>
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<a class="sourceLine" id="cb1-6" title="6"><span class="kw">end</span></a>
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<a class="sourceLine" id="cb1-7" title="7"></a>
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<a class="sourceLine" id="cb1-8" title="8">v = eigvals(ginibre(<span class="fl">2000</span>))</a>
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<a class="sourceLine" id="cb1-9" title="9"></a>
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<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>
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<p>I like using <code>UnicodePlots</code> for this kind of quick-and-dirty plots, directly in the terminal. Here is the output:</p>
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<p><img src="../images/ginibre.png" /></p>
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<h2 id="references">References</h2>
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<ol>
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<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>
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<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>
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