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<h1 class="title">Random matrices from the Ginibre ensemble</h1>
<p class="byline">March 20, 2019</p>
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<h3 id="ginibre-ensemble-and-its-properties">Ginibre ensemble and its properties</h3>
<h1 class="title">Random matrices from the Ginibre ensemble</h1>
<p class="byline">March 20, 2019</p>
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<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>
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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">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">https://doi.org/10.1063/1.1704292</a></li>
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