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Dimitri Lozeve 2020-08-27 15:01:49 +02:00
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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>
<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>
<div class="sourceCode" id="cb1"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true"></a><span class="kw">using</span> LinearAlgebra</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true"></a><span class="kw">using</span> UnicodePlots</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true"></a></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true"></a><span class="kw">function</span> ginibre(n)</span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true"></a> <span class="kw">return</span> randn((n<span class="op">,</span> n)) <span class="op">*</span> sqrt(<span class="fl">1</span><span class="op">/</span><span class="fl">2</span>n) <span class="op">+</span> <span class="cn">im</span> <span class="op">*</span> randn((n<span class="op">,</span> n)) <span class="op">*</span> sqrt(<span class="fl">1</span><span class="op">/</span><span class="fl">2</span>n)</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true"></a><span class="kw">end</span></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true"></a></span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true"></a>v <span class="op">=</span> eigvals(ginibre(<span class="fl">2000</span>))</span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true"></a></span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true"></a>scatterplot(real(v)<span class="op">,</span> imag(v)<span class="op">,</span> xlim<span class="op">=</span>[<span class="op">-</span><span class="fl">1.5</span><span class="op">,</span><span class="fl">1.5</span>]<span class="op">,</span> ylim<span class="op">=</span>[<span class="op">-</span><span class="fl">1.5</span><span class="op">,</span><span class="fl">1.5</span>])</span></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>
<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>
</ol>
</section>
</article>