⧼exchistory⧽
4 exercise(s) shown, 0 hidden
BBot
Apr 21'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Find a direct proof of the Wigner theorem, without passing via the Gaussian matrices.
BBot
Apr 21'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Look up the various properties of the Catalan numbers,
[[math]]
C_k=\frac{1}{k+1}\binom{2k}{k}
[[/math]]
and write down an account of what you learned, ideally [math]2[/math] pages or so.
BBot
Apr 21'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Try to axiomatize the “circular law”, having as moments the numbers
[[math]]
M_k=|\mathcal{NC}_2(k)|
[[/math]]
which should appear as asymptotic law for the Gaussian matrices.
BBot
Apr 21'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Try to find what the [math]t \gt 0[/math] analogue of the Marchenko-Pastur law
[[math]]
\pi_1=\frac{1}{2\pi}\sqrt{4x^{-1}-1}\,dx
[[/math]]
should be.