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Random matrix theory course of the University of Oxford

Past exam of the random matrix theory course of the University of Oxford

2025

Evaluate the <orthogonal polynomial projection kernel> on its diagonal:
$$
\begin{aligned}
R_3^{(1)}(\lambda)
&=\sum_{\ell=0}^2\phi_\ell(\lambda)^2\\
&=\frac{e^{-\lambda^2/2}}{\sqrt{2\pi}}
\left[1+\lambda^2+\frac{(\lambda^2-1)^2}{2}\right]\\
&=\boxed{\frac{\lambda^4+3}{2\sqrt{2\pi}}e^{-\lambda^2/2}.}
\end{aligned}
$$
The $\lambda^2$ terms cancel. Integrating against <Lebesgue measure> gives $(\mathbb EZ^4+3)/2=3$ for a <standard normal random variable> $Z$, checking the <correlation function of a point process> normalization.

Solution

Past exam of the random matrix theory course of the University of Oxford / 2025 / 2 / c / ii / Solution

Past exam of the random matrix theory course of the University of Oxford / 2025 / 2 / c / ii / Solution

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Past exam of the random matrix theory course of the University of Oxford / 2025 / 2 / c / ii / Solution

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