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

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- Solution ii

Solution

Evaluate the orthogonal polynomial projection kernel on its diagonal:

R_{3}^{(1)} (λ) = ℓ = 0 \sum 2 ϕ_{ℓ} (λ)^{2} = \frac{e ^{- λ^{2} /2}}{2 π} [1 + λ^{2} + \frac{( λ ^{2} - 1 ) ^{2}}{2}] = \frac{λ ^{4} + 3}{2 2 π} e^{- λ^{2} /2} .

The

λ^{2}

terms cancel. Integrating against Lebesgue measure gives

(E Z^{4} + 3) /2 = 3

for a standard normal random variable

Z

, checking the correlation function of a point process normalization.

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