 Solution

ID: past-exam-of-the-random-matrix-theory-course-of-the-university-of-oxford/2025/2/a/ii/solution

⟨ f, g ⟩ = E [f (Z) g (Z)]

Z

. The degree-zero monic polynomial is

p_{0} = 1

. Write

p_{1} = x + a

and

p_{2} = x^{2} + b x + c

. The conditions

0 = ⟨ p_{1}, 1 ⟩ = a, 0 = ⟨ p_{2}, 1 ⟩ = 1 + c, 0 = ⟨ p_{2}, x ⟩ = b

use

E Z = E Z^{3} = 0

and

E Z^{2} = 1

p_{0} (x) = 1, p_{1} (x) = x, p_{2} (x) = x^{2} - 1.

E Z^{4} = 3

, so their squared norms are

h_{0} = 1, h_{1} = 1, h_{2} = E (Z^{2} - 1)^{2} = 3 - 2 + 1 = 2, h_{ℓ} = \int_{R} p_{ℓ} (x)^{2} \frac{e ^{- x^{2} /2}}{2 π} d x .

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