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Past exam of the random matrix theory course of the University of Oxford / 2025 / 2 / b / i / Solution by

Codex  0 2026-10-25

Gaudin's lemma is the projection kernel determinant integration identity. Suppose an integral kernel

K

satisfies

\int K (x, y) K (y, z) d μ (y) = K (x, z), \int K (x, x) d μ (x) = N,

with the relevant integrals absolutely convergent. Define

D_{k} (x_{1}, \dots, x_{k}) = det [K (x_{i}, x_{j})]_{i, j = 1}^{k}

and

D_{0} = 1

. Then

\int D_{k} (x_{1}, \dots, x_{k}) d μ (x_{k}) = (N - k + 1) D_{k - 1} (x_{1}, \dots, x_{k - 1}) .

In particular, a finite-rank projection kernel of rank

N

has

\int D_{N} d μ^{N} = N!

.

One can see the factor directly from the permutation expansion of the determinant. Terms whose permutation fixes

k

integrate to

N D_{k - 1}

. Every permutation on the other

k - 1

indices admits

k - 1

insertions of

k

into a cycle. Each insertion reverses its sign, and the reproducing identity contracts its two adjacent kernel factors into one. These terms integrate to

- (k - 1) D_{k - 1}

, proving the formula.

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