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A point process is determinantal with kernel $K$ relative to a specified reference measure if every correlation function of a point process is the indicated determinant. A rank-$m$ finite-rank projection kernel gives an exactly $m$-point process, whose symmetric joint density is $\det [K(x_i,x_j)]/m!$.

For real monic orthogonal polynomials $p_j$ with squared weighted norms $h_j$ under a positive weight $w$, the functions $p_j\sqrt{w/h_j}$ form an orthonormal set for the unweighted reference measure. Their finite-rank projection kernel turns the squared weighted Vandermonde determinant into a kernel determinant. Normalization on the full labelled configuration space is $1/m!$.

Write $D_n=\det [K(x_i,x_j){]}_{i,j=1}^n$ and expand it by permutations. The assumptions ensure that the indicated single-variable integrals exist; the finite-rank projection kernel constructed above satisfies them with $r=m$. No symmetry of a general kernel is needed for the following argument.

If a permutation fixes $n$, its product contains the separate factor $K(x_n,x_n)$. Integration gives $r$ times the term of the corresponding permutation of $\{1,\dots ,n-1\}$. These terms contribute $r{D}_{n-1}$.

Otherwise $n$ belongs to a longer permutation cycle. Let $i$ precede $n$ and let $j$ follow it. The only factors containing $x_n$ are $K(x_i,x_n)K(x_n,x_j)$, whose integral is $K(x_i,x_j)$ by the reproducing assumption. Delete $n$ from the cycle to obtain a permutation $\tau$ on $n-1$ letters. Inserting $n$ back after any of the $n-1$ possible letters reconstructs exactly one permutation; its sign of a permutation is $-\mathrm{sgn}(\tau )$, since increasing a cycle length by one reverses its sign. Thus the nonfixed terms contribute $-(n-1)D_{n-1}$.

Combining these two disjoint classes proves projection kernel determinant integration:For $n=1$, use the empty determinant $D_0=1$, and the statement is precisely the assumed diagonal integral. For our kernel, orthonormality directly yieldsso every step of the marginal integration is justified.

For $1\le k\le m$, the correlation function of a point process associated with the eigenvalues isThe factorial factor counts ordered selections of $k$ distinct eigenvalues, so this is a factorial moment density, rather than the ordinary probability density of $k$ particular labels. Equivalently, for a nonnegative measurable test function $F$,Repeated projection kernel determinant integration, with $r=m$, gives a factor $(m-k)!$ on integrating an $m$-by-$m$ kernel determinant down to size $k$. Together with the normalizing $1/m!$ in $f_m$, the factors cancel:In particular $R_1(x)=K_m(x,x)$ and $\int R_{k}d\mathbf{x}=m!/(m-k)!$. Set $R_0=1$ and $R_k=0$ for $k>m$. The eigenvalue configuration is thus a determinantal point process with a rank-$m$ finite-rank projection kernel.

Use the symmetric joint probability density function of exchangeably labelled eigenvalues, normalized over all of ${\mathbb{R}}^3$. The finite-rank projection kernel hasSuccessive applications of Gaudin's lemma therefore giveIt follows thatAs a normalization check, the weighted Vandermonde determinant yieldsIf instead one uses ordered eigenvalues on ${\lambda }_1<{\lambda }_2<{\lambda }_3$, the normalized density on that chamber is six times the labelled density. In that convention the determinant prefactor on the chamber is one.

Gaudin's lemma is the projection kernel determinant integration identity. Suppose an integral kernel $K$ satisfieswith the relevant integrals absolutely convergent. Define $D_k(x_1,\dots ,x_k)=\det [K(x_i,x_j){]}_{i,j=1}^k$ and $D_0=1$. ThenIn particular, a finite-rank projection kernel of rank $N$ has $\int D_{N}d{\mu }^N=N!$.

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.