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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 $1\le k\le m$, 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.

The correlation function of a point process is even and tends to zero as $|\lambda |\to \infty$. Its derivative factors asConsequently its local maxima occur at $0,\pm \sqrt{3}$, and its local minima at $\pm 1$. The heights are

For an interval $I$, ${\int }_I{R}_3^{(1)}(\lambda )d\lambda$ is the expected value of the number of eigenvalues in $I$. Thus equally short intervals have the highest expected counts nearwith the central peak highest. Each exchangeably labelled eigenvalue has the same marginal probability density function $R_3^{(1)}/3$. The three peaks should not be interpreted as three separate marginal densities for ordered eigenvalues.

In this case the most likely joint configuration also lies at these three positions. On the ordered chamber, the logarithm of the joint probability density function, up to a constant, isIts stationary equations are ${\lambda }_i=2{\sum }_{j\ne i}({\lambda }_i-{\lambda }_j{)}^{-1}$, satisfied by $(-\sqrt{3},0,\sqrt{3})$. For every nonzero real vector $v$, the second directional derivative isThus $L$ is strictly concave on this convex chamber, proving that the stationary configuration is its unique global maximum. Permuting the labels gives the other maxima on ${\mathbb{R}}^3$.

Evaluate the orthogonal polynomial projection kernel on its diagonal:The ${\lambda }^2$ terms cancel. Integrating against Lebesgue measure gives $(\mathbb{E}{Z}^4+3)/2=3$ for a standard normal random variable $Z$, checking the correlation function of a point process normalization.