= Solution
The <correlation function of a point process> is even and tends to zero as $|\lambda|\to\infty$. Its <derivative> factors as
$$
\frac{d}{d\lambda}R_3^{(1)}(\lambda)
=-\frac{\lambda(\lambda^2-1)(\lambda^2-3)}{2\sqrt{2\pi}}
e^{-\lambda^2/2}.
$$
Consequently its local maxima occur at $0,\pm\sqrt3$, and its local minima at $\pm1$. The heights are
$$
\begin{aligned}
R_3^{(1)}(0)&=\frac3{2\sqrt{2\pi}}\simeq0.5984,\\
R_3^{(1)}(\pm1)&=\frac{2e^{-1/2}}{\sqrt{2\pi}}\simeq0.4839,\\
R_3^{(1)}(\pm\sqrt3)&=\frac{6e^{-3/2}}{\sqrt{2\pi}}\simeq0.5341.
\end{aligned}
$$
\Image[/past-exam-of-the-random-matrix-theory-course-of-the-university-of-oxford/2025-one-point-intensity.png]
{title=The three peaks of the GUE one-point correlation function}
{description=Original plot of the exact <correlation function of a point process> calculated above. Filled markers indicate local maxima at $-\sqrt3,0,\sqrt3$; hollow markers indicate the local minima at $-1,1$. The total area is three.}
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 near
$$
\boxed{\lambda=-\sqrt3,\quad0,\quad\sqrt3,}
$$
with 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, is
$$
L(\boldsymbol\lambda)=2\sum_{i<j}\log(\lambda_j-\lambda_i)
-\frac12\sum_i\lambda_i^2.
$$
Its stationary equations are $\lambda_i=2\sum_{j\ne i}(\lambda_i-\lambda_j)^{-1}$, satisfied by $(-\sqrt3,0,\sqrt3)$. For every nonzero real <vector> $v$, the second directional <derivative> is
$$
D^2L[v,v]=-\sum_i v_i^2
-2\sum_{i<j}\frac{(v_i-v_j)^2}{(\lambda_i-\lambda_j)^2}<0.
$$
Thus $L$ 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 $\mathbb R^3$.
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