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In the unscaled convention with diagonal variance one, a Gaussian unitary ensemble matrix $H$ is a Hermitian matrix with independent real diagonal entries $N(0,1)$ and upper off-diagonal entries $(A_{ij}+i{B}_{ij})/\sqrt{2}$, where all the $A_{ij},B_{ij}$ and diagonal entries are independent and have the standard normal distribution. Its density on the real coordinates of Hermitian matrices is proportional to $\exp [-\mathrm{tr}(H^2)/2]$. Its exchangeably labelled real eigenvalues have symmetric joint probability density function proportional to ${\prod }_{i<j}({\lambda }_i-{\lambda }_j{)}^2{e}^{-{\sum }_i{\lambda }_i^2/2}$. The squared Vandermonde determinant suppresses coincident eigenvalues; orthogonal polynomial projection kernels express the correlation functions of a point process as determinants.

For $w(x)=x^a{e}^{-x}$ on $[0,\infty )$, $a>-1$, the monic Generalized Laguerre polynomials have squared norms $h_j=j!\Gamma (j+a+1)$. The corresponding orthogonal polynomial projection kernel is $(xy{)}^{a/2}{e}^{-(x+y)/2}{\sum }_{j=0}^{m-1}j!L_j^{(a)}(x)L_j^{(a)}(y)/\Gamma (j+a+1)$ for nonnegative $x,y$, and zero otherwise.

Choose real monic orthogonal polynomials $p_j$ of degree $j$, $j=0,1,\dots$, for $w(x)=x^a{e}^{-x}$ on $[0,\infty )$, and letExtend each ${\varphi }_j$ by zero to negative $x$. The resulting functions form an orthonormal set in $L^2(\mathbb{R},dx)$. Define the orthogonal polynomial projection kernelThe formula on the right is for nonnegative arguments; the kernel is zero when either argument is negative. With ${\Phi }_{ij}={\varphi }_{j-1}(x_i)$, the kernel evaluation matrix is $\Phi {\Phi }^T$. By the determinant multiplication identity and the preceding monic basis change,Thus initially ${\hat{c}}_m=c_m{\prod }_j{h}_j$.

Use the symmetric labelled joint probability density on the full orthant. Expanding the two copies of $\det \Phi$ into permutations and integrating all variables, orthonormality makes a term vanish unless the two permutations agree. Each of the $m!$ surviving terms integrates to $1$. HenceIf the eigenvalues are instead listed in increasing order on a single ordered chamber, that chamber's density is $m!$ times this symmetric density. We keep the full-orthant convention throughout.

An explicit choice is $p_j(x)=(-1{)}^{j}j!L_j^{(a)}(x)$ in terms of the Generalized Laguerre polynomials. To verify the normalization independently, their Rodrigues' formula givesIntegrating by parts $j$ times proves orthogonality to every lower-degree polynomial. The boundary terms vanish at zero and infinity for $a\ge 0$. Taking the other factor to be $p_j$ and using $p_j^{(j)}=j!$ gives $h_j=j!\Gamma (a+j+1)$, where $\Gamma$ is the Gamma function. The Laguerre projection kernel is thereforefor $x,y\ge 0$.

The squared norm of $p_{\ell }(x)e^{-x^2/4}$ in $L^2(\mathbb{R},dx)$ is $\sqrt{2\pi }{h}_{\ell }$, where $(h_0,h_1,h_2)=(1,1,2)$. Choose positive normalizing constantsThen ${\varphi }_0,{\varphi }_1,{\varphi }_2$ are an orthonormal set for Lebesgue measure. Expanding the finite sums givesThus this is an orthogonal polynomial projection kernel, explicitlyThe positive constants above are forced by the reproducing identity: before normalization, its convolution multiplies each independent term ${\varphi }_{\ell }(x){\varphi }_{\ell }(z)$ by $\sqrt{2\pi }{h}_{\ell }/{\kappa }_{\ell }^2$. Each multiplier must equal one. Changing the sign of a real ${\kappa }_{\ell }$ leaves the kernel unchanged.

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.