Orthogonal polynomials and determinantal kernels

Consider a unitary polynomial ensemble whose unordered joint density is proportional to the squared Vandermonde times $\prod_j w(x_j)$. Assume the positive weight has all moments required through the polynomial degrees being used, and the polynomial inner product is nondegenerate. Choose real orthonormal polynomials $p_j$ in the measure $w(x)dx$. Multiplying each polynomial by the square root of the weight gives functions $\phi_j$ orthonormal in ordinary Lebesgue measure. Their finite sum defines the kernel in the display.

This kernel is the integral kernel of the orthogonal projection onto the span of those first N functions. Orthonormality proves the projection identity by integrating and collapsing the double sum. Its diagonal integrates to N, the rank of the projection. One can instead keep the weight outside and use the polynomial sum as a kernel relative to weighted measure, but mixing that expression with Lebesgue integration loses the weight and changes every correlation formula. Here all correlation kernels use Lebesgue reference measure.

Replacing monomials in the Vandermonde by orthogonal polynomials changes only a constant. Expanding the two determinants and integrating with orthonormality yields the normalized unordered density as the determinant of the full kernel matrix divided by N factorial. Integrating out coordinates gives factorial correlation functions: they weight ordered tuples of distinct particles. In particular the first correlation is expected number density, not the probability density of one uniformly selected eigenvalue. Divide it by N to obtain that latter normalized density.

For two particles the determinant subtracts a squared cross-kernel term from the product of diagonal intensities. At coincidence the two rows agree and the determinant vanishes, expressing repulsion. Integrating the first correlation over an interval gives the expected count there. Projection also gives count variance as this expectation minus the double integral of the squared kernel over that interval, hence a variance no larger than the expected count. These scalar determinantal formulas describe the beta-two ensemble; GOE correlations require a different, Pfaffian structure.

For the Gaussian weight $w(x)=e^{-x^2/2}$, the polynomial system is the probabilists’ Hermite system with its appropriate normalization. These coordinates are unscaled Gaussian eigenvalues: to recover the normalized GUE convention of the previous lesson, divide each coordinate by the square root of N. A kernel transforms under this rescaling with a Jacobian factor. Merely replacing its arguments while leaving its amplitude unchanged would produce the wrong expected count.

Christoffel–Darboux identities compress a long polynomial sum into an expression involving two adjacent degrees. This becomes a practical starting point for asymptotic analysis: different spatial rescalings of the Hermite functions lead to bulk and edge kernels. The limiting formulas require asymptotic estimates, not just finite-dimensional orthogonality. The finite projection calculation already explains exact normalization and repulsion, while locating a sine or Airy limit needs additional information about how the polynomial degree and spatial coordinate grow together.