A Gaussian integral formula for the Hermite polynomials: Combinatorics, Asymptotics and Applications

TL;DR

This work presents a Gaussian integral (Gaussian-expectation) representation for probabilists’ Hermite polynomials, enabling streamlined derivations of fundamental properties and broad multivariate generalizations. Through Wick/Isserlis combinatorics, it provides new combinatorial proofs of orthogonality and recurrences, and via Laplace/Eulerian methods yields refined Plancherel–Rotach-type asymptotics across multiple regimes, including edge scaling with Airy behavior. The framework unifies oscillator wave functions, Hermite kernels, and random-matrix theory, deriving bulk semicircle limits, Airy-edge limits, and Tracy–Widom fluctuations, as well as BBP-like spiked transitions in Dyson Brownian motion. It also delivers Edgeworth expansions (1D and multi-D) and Fourier/IBP identities that illuminate connections to Fourier analysis, cumulants, and Gaussian integration by parts. The exposition emphasizes generalization to arbitrary variance and higher dimensions, linking probabilistic representations to deterministic spectral limits with broad applicability in combinatorics, asymptotics, and random matrix theory.

Abstract

The Hermite polynomials are ubiquitous but can be difficult to work with due to their unwieldy definition in terms of derivatives. To remedy this, we showcase an underappreciated Gaussian integral formula for the Hermite polynomials, which is especially useful for generalizing to multivariable Hermite polynomials. Taking this as our definition, we prove many useful consequences, including: 1. Combinatorial interpretations for the Hermite polynomials, including a proof of orthogonality. 2. A more elementary proof of Plancherel--Rotach asymptotics that does not involve residues. 3. Limit theorems for GUE random matrices and Dyson's Brownian motion, including bulk convergence to the semi-circle law and edge convergence to the Airy limit/Tracy-Widom law. 4. An analysis of a phase transition in the spiked GUE random matrix as the top eigenvalue goes from well-separated to attached to the bulk, analogous to the BBP phase transition. 5. Elementary derivations of Edgeworth expansions and multivariable Edgeworth expansions. This article is primarily expository and features many illustrative figures.

Figures (14)

• Figure 1: Two examples of the Wick/Isserlis theorem when $n=4$ and $n=6$. When $n=4$, $\mathbb{E}[Z_1 Z_2 Z_3 Z_4]$ is a sum of $3$ terms, each of which is a product of 2 expected values, and when when $n=6$ , $\mathbb{E}[Z_1 Z_2 Z_3 Z_4 Z_5 Z_6]$ is a sum of 15 terms, each of which is a product of 3.
• Figure 2: The combinatorics of the Hermite polynomial $\widetilde{H}_4(x,t)$. There are $2^4 = 16$ possible monomials in the expansion of $(x+\imath \sqrt{t} Z)^4$, but only $1+6+1=8$ monomials have an even power of $Z$, and therefore have a non-zero contribution. The monomial with $Z^4$ becomes 3 pairings due to the Wick/Isserlis theorem for the expectation of Gaussian monomials, yielding 10 total terms. Each pair contributes a multiplicative factor of $(\imath \sqrt{t})^2=-t$ .
• Figure 3: Three examples of diagrams for $\mathbb{E}[H\!\!\operatorname{e}_3(X)H\!\!\operatorname{e}_3(X) ]$ as defined in Definition \ref{['def:diagrams']} when $n_{} = 3$, $m_{}=3$. The product of the 6 terms in $(X+\imath L)^3(X+\imath R)^3$ is illustrated here with a dotted line for greater visual separation of the left/right halves. In this example: $\delta_1 = (C_1,\pi_1)$ has $C_1 = (X,\imath L,\imath L,X,\imath R, \imath R)$ and $\pi_1 = \{ \{1_L, 1_R\}, \{2_L,3_L\},\{2_R,3_R\}\}$ ; $\delta_2 = (C_2, \pi_2)$ where $C_2 = (X,X,X,X,\imath R, \imath R)$ and $\pi_2 = \{ \{1_L, 1_R\}, \{2_L,3_L\},\{2_R,3_R\}\}$ ; $\delta_3 = (C_3,\pi_3)$ has $C_3 = (X,X,X,X,X,X)$ and $\pi_3 =$$\{ \{1_L,2_R\},\{2_L,3_R\},\{3_L,1_R\}\}$ . The involution $\tau$ defined in Definition \ref{['def:involution']} swaps the labels of the first entry which is entirely on one half. In this example, $\tau(\delta_1)=\delta_2$ and $\tau(\delta_2)=\delta_1$ . If there are no pairs entirely on one half, then $\tau$ does nothing, so $\tau(\delta_3)=\delta_3$. When $\tau(\delta)=\delta$, since every pair has one end on the left half and one end on the right half, the diagram can be interpreted as a bijection. For example, the permutation for $\delta_3$ is $(2, 3, 1)$ when written in one line notation.
• Figure 4: The approximation formulas for various regimes from Propositions \ref{['prop:unscaled']}, \ref{['prop:superlarge']} and \ref{['thm:main-asymptotics']} compared to $H\!\!\operatorname{e}_n(x)$ in the case $n=25$. The $y$-axis is a modified log scale, so that the sign of each $y$ value is preserved and $0$ is mapped to $0$; the map is $y \to sgn(y) \log_{10}(|y|+1)$. The relatively simple asymptotics when $x=O(1)$ and $x=O(n)$ are only accurate when $x \ll \sqrt{n}$ and $x \gg \sqrt{n}$ (and in fact can also be recovered as limits of the more complicated $x=O(\sqrt{n})$ asymptotics). The $x=O(\sqrt{n})$ asymptotics are divided into three regimes; $x=2\sqrt{n}\cos(\alpha) < 2\sqrt{n}$ for small $x$ values, $x=2\sqrt{n}\cosh(\gamma) > 2\sqrt{n}$ for large $x$ values, and $x=2\sqrt{n} + n^{-1/6}u$ for $x$ values on the edge near $x=2\sqrt{n}$. Notice from the zoomed insert that the small-x and large-x approximations are both singular at the edge $x=2\sqrt{n}$, and this is the regime where the edge approximation with the Airy function is accurate.
• Figure 5: The location of critical points $w^\ast_\pm$ and constant phase curves $\{w : Im(F_a(w)) = Im(F_a(w^\ast_\pm)) \}$ shown in grey. The location of the critical points changes from conjugate pair of roots when $a<1$ (small x regime), to a double critical point when $a=1$, to two distinct roots when $a>1$ (large $x$ regime). The blue curves $C_\pm$ and $C_1$ are used in the proofs: we deform the integral from the real axis to these curves so that we can apply the Laplace appropriation method through the appropriate critical points.

Theorems & Definitions (146)

• Definition 1.1
• Definition 1.2
• Example 1.3
• Definition 1.4
• Definition 1.5
• Example 1.6
• Remark 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4