---
tag: 4N53T1BZ9S
kind: statement
role: theorem
name: Mehta integral
msc: [33B15, 15B52, 60B20]
uses: [0H5SYBSED9, S8VTVSJ6EX, 0DFT1GP1JP]
about: [0H5SYBSED9, 1HS4RT7HK5]
sources:
  - url: https://en.wikipedia.org/wiki/Selberg_integral
---

## Statement

Let $n \ge 1$ be an integer and let $\gamma \in \C$ with $\operatorname{Re} \gamma > -\frac{1}{n}$.
Let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
Then
$$\frac{1}{(2\pi)^{n/2}} \int_{\R^n} \prod_{1 \le i < j \le n} \abs{x_i - x_j}^{2\gamma}\, e^{-(x_1^2 + \cdots + x_n^2)/2} \, dx_1 \cdots dx_n = \prod_{j=1}^{n} \frac{\Gamma(1 + j\gamma)}{\Gamma(1 + \gamma)}.$$

## Remarks

The formula follows from the [[Selberg integral|S8VTVSJ6EX]] by a limit.
Put $\alpha = \beta = \lambda$ and $t_i = \frac12 + \frac{x_i}{2\sqrt{2\lambda}}$; then $(t_i(1 - t_i))^{\lambda - 1}$ tends, after rescaling, to $e^{-x_i^2/2}$ as $\lambda \to \infty$, and Stirling's formula evaluates the limit of the right side.
For $\gamma = 0$ both sides equal $1$, by the [[Gaussian integral|0DFT1GP1JP]].
Example: $n = 2$ and $\gamma = 1$.
Then $x_1 - x_2$ is normal with mean $0$ and variance $2$ under the Gaussian weight, so the left side is $\mathbb{E}[(x_1 - x_2)^2] = 2$, and the right side is $\frac{\Gamma(2)}{\Gamma(2)} \cdot \frac{\Gamma(3)}{\Gamma(2)} = 2$.
The bound $\operatorname{Re} \gamma > -\frac1n$ is needed: for smaller $\gamma$ the singularity where all $x_i$ coincide makes the integral diverge.

## Explanations

### L1 human

Take $n$ independent standard normal numbers and look at the product of all their pairwise distances, raised to the power $2\gamma$.
The Mehta integral gives the average of this quantity as a short product of gamma values.

### L2 human

For $\gamma = \frac12, 1, 2$ the integrand is, up to the Gaussian factor, the joint density of the eigenvalues of the Gaussian orthogonal, unitary and symplectic ensembles of [[random matrices|1HS4RT7HK5]].
So the formula gives the normalizing constants of these densities.
It is the Gaussian member of a family of integrals over root systems, the Macdonald–Mehta integrals, of which this is the case of type $A_{n-1}$.