Selberg integral

Short description: Mathematical function

In mathematics, the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg. It has applications in statistical mechanics, multivariable orthogonal polynomials, random matrix theory, Calogero–Moser–Sutherland model, and Knizhnik–Zamolodchikov equations.

Selberg's integral formula

When $Re(\alpha )>0,Re(\beta )>0,Re(\gamma )>-\min (\frac{1}{n},\frac{Re(\alpha )}{n-1},\frac{Re(\beta )}{n-1})$, we have

$\begin{array}{rl}{S}_n(\alpha ,\beta ,\gamma )& =\underset{0}{\overset{1}{\int }}\cdots \underset{0}{\overset{1}{\int }}\prod _{i=1}^n{t}_i^{\alpha -1}(1-t_i{)}^{\beta -1}\prod _{1\le i<j\le n}|t_i-t_j{|}^{2\gamma }d{t}_1\cdots d{t}_n\\ & =\prod _{j=0}^{n-1}\frac{\mathrm{\Gamma }(\alpha +j\gamma )\mathrm{\Gamma }(\beta +j\gamma )\mathrm{\Gamma }(1+(j+1)\gamma )}{\mathrm{\Gamma }(\alpha +\beta +(n+j-1)\gamma )\mathrm{\Gamma }(1+\gamma )}\end{array}$

Selberg's formula implies Dixon's identity for well poised hypergeometric series, and some special cases of Dyson's conjecture. This is a corollary of Aomoto.

Aomoto's integral formula

Aomoto proved a slightly more general integral formula. With the same conditions as Selberg's formula,

$\underset{0}{\overset{1}{\int }}\cdots \underset{0}{\overset{1}{\int }}\left(\prod _{i=1}^k{t}_i\right)\prod _{i=1}^n{t}_i^{\alpha -1}(1-t_i{)}^{\beta -1}\prod _{1\le i<j\le n}|t_i-t_j{|}^{2\gamma }d{t}_1\cdots d{t}_n$

$=S_n(\alpha ,\beta ,\gamma )\prod _{j=1}^k\frac{\alpha +(n-j)\gamma }{\alpha +\beta +(2n-j-1)\gamma }.$

A proof is found in Chapter 8 of (Andrews Askey).

Mehta's integral

When $Re(\gamma )>-1/n$,

$\frac{1}{(2\pi {)}^{n/2}}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\cdots \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\prod _{i=1}^n{e}^{-t_i^2/2}\prod _{1\le i<j\le n}|t_i-t_j{|}^{2\gamma }d{t}_1\cdots d{t}_n=\prod _{j=1}^n\frac{\mathrm{\Gamma }(1+j\gamma )}{\mathrm{\Gamma }(1+\gamma )}.$

It is a corollary of Selberg, by setting $\alpha =\beta$, and change of variables with $t_i=\frac{1+t{\text{'}}_i/\sqrt{2\alpha }}{2}$, then taking $\alpha \to \mathrm{\infty }$.

This was conjectured by (Mehta Dyson), who were unaware of Selberg's earlier work.

It is the partition function for a gas of point charges moving on a line that are attracted to the origin.

In particular, when $\gamma =1$, the term on the right is $\prod _{j=1}^{n}j!$.

Macdonald's integral

(Macdonald 1982) conjectured the following extension of Mehta's integral to all finite root systems, Mehta's original case corresponding to the A_n−1 root system.

$\frac{1}{(2\pi {)}^{n/2}}\int \cdots \int {\left|\prod _r\frac{2(x,r)}{(r,r)}\right|}^{\gamma }{e}^{-(x_1^2+\cdots +x_n^2)/2}d{x}_1\cdots d{x}_n=\prod _{j=1}^n\frac{\mathrm{\Gamma }(1+d_j\gamma )}{\mathrm{\Gamma }(1+\gamma )}$

The product is over the roots r of the roots system and the numbers d_j are the degrees of the generators of the ring of invariants of the reflection group. (Opdam 1989) gave a uniform proof for all crystallographic reflection groups. Several years later he proved it in full generality, making use of computer-aided calculations by Garvan.

Further reading

• Forrester, Peter (2010). "4. The Selberg integral". Log-gases and random matrices. London Mathematical Society monographs. Princeton: Princeton University Press. ISBN 978-0-691-12829-0.