# Definition¶

Let \(\beta>0\), the joint distribution of the \(\beta\)-Ensemble associated to the reference measure \(\mu\) writes

Hint

\(|\Delta(x_1,\dots,x_N)| = \prod_{i<j} |x_i - x_j|\) is the absolute value of the determinant of the Vandermonde matrix,

(32)¶\[\begin{split}\Delta(x_1,\dots,x_N) = \det \begin{bmatrix} 1 & \dots & 1 \\ x_1 & \dots & x_N \\ \vdots & & \vdots \\ x_1^{N-1} & &x_N^{N-1} \end{bmatrix},\end{split}\]encoding the repulsive interaction. The

*closer*the points are the lower the density.\(\beta\) is the inverse temperature parameter quantifying the strength of the repulsion between the points.

Important

For Gaussian, Gamma and Beta reference measures, the \(\beta=1,2\) and \(4\) cases received a very special attention in the random matrix literature, e.g. [DE02].

The associated ensembles actually correspond to the eigenvalues of random matrices whose distribution is invariant to the action of the orthogonal (\(\beta=1\)), unitary (\(\beta=2\)) and symplectic (\(\beta=4\)) group respectively.

\(\mu\) |
\(\mathcal{N}\) |
\(\Gamma\) |
\(\operatorname{Beta}\) |
---|---|---|---|

Ensemble name |
Hermite |
Laguerre |
Jacobi |

support |
\(\mathbb{R}\) |
\(\mathbb{R}^+\) |
\([0,1]\) |

Note

The study of the distribution of the eigenvalues of random orthogonal, unitary and symplectic matrices lying on the unit circle is also very thorough [KN04].

## Orthogonal Polynomial Ensembles¶

The case \(\beta=2\) corresponds a specific type of *projection* DPPs also called Orthogonal Polynomial Ensembles (OPEs) [Konig04] with associated kernel

where \((P_n)\) are the orthonormal polynomials w.r.t. \(\mu\) i.e. \(\operatorname{deg}(P_n)=n\) and \(\langle P_k, P_l \rangle_{L^2(\mu)}=\delta_{kl}\).

Note

OPEs (with \(N\) points) correspond to *projection* DPPs onto
\(\operatorname{Span}\{P_n\}_{n=0}^{N-1} = \mathbb{R}^{N-1}[X]\)

Hint

First, linear combinations of the rows of \(\Delta(x_1,\dots,x_N)\) allow to make appear the orthonormal polynomials \((P_n)\) so that

Then,

Finally, the joint distribution of \((x_1, \dots, x_N)\) reads