Definition

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

(31)\[(x_1,\dots,x_N) \sim \frac{1}{Z_{N,\beta}} \left|\Delta(x_1,\dots,x_N)\right|^{\beta} \prod_{i= 1}^N \mu(d x_i).\]

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

\[K_N(x, y) = \sum_{n=0}^{N-1} P_n(x) P_n(y),\]

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

\[\begin{split}|\Delta(x_1,\dots,x_N)| \propto \begin{vmatrix} P_0(x_1) & \dots & P_0(x_N) \\ P_1(x_1) & \dots & P_1(x_N) \\ \vdots & & \vdots \\ P_{N-1}(x_1) & & P_{N-1}(x_N) \end{vmatrix}.\end{split}\]

Then,

\[|\Delta|^2 = | \Delta^{\top} \Delta | \propto \det \left[ K_N(x_i, x_j)\right]_{i,j=1}^N.\]

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

(33)\[(x_1,\dots,x_N) \sim \frac{1}{N!} \det \left[ K_N(x_i, x_j)\right]_{i,j=1}^N \prod_{i= 1}^N \mu(d x_i).\]

See also

[Konig04], [Joh06]