Definition¶

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

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

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,

(33)¶\[\begin{split}\Delta(x_1,\dots,x_N) = \det \begin{bmatrix} 1 & \dots & 1 \\ x_1 & \dots & x_N \\ \vdots & & \vdots \\ x_1^{N-1} & &x_N^{N-1} \end{bmatrix},\end{split}\]

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

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