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