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Aspects of Beta Ensembles

Aspirateur sans fil Dyson Absolute
Ceci n’est pas un ensemble de Dyson.

This post is devoted to computations for beta ensembles from random matrix theory.

Real case. Following MR2813333, or MR2325917 MR1936554 with a different scaling, for all $\beta>0$ and $n\geq2$, the $\beta$ Hermite ensemble is the probability measure on $\mathbb{R}^n$ defined by $$
\mathrm{d}P_{\beta,n}(x)
=\frac{\mathrm{e}^{-n\frac{\beta}{4}(x_1^2+\cdots+x_n^2)}}{C_{\beta,n}}\prod_{i<j}|x_i-x_j|^\beta\mathrm{d}x.
$$ Let $X_n=(X_{n,1},\ldots,X_{n,n})\sim P_{\beta,n}$. The normalization $C_{\beta,n}$ can be explicitly computed in terms of Gamma functions via a Selberg integral. Following MR1936554, the law $P_{\beta,n}$ is the distribution of the ordered eigenvalues of the random tridiagonal symmetric $n\times n$ matrix
$$
M_{\beta,n}=\frac{1}{\sqrt{\beta n}}
\begin{pmatrix}
\mathcal{N}(0, 2) & \chi_{(n-1) \beta} & & &\\
\chi_{(n-1) \beta} & \mathcal{N}(0, 2) & \chi_{(n-2) \beta} & & \\
& \ddots & \ddots & \ddots & \\
& & \chi_{2\beta} & \mathcal{N}(0,2) & \chi_{\beta} \\
& & & \chi_{\beta} & \mathcal{N}(0,2)
\end{pmatrix}
$$
where, up to the scaling prefactor $1/\sqrt{\beta n}$, the entries in the upper triangle including the diagonal are independent, follow a Gaussian law $\mathcal{N}(0,2)$ on the diagonal, and $\chi$-laws just above the diagonal with a decreasing parameter with step $\beta$ from $(n-1)\beta$ to $\beta$. In particular $$X_{n,1}+\cdots+X_{n,n}=\mathrm{Trace}(M_{\beta,n})\sim\mathcal{N}\left(0,\frac{2}{\beta}\right).$$

Using standard algebra on Gamma distributions we also get
$$
X_{n,1}^2+\cdots+X_{n,n}^2
=\mathrm{Tr}(M_{\beta,n}^2)
\sim\mathrm{Gamma}\left(\frac{n}{2}+\frac{\beta n(n-1)}{4},\frac{\beta n}{4}\right).
$$ In particular, we have $$\mathbb{E}((X_{n,1}+\cdots+X_{n,n})^2)=\frac{2}{\beta}\quad\text{and}\quad\mathbb{E}(X_{n,1}^2+\cdots+X_{n,n}^2)=\frac{2}{\beta}+n-1.$$

The mean and covariance of the random vector $X_n$ are given, for all $1\leq i\neq j\leq n$, by
$$
\int x_i \, \mathrm{d}P_{\beta,n} =0, \quad
\int x_i^2 \, \mathrm{d}P_{\beta,n} = \frac{n-1}{n} + \frac{2}{n\beta},
\quad \int x_ix_j \, \mathrm{d}P_{\beta,n} = – \frac{1}{n} .
$$

Dynamics. It is also possible to compute using the overdamped Langevin dynamics associated to the Boltzmann-Gibbs measure $P_{\beta,n}$. This is known as the Dyson-Ornstein-Uhlenbeck dynamics. Namely, the law $P_{\beta,n}$ is invariant for the operator $$Lf(x)=\Delta f(x)-\nabla H(x)\cdot\nabla f(x)$$ where $$H(x)=n\frac{\beta}{4}(x_1^2+\cdots+x_n^2)-\frac{\beta}{2}\sum_{i\neq j}\log|x_i-x_j|.$$ Since $\partial_{x_i}H(x)=n\frac{\beta}{2}x_i-\beta\sum_{j\neq i}\frac{1}{x_i-x_j}$, we find $$L=\sum_{i=1}^n\partial_{x_i}^2-n\frac{\beta}{2}\sum_{i=1}^nx_i\partial_{x_i}+\frac{\beta}{2}\sum_{i\neq j}\frac{\partial_{x_i}-\partial_{x_j}}{x_i-x_j}.$$ The first two terms form an Ornstein-Uhlenbeck operator, while the last term leaves globally invariant symmetric polynomials. Certain symmetric polynomials are eigenvectors. For instance the function $x_1+\cdots+x_n$ is an eigenvector, indeed we find $$L(x_1+\cdots+x_n)=-n\frac{\beta}{2}(x_1+\cdots+x_n).$$. Similarly, the function $x_1^2+\cdots+x_n^2+c$ is, for a choice of $c$, an eigenvector since $$L(x_1^2+\cdots+x_n^2)=2n+\beta n(n-1)-n\beta(x_1^2+\cdots+x_n^2).$$

Let $S={(S_t)}_{t\geq0}$ be the stochastic process on $\mathbb{R}^n$ solution of the stochastic differential equation $$\mathrm{d}S_t=\sqrt{2}\mathrm{d}B_t-\nabla H(S_t)\mathrm{d}t$$ where $B={(B_t)}_{t\geq0}$ is a standard Brownian motion. The Itô formula gives, for $f\in\mathcal{C}^2(\mathbb{R}^n,\mathbb{R})$, $$\mathrm{d}f(S_t)=\sqrt{2}\nabla f(S_t)\cdot \mathrm{d}B_t+(Lf)(S_t)\mathrm{d}t.$$

With $f(x_1,\ldots,x_n)=x_1+\cdots+x_n$ we get that $U_t:=S_{t,1}+\cdots+S_{t,n}$ solves $$\mathrm{d}U_t=\sqrt{2 n}\mathrm{d}W_t-n\frac{\beta}{2}U_t\mathrm{d}t$$ where $W={(\frac{B_{t,1}+\cdots+B_{t,n}}{\sqrt{n}})}_{t\geq0}$ is a standard Brownian motion. Thus $U$ is an Ornstein-Uhlenbeck process. Since $U_\infty$ has the law of $X_n$, we recover by this way the formula $X_n\sim\mathcal{N}(0,2\beta^{-1})$.

With $f(x_1,\ldots,x_n)=x_1^2+\cdots+x_n^2$, we get that $V_t:=S_{t,1}^2+\cdots+S_{t,n}^2=|S_t|^2$ solves $$\mathrm{d}V_t=\sqrt{2}2\sqrt{V_t}\mathrm{d}W_t+n\beta\left(\frac{2}{\beta}+n-1-V_t\right)\mathrm{d}t$$ where $W={(\frac{S_t}{|S_t|}B_t)}_{t\geq0}$ is a standard Brownian motion. Thus $V$ is a Cox-Ingersoll-Ross process. The generator of such a process is a Laguerre operator, and the invariant distribution is a Gamma law. Since $V_\infty$ has the law of $|X_n|^2$, we recover the formula $|X_n|^2\sim\mathrm{Gamma}(…)$.

Complex case. For all $\beta>0$ and $n\geq2$, we consider the probability measure on $\mathbb{C}^n$ defined by
$$
\mathrm{d}P_{\beta,n}(x)
=\frac{\mathrm{e}^{-n\frac{\beta}{2}(|x_1|^2+\cdots+|x_n|^2)}}{C_{\beta,n}}\prod_{i<j}|x_i-x_j|^\beta\mathrm{d}x.
$$ Let $X_n=(X_{n,1},\ldots,X_{n,n})\sim P_{\beta,n}$. Up to our knowledge, there is no useful matrix model with independent entries valid for all $\beta$. However, it is possible as in the real case to use the eigenvectors of an (overdamped) Langevin dynamics, namely
$$
Lf(x)=\Delta f(x)-\nabla f(x)\cdot \nabla H(x)
$$ where $$
H(x)=n\frac{\beta}{2}(|x_1|^2+\cdots+|x_n|^2)-\frac{\beta}{4}\sum_{i\neq j}\log|x_i-x_j|^2.
$$
Now $\nabla_{x_i}H(x)=n\beta x_i-\beta\sum_{j\neq i}\frac{x_i-x_j}{|x_i-x_j|^2}$, which gives $$L=\sum_{i=1}^n\partial^2_{x_i}-n\beta\sum_{i=1}^nx_i\cdot\partial_{x_i}+\frac{\beta}{2}\sum_{j\neq i}\frac{(x_i-x_j)\cdot(\partial_{x_i}-\partial_{x_j})}{|x_i-x_j|^2}.$$ As in the real case, the first two terms still form an Ornstein-Uhlenbeck operator. Certain special symmetric polynomials are eigenvectors, such as $\Re(x_1+\cdots+x_n)$, $\Im(x_1+\cdots+x_n)$ and $|x_1|^2+\cdots+|x_n|^2+c$ for a suitable constant $c$. More precisely, we have $$ L(\Re(x_1+\cdots+x_n))=-n\beta\Re(x_1+\cdots+x_n)$$ and $$L(\Im(x_1+\cdots+x_n))=-n\beta\Im(x_1+\cdots+x_n).$$ Similarly we find
$$
L(|x_1|^2+\cdots+|x_n|^2)=4n-2n\beta(|x_1|^2+\cdots+|x_n|^2)+\beta n(n-1).
$$ Let $S={(S_t)}_{t\geq0}$ be the stochastic process on $\mathbb{R}^{2n}$ solution of the stochastic differential equation $$\mathrm{d}S_t=\sqrt{2}\mathrm{d}B_t-\nabla H(S_t)\mathrm{d}t$$ where $B={(B_t)}_{t\geq0}$ is a standard Brownian motion. By Itô formula, for all $f\in\mathcal{C}^2(\mathbb{R}^{2n},\mathbb{R})$, $$\mathrm{d}f(S_t)=\sqrt{2}\nabla f(S_t)\cdot \mathrm{d}B_t+(Lf)(S_t)\mathrm{d}t.$$

If $f(x_1,\ldots,x_n)=\Re(x_1+\cdots+x_n)$ then $U_t:=\Re(S_{t,1}+\cdots+S_{t,n})$ solves $$\mathrm{d}U_t=\sqrt{2 n}\mathrm{d}W_t-n\beta U_t\mathrm{d}t$$ where $W={(\frac{\Re(B_{t,1}+\cdots+B_{t,n})}{\sqrt{n}})}_{t\geq0}$ is a standard Brownian motion. Thus $U$ is an Ornstein-Uhlenbeck process. Since $U_\infty$ has the law of $\Re(X_{n,1}+\cdots+X_{n,n})$, we get that $$\Re(X_{n,1}+\cdots+X_{n,n})\sim\mathcal{N}(0,\beta^{-1}).$$ By doing the same for $\Im$, we get that $\Re(X_{n,1}+\cdots+X_{n,n})$ and $\Im(X_{n,1}+\cdots+X_{n,n})$ are independent and of same law $\mathcal{N}(0,\beta^{-1})$ and therefore $$X_{n,1}+\cdots+X_{n,n}\sim\mathcal{N}(0,\beta^{-1}I_2).$$

If $f(x_1,\ldots,x_n)=|x_1|^2+\cdots+|x_n|^2$ then $V_t:=|S_{t,1}|^2+\cdots+|S_{t,n}|^2=|S_t|^2$ solves $$\mathrm{d}V_t=\sqrt{2}2\sqrt{V_t}\mathrm{d}W_t+2n\beta\left(\frac{2}{\beta}+\frac{n-1}{2}-V_t\right)\mathrm{d}t$$ where $W={(\frac{S_t}{|S_t|}B_t)}_{t\geq0}$ is a standard Brownian motion. Thus $V$ is a Cox-Ingersoll-Ross process. The generator of such a process is a Laguerre operator, and the invariant distribution is a Gamma law. Since $V_\infty$ has the law of $|X_n|^2$, we obtain the formula $$|X_n|^2\sim\mathrm{Gamma}\left(n+\frac{\beta n(n-1)}{4},\frac{n\beta}{2}\right).$$ When $\beta=2$, this is compatible with the observation of Kostlan that $n|X_n|^2$ has the law of a sum of $n$ independent random variables of law $\mathrm{Gamma}(1,1),\ldots,\mathrm{Gamma}(n,1)$.

We can compute quickly the mean using the invariance of $P_{\beta,n}$ with respect to $L$, namely
$$
0=4n+\beta n(n-1)-2n\beta\mathbb{E}(|X_{n,1}|^2+\cdots+|X_{n,n}|^2)
$$
gives
$$
\mathbb{E}(|X_n|^2)=\mathbb{E}(|X_{n,1}|^2+\cdots+|X_{n,n}|^2)=\frac{2}{\beta}+\frac{n-1}{2}.
$$ 

Note. A squared Ornstein-Uhlenbeck (OU) process is a Cox-Ingersoll-Ross (CIR) process. In this sense CIR processes play for OU processes the role played for BM by squared Bessel processes.

Further reading. 

  • Chafaï & Lehec, On Poincaré and logarithmic Sobolev inequalities for a class of singular Gibbs measures, arXiv:1805.00708
  • Bolley & Chafaï & Fontbona, Dynamics of a planar Coulomb gas, arXiv:​​1706.08776

 

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