{"id":11204,"date":"2019-02-15T10:12:44","date_gmt":"2019-02-15T09:12:44","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=11204"},"modified":"2019-03-11T15:54:57","modified_gmt":"2019-03-11T14:54:57","slug":"aspects-of-beta-ensembles","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2019\/02\/15\/aspects-of-beta-ensembles\/","title":{"rendered":"Aspects of Beta Ensembles"},"content":{"rendered":"\r\n
\r\n
\"Aspirateur\r\n
Ceci n'est pas un ensemble de Dyson.<\/figcaption>\r\n<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

This post is devoted to computations for beta ensembles from random matrix theory.<\/p>\r\n\r\n\r\n\r\n

Real case.<\/strong> Following MR2813333<\/a>, or MR2325917<\/a> MR1936554<\/a> 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<\/a>. Following MR1936554<\/a>, 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 $$X_{n,1}+\\cdots+X_{n,n}=\\mathrm{Trace}(M_{\\beta,n})\\sim\\mathcal{N}\\left(0,\\frac{2}{\\beta}\\right).$$<\/p>\r\n\r\n\r\n\r\n

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 $$\\mathbb{E}((X_{n,1}+\\cdots+X_{n,n})^2)=\\frac{2}{\\beta}\\quad\\text{and}\\quad\\mathbb{E}(X_{n,1}^2+\\cdots+X_{n,n}^2)=\\frac{2}{\\beta}+n-1.$$<\/p>\r\n\r\n\r\n\r\n

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

Dynamics.<\/strong> 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 $$L(x_1^2+\\cdots+x_n^2)=2n+\\beta n(n-1)-n\\beta(x_1^2+\\cdots+x_n^2).$$<\/p>\r\n\r\n\r\n\r\n

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 It\u00f4 formula gives, for $f\\in\\mathcal{C}^2(\\mathbb{R}^n,\\mathbb{R})$, $$\\mathrm{d}f(S_t)=\\sqrt{2}\\nabla f(S_t)\\cdot \\mathrm{d}B_t+(Lf)(S_t)\\mathrm{d}t.$$<\/p>\r\n\r\n\r\n\r\n

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<\/a>. Since $U_\\infty$ has the law of $X_n$, we recover by this way the formula $X_n\\sim\\mathcal{N}(0,2\\beta^{-1})$.<\/p>\r\n\r\n\r\n\r\n

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<\/a>. 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 $|X_n|^2\\sim\\mathrm{Gamma}(...)$.<\/p>\r\n\r\n\r\n\r\n

Complex case.<\/strong> 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\u00a0eigenvectors 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 It\u00f4 formula, for all $f\\in\\mathcal{C}^2(\\mathbb{R}^{2n},\\mathbb{R})$, $$\\mathrm{d}f(S_t)=\\sqrt{2}\\nabla f(S_t)\\cdot \\mathrm{d}B_t+(Lf)(S_t)\\mathrm{d}t.$$<\/p>\r\n

\r\n\r\n<\/p>\r\n

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<\/a>. 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 $$X_{n,1}+\\cdots+X_{n,n}\\sim\\mathcal{N}(0,\\beta^{-1}I_2).$$<\/p>\r\n

\r\n\r\n<\/p>\r\n

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<\/a>. 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 $\\mathrm{Gamma}(1,1),\\ldots,\\mathrm{Gamma}(n,1)$.<\/p>\r\n

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
$$
\\mathbb{E}(|X_n|^2)=\\mathbb{E}(|X_{n,1}|^2+\\cdots+|X_{n,n}|^2)=\\frac{2}{\\beta}+\\frac{n-1}{2}.
$$\u00a0<\/p>\r\n\r\n\r\n\r\n

Note.<\/strong>\u00a0A 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.<\/p>\r\n

Further reading.\u00a0<\/strong><\/p>\r\n\r\n\r\n\r\n