{"id":11300,"date":"2019-02-15T13:51:27","date_gmt":"2019-02-15T12:51:27","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=11300"},"modified":"2019-02-15T16:55:17","modified_gmt":"2019-02-15T15:55:17","slug":"projections","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2019\/02\/15\/projections\/","title":{"rendered":"Projections"},"content":{"rendered":"\r\n
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\"\"<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

This post is about a remarkable projective property of certain Boltzmann-Gibbs measures.<\/p>\r\n\r\n\r\n\r\n

The model.<\/strong> Let $d\\geq1$ and $g:\\mathbb{R}^d\\to(-\\infty,+\\infty]$ be continuous with $g(x)<\\infty$ for all $x\\neq0$. For all $\\beta>0$, $n\\geq2$, let $P_n$ be the probability measure on $(\\mathbb{R}^d)^n$ with density proportional to $$(x_1,\\ldots,x_n)\\in(\\mathbb{R}^d)^n\\mapsto\\exp(-\\beta H(x_1,\\ldots,x_n))$$ where $$H(x_1,\\ldots,x_n)=\\sum_{i=1}^n\\frac{1}{2}|x_i|^2+\\sum_{i\\neq j}g(x_i-x_j).$$ Let $$X=(X_{n,1},\\ldots,X_{n,n})\\sim P_n.$$<\/p>\r\n

A projection. <\/strong>Let $p:\\mathbb{R}^d\\to\\mathbb{R}^d$ be an orthogonal projection on a subspace $E\\subset\\mathbb{R}^d$. Let $\\pi$ and $\\pi^\\perp$ be the orthogonal projections on the subspaces $$L=\\{(p(z),\\ldots,p(z))\\in(\\mathbb{R}^d)^n:z\\in\\mathbb{R}^d\\}\\quad\\text{and}\\quad L^\\perp.$$ We have, for all $x\\in(\\mathbb{R}^d)^n$,<\/p>\r\n

$$\\pi(x)=(p(s(x)),\\ldots,p(s(x)))\\quad\\text{where}\\quad s(x):=\\frac{x_1+\\cdots+x_n}{n}\\in\\mathbb{R}^d.$$<\/p>\r\n

Indeed, we have
$$
L^\\perp=\\{(x_1,\\ldots,x_n)\\in(\\mathbb{R}^d)^n:(x_1+\\cdots+x_n)\\cdot
p(z)=0\\text{ for all }z\\in\\mathbb{R}^d\\},
$$
and for all $x=(x_1,\\ldots,x_n)\\in(\\mathbb{R}^d)^n$ and all $(z,\\ldots,z)\\in L$, $z\\in E$, we have
\\begin{align}((x_1,\\ldots,x_n)-(p(s(x)),\\ldots,p(s(x))))\\cdot(z,\\ldots,z) &=\\sum_{i=1}^nx_i\\cdot z-np(s(x))\\cdot z\\\\&=\\sum_{i=1}^nx_i\\cdot z-\\sum_{i=1}^np(x_i)\\cdot z\\\\&=\\sum_{i=1}^n(x_i-p(x_i))\\cdot z\\\\&=\\sum_{i=1}^n0=0.\\end{align}\u00a0<\/em><\/p>\r\n\r\n\r\n\r\n

Example 1: if $E=\\mathbb{R}^d$ then $p(x)=x$.<\/p>\r\n

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

Example 2 : $E=\\mathbb{R}z$ for $z\\in\\mathbb{R}^d$ with $|z|=1$. Then $p(x)=(x\\cdot z)z$.\u200b\u200b<\/p>\r\n

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

The statement.<\/strong>\u00a0 If all the ingredients are as above, then:<\/p>\r\n\r\n\r\n\r\n