{"id":12008,"date":"2019-12-15T14:09:57","date_gmt":"2019-12-15T13:09:57","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=12008"},"modified":"2021-12-17T22:37:59","modified_gmt":"2021-12-17T21:37:59","slug":"an-unexpected-distribution","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2019\/12\/15\/an-unexpected-distribution\/","title":{"rendered":"An unexpected distribution"},"content":{"rendered":"

\u03a3<\/span><\/a><\/p>\n

Let $X=(X_1,\\ldots,X_n)$ be a random vector of $(\\mathbb{R}^d)^n$ with density proportional to $$(x_1,\\ldots,x_n)\\in(\\mathbb{R}^d)^n\\mapsto\\mathrm{e}^{-\\beta\\sum_{i=1}^nV(x_i)}\\prod_{i<j}W(x_i-x_j),$$ where $V,W:\\mathbb{R}^d\\to\\mathbb{R}$ are homogeneous functions, with $W\\geq0$. This means that there exist $a,b\\geq0$ such that for all $\\lambda\\geq0$ and $x\\in\\mathbb{R}^d$, $V(\\lambda x)=\\lambda^a V(x)$ and $W(\\lambda x)=\\lambda^bW(x)$. Now, for all $\\theta>0$, by the change of variable $x_i=\\sqrt[a]{\\beta\/(\\theta+\\beta)}y_i$,
\n\\begin{multline*}
\n\\int_{(\\mathbb{R}^d)^n}\\mathrm{e}^{-(\\theta+\\beta)\\sum_iV(x_i)}\\prod_{i<j}W(x_i-x_j)\\mathrm{d}x\\\\
\n=\\Bigr(\\frac{\\beta}{\\theta+\\beta}\\Bigr)^{\\frac{nd}{a}+\\frac{n(n-1)a}{2b}}
\n\\int_{(\\mathbb{R}^d)^n}\\mathrm{e}^{-\\beta\\sum_iV(y_i)}\\prod_{i<j}W(y_i-y_j)\\mathrm{d}y.
\n\\end{multline*}
\nWe recognize the Laplace transform of a Gamma distribution, since
\n\\[
\n\\int_0^\\infty\\mathrm{e}^{-\\theta u}u^{\\alpha-1}\\mathrm{e}^{-\\beta u}\\mathrm{d}u
\n=\\int_0^\\infty u^{\\alpha-1}\\mathrm{e}^{-(\\theta+\\beta)u}\\mathrm{d}u
\n=\\Bigr(\\frac{\\beta}{\\theta+\\beta}\\Bigr)^\\alpha\\frac{\\Gamma(\\alpha)}{\\beta^\\alpha},
\n\\]and we obtain
\n\\[
\n\\sum_iV(X_i)\\sim\\mathrm{Gamma}\\Bigr(\\frac{nd}{a}+\\frac{n(n-1)b}{2a},\\beta\\Bigr).
\n\\]
\nA remarkable general fact! The case $V=\\frac{1}{2}\\left|\\cdot\\right|^2$ and $W=\\left|\\cdot\\right|^\\beta$ corresponds to the beta Ginibre gas of random matrix theory. The case $V=\\frac{n+1}{2}\\log(1+\\left|\\cdot\\right|^2)$ and $W=\\left|\\cdot\\right|^2$ corresponds to the Forrester--Krishnapur spherical gas of random matrix theory.<\/p>\n

We could generalize even more,\u00a0 and replace $(x_1,\\ldots,x_n)\\mapsto\\sum_iV(x_i)$ by a homogenenous $(x_1,\\ldots,x_n)\\mapsto V(x_1,\\ldots,x_n)$ and $(x_1,\\ldots,x_n)\\mapsto\\prod_{i<j}W(x_i-x_j)$ by a homogeneous $(x_1,\\ldots,x_n)\\mapsto W(x_1,\\ldots,x_n)$, in the sense that for some $a,b\\geq0$ and all $\\lambda\\geq0$, $x\\in(\\mathbb{R}^d)^n$, $V(\\lambda x)=\\lambda^aV(x)$ and $W(\\lambda x)=\\lambda^bW(x)$. In this case $X=(X_1,\\ldots,X_n)$ has density proportional to $x\\in(\\mathbb{R}^d)^n\\mapsto\\mathrm{e}^{-\\beta V(x)}W(x)$. This would hide the structure of exchangeable gas with pair-interaction that we had in mind for the examples. But this would give $$V(X)=V(X_1,\\ldots,X_n)\\sim\\mathrm{Gamma}\\Bigr((n+b)\\frac{d}{a},\\beta\\Bigr).$$<\/p>\n

Epilogue.<\/strong> My first successful attempt to compute the law of $\\sum_i V(x_i)$ for Coulomb gases was by using a Langevin dynamics in the beta Ginibre case, in relation with a Cox-Ingersoll-Ross process, an observation that goes back to my work in collaboration<\/a> with Fran\u00e7ois Bolley<\/a> and Joaqu\u00edn Fontbona<\/a>. I included this computation in a talk at Oberwolfach<\/a>, and mentioned that I do not have any other proof. After my talk, during the break, B\u00e1lint Vir\u00e1g<\/a> looked at me and said Are you sure that there is no other proof<\/em>? This remark excited my mind and I realized after few hours of intense thinking that there is a direct proof without dynamics and that it is a fairly general fact for gases, far beyond Coulomb gases. The key point is a normalizing constant trick for probability distributions, learnt twenty years ago from G\u00e9rard Letac<\/a>. This is a typical story in the life of a mathematician.<\/p>\n","protected":false},"excerpt":{"rendered":"

Σ Let $X=(X_1,\\ldots,X_n)$ be a random vector of $(\\mathbb{R}^d)^n$ with density proportional to $$(x_1,\\ldots,x_n)\\in(\\mathbb{R}^d)^n\\mapsto\\mathrm{e}^{-\\beta\\sum_{i=1}^nV(x_i)}\\prod_{i<j}W(x_i-x_j),$$ where $V,W:\\mathbb{R}^d\\to\\mathbb{R}$ are homogeneous functions, with $W\\geq0$. This means that there…<\/p>\n

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