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An unexpected distribution

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 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 $x_i=\sqrt[a]{\beta/(\theta+\beta)}y_i$,
\begin{multline*}
\int_{(\mathbb{R}^d)^n}\mathrm{e}^{-(\theta+\beta)\sum_iV(x_i)}\prod_{i<j}W(x_i-x_j)\mathrm{d}x\\
=\Bigr(\frac{\beta}{\theta+\beta}\Bigr)^{\frac{nd}{a}+\frac{n(n-1)a}{2b}}
\int_{(\mathbb{R}^d)^n}\mathrm{e}^{-\beta\sum_iV(y_i)}\prod_{i<j}W(y_i-y_j)\mathrm{d}y.
\end{multline*}
We recognize the Laplace transform of a Gamma distribution, since
\[
\int_0^\infty\mathrm{e}^{-\theta u}u^{\alpha-1}\mathrm{e}^{-\beta u}\mathrm{d}u
=\int_0^\infty u^{\alpha-1}\mathrm{e}^{-(\theta+\beta)u}\mathrm{d}u
=\Bigr(\frac{\beta}{\theta+\beta}\Bigr)^\alpha\frac{\Gamma(\alpha)}{\beta^\alpha},
\]and we obtain
\[
\sum_iV(X_i)\sim\mathrm{Gamma}\Bigr(\frac{nd}{a}+\frac{n(n-1)b}{2a},\beta\Bigr).
\]
A 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.

We could generalize even more,  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 $$V(X)=V(X_1,\ldots,X_n)\sim\mathrm{Gamma}\Bigr((n+b)\frac{d}{a},\beta\Bigr).$$

Epilogue. 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 with François Bolley and Joaquín Fontbona. I included this computation in a talk at Oberwolfach, and mentioned that I do not have any other proof. After my talk, during the break, Bálint Virág looked at me and said Are you sure that there is no other proof? 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 Gérard Letac. This is a typical story in the life of a mathematician.

Last Updated on 2021-04-14

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