Press "Enter" to skip to content

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
\[
\sum_iV(X_i)\sim\mathrm{Gamma}\Bigr(\frac{nd}{a}+\frac{n(n-1)bd}{2a},\beta\Bigr).
\]
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 $$V(X)=V(X_1,\ldots,X_n)\sim\mathrm{Gamma}\Bigr((n+b)\frac{d}{a},\beta\Bigr).$$

One Comment

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Syntax · Style · .