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A suspension of a point process on $\mathbb{R}^d$ is a measure preserving automorphism of the (distribution of the) point process which is determined by a map $T:\mathbb{R}^d\to\mathbb{R}^d$. The suspension takes $\nu=\left\{x\right\}_{x\in\nu}$ to $T_*\nu=\left\{Tx\right\}_{x\in\nu}$. The most famous example is in the case of Poisson point processes and it is called Poisson suspensions. An account of it is given in the ergodic theory book of Kornfeld, Fomin and Sinai. Here every $T$ which preserves Lebesgue measure gives a Poisson suspension.

My question is if there are other natural examples of translation invariant point processes which have other suspensions then the ones from $T$ which are translations/rotations of $\mathbb{R}^d$?

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  • $\begingroup$ I’m not quite clear what you’re asking for. One example of a point process takes values in the translates of $\Z^d$ and $\R^d$ acts by translation. Or, more generally, the Ambrose-Kakutani theorem can be interpreted as saying that for any measure-preserving action of $\R^d$ on a space $X$, there is a “section” $S$ of $X$, such that the values of $t\in\R^d$ for which $T^tx\in S$ is a discrete set. $\endgroup$
    – Anthony Quas
    Commented Feb 13, 2018 at 18:26
  • $\begingroup$ I mean Point processes such as zeroes set of Gaussian analytic functions/ determinental point processes (eigenvalue of large random matrices), Gibbs measures and so on. $\endgroup$
    – user103342
    Commented Feb 14, 2018 at 18:13

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