Simon Machado

ASL_d(Z)-invariant point processes and their classification

A point process $X$ in $\mathbb{R}^d$ is a random variable taking values in the space of discrete subsets of $\mathbb{R}^d$. We are interested in those whose laws are invariant under large transformation groups, in particular under the affine special linear group $\mathrm{ASL}_d(\mathbb{R}) = \mathrm{SL}_d(\mathbb{R}) \ltimes \mathbb{R}^d$.

Motivated by questions from mathematical physics, Marklof asked whether one can classify all $\mathrm{ASL}_d(\mathbb{R})$-invariant point processes. The known examples fall into two essentially different families: Poisson point processes, which are chaotic, and those built from randomly shifted lattices, which are highly structured. No other examples are known.

A provocative conjecture proposes that every $\mathrm{ASL}_d(\mathbb{R})$-invariant point process arises from these two families. In this talk, I will discuss a discrete analogue of this problem for $\mathrm{ASL}_d(\mathbb{Z})$-invariant processes on $\mathbb{Z}^d$. Somewhat surprisingly, while the conjecture is expected to fail in $\mathbb{R}^d$, it holds in the discrete setting: every such process arises from congruence relations and Bernoulli thinnings.

The proof combines ideas from the Host--Kra--Ziegler theory of structure factors, unipotent dynamics, and the study of $\mathrm{SL}_d(\mathbb{Z})$-cocycles. Based on joint works with Mikolaj Fraczyk and with Asgar Jamneshan.