Rigidity and non-rigidity for uniform perturbed lattice

TL;DR

This paper investigates the properties of rigidity and tolerance in uniform perturbed lattices, demonstrating phase transitions in higher dimensions and establishing measure absolute continuity in certain cases.

Contribution

It extends the study of phase transitions in perturbed lattices to the uniform distribution case, showing measure absolute continuity and rigidity-tolerance phase transitions in dimensions four and higher.

Findings

Mutual absolute continuity of measures without one point in $d ext{geq}4$

Existence of phase transitions related to tolerance in $d ext{geq}4$

Extension of rigidity results from Gaussian to uniform perturbations

Abstract

A point process on the topological space S is at most countable subset without a random accumulation point in S. In studies of the point processes, there is a problem of seeing the properties of rigidity and tolerance, and this problem is studied actively in recent years. When let $\displaystyle\mathbb{Z}(\mathbf{X}):=(z+X_z)_{z\in\mathbb{Z}^d}$ be the perturbed lattice that is the lattice $\mathbb{Z}^d$ perturbed by independent and identically random variables $(X_z)_{z\in\mathbb{Z}^d}$ taking values in $\mathbb{R}^d$, regarding the Gaussian perturbed lattice, Peres and Sly showed that there exist the phase transitions with respect to the rigidity and the tolerance when $d\geq 3$ in recent paper. In this paper, when random variables $(X_z)_{z\in\mathbb{Z}^d}$ follow uniform distribution, we show the mutually absolute continuity of the measure without one point and the original measure on a restricted set of spaces of the point process in $d\geq 4$. Also, as a consequence of the above, we show that when random variables $(X_z)_{z\in\mathbb{Z}^d}$ follow the uniform distribution, phase transitions related to the tolerance can be seen in $d\geq 4$.