2019/09/04 by Yuta Arai, Arai, Yuta
Mathematics · #60G55 #82B43 #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1909.01555
openalex publication_date 2019/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℤ(X):=(z+Xz)z∈ℤd be the perturbed lattice that is the lattice ℤd perturbed by independent and identically random variables (Xz)z∈ℤd taking values in ℝd, 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≥ 3 in recent paper. In this paper, when random variables (Xz)z∈ℤd 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≥ 4. Also, as a consequence of the above, we show that when random variables (Xz)z∈ℤd follow the uniform distribution, phase transitions related to the tolerance can be seen in d≥ 4.