×

zbMATH — the first resource for mathematics

How to find an extra head: Optimal random shifts of Bernoulli and Poisson random fields. (English) Zbl 1019.60048
From the authors’ summary: We consider the following problem: given an i.i.d. family of Bernoulli random variables indexed by \(Z^d\), find a random occupied site \(X\in Z^d\) such that relative to \(X\), the other random variables are still i.i.d. Bernoulli. Results of Thorisson imply that such an \(X\) exists for all \(d\). Liggett proved that for \(d=1\), there exists an \(X\) with tails \(P(|X|\geq t)\) of order \(ct^{-1/2}\), but none with finite 1/2th moment. We prove that for general \(d\) there exists a solution with tails of order \(ct^{-d/2}\), while for \(d=2\) there is none with finite first moment. We also prove analogous results for a continuum version of the same problem. Finally we prove a result which strongly suggests that the tail behavior mentioned above is the best possible for all \(d\).

MSC:
60G60 Random fields
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDF BibTeX XML Cite