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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 \(d\).

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