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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 $$d$$.

##### MSC:
 60G60 Random fields 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60K35 Interacting random processes; statistical mechanics type models; percolation theory
##### Keywords:
random shift; product measure; Poisson process; shift coupling