The Annals of Probability

How to Find an extra Head: Optimal Random Shifts of Bernoulli and Poisson Random Fields

Alexander E. Holroyd and Thomas M. Liggett

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We consider the following problem:given an i.i.d. family of Bernoulli random variables indexed by $\mathbb{Z}^d$, find a random occupied site $X \in \mathbb{Z}^d$ 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 $1/2$th 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$d$.

Article information

Ann. Probab., Volume 29, Number 4 (2001), 1405-1425.

First available in Project Euclid: 5 March 2002

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Zentralblatt MATH identifier

Primary: 60G60: Random fields
Secondary: 60G55: Point processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

product measure random shift Poisson process tagged particle shift coupling


Holroyd, Alexander E.; Liggett, Thomas M. How to Find an extra Head: Optimal Random Shifts of Bernoulli and Poisson Random Fields. Ann. Probab. 29 (2001), no. 4, 1405--1425. doi:10.1214/aop/1015345755.

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