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

Abstract

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$d$.