Celebratio Mathematica — Liggett — Complete Bibliography

Complete Bibliography

Works connected to Vladas Sidoravicius

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T. M. Liggett: “Tagged particle distributions or how to choose a head at random,” pp. 133–162 in In and out of equilibrium: Probability with a physics flavor (Mambucaba, Brazil, 14–19 August 2000). Edited by V. Sidoravicius. Progress in Probability 51. Birkhäuser (Boston), 2002. MR 1901951 Zbl 1108.60319 incollection

Thorisson and others have proved results that imply the following: given an i.i.d. family of Bernoulli random variables indexed by \( \mathbb{Z}^d \) there exists an occupied site \( X\in \mathbb{Z}^d \) with the property that relative to its location, the other variables are still i.i.d. We raise the question of how large such an \( X \) must be. For \( d = 1 \), we prove that any \( X \) with this property satisfies \( E|X|^{\frac{1}{2}} = \infty \). Moreover, there does exist such an \( X \) with tails \( P(|X|\geq n) \) of order \( Cn^{-\frac{1}{2}} \) so these results are essentially best possible. Analogous results for the Poisson process in one dimension are given. The corresponding problem for stationary ergodic sequences is considered also. This project was motivated by some tagged particle problems.