Abstract
In this chapter, we prove a central limit theorem for the position of a tagged particle in exclusion processes. This problem is a special case of a random walk in a random environment. We adopt the approach of the environment as seen from the particle introduced in Sect. 1.3. It is first shown that this position can be written as the sum of a martingale and an additive functional of the exclusion process as seen from the particle. The techniques developed in the first part of the book applied to the present context permit to show that the additive functional can be itself expressed as the sum of a martingale and a remainder which vanishes asymptotically. This observation reduces the proof of the central limit theorem for the tagged particle to a central limit theorem for martingales which has been presented in the first part of the book. A variational formula for the asymptotic variance as well as bounds are given in the last section of the chapter.
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Komorowski, T., Landim, C., Olla, S. (2012). Self-diffusion. In: Fluctuations in Markov Processes. Grundlehren der mathematischen Wissenschaften, vol 345. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29880-6_6
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