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A Vector-Valued Almost Sure Invariance Principle for Sinai Billiards with Random Scatterers

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Abstract

Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. Here we study a greatly simplified but related model, proposed by Arvind Ayyer and popularized by Joel Lebowitz, in which a scatterer configuration on the torus is randomly updated between collisions. Taking advantage of recent progress in the theory of time-dependent billiards on the one hand and in probability theory on the other, we prove a vector-valued almost sure invariance principle for the model. Notably, the configuration sequence can be weakly dependent and non-stationary. We provide an expression for the covariance matrix, which in the non-stationary case differs from the traditional one. We also obtain a new invariance principle for Sinai billiards (the case of fixed scatterers) with time-dependent observables, and improve the accuracy and generality of existing results.

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Authors and Affiliations

  1. Department of Mathematics, University of Rome “Tor Vergata”, Via della Ricerca Scientifica, 00133, Rome, Italy

    Mikko Stenlund

  2. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsinki, Finland

    Mikko Stenlund

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  1. Mikko Stenlund
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Correspondence to Mikko Stenlund.

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Communicated by F. Toninelli

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Stenlund, M. A Vector-Valued Almost Sure Invariance Principle for Sinai Billiards with Random Scatterers. Commun. Math. Phys. 325, 879–916 (2014). https://doi.org/10.1007/s00220-013-1870-3

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  • DOI: https://doi.org/10.1007/s00220-013-1870-3

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