Periodic Lorentz gas with small scatterers

Probability Theory and Related Fields, vol. 186, pp. 159–219

Abstract

Abstract We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time n tends to infinity, the scatterer size $$\rho $$ ρ may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive $$\sqrt{n\log n}$$ n log n scaling (i) for fixed infinite horizon configurations—letting first $$n\rightarrow \infty $$ n → ∞ and then $$\rho \rightarrow 0$$ ρ → 0 —studied e.g. by Szász and Varjú (J Stat Phys 129(1):59–80, 2007) and (ii) Boltzmann–Grad type situations—letting first $$\rho \rightarrow 0$$ ρ → 0 and then $$n\rightarrow \infty $$ n → ∞ —studied by Marklof and Tóth (Commun Math Phys 347(3):933–981, 2016) .

Authors 3

1. Péter Bálint corresponding

   Budapest University of Technology and Economics

   Affiliation as printed

   Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., Budapest, 1111 Hungary

   MTA-BME Stochastics Research Group, Budapest University of Technology and Economics, Műegyetem rkp. 3., Budapest, 1111 Hungary

   Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Budapest, Hungary

   MTA-BME Stochastics Research Group, Budapest University of Technology and Economics, Budapest, Hungary

2. Henk Bruin

   University of Vienna

   Affiliation as printed

   Faculty of Mathematics, University of Vienna, Oskar Morgensternplatz 1, 1090 Vienna, Austria

   Faculty of Mathematics, University of Vienna, Vienna, Austria

3. Dalia Terhesiu Aachen Institute of Mathematics, University of Leiden Institute of Mathematics, University of Leiden, Leiden

   Leiden University

   Affiliation as printed

   Institute of Mathematics, University of Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands

   Institute of Mathematics, University of Leiden, Leiden, The Netherlands