We prove limit laws for infinite horizon planar periodic Lorentz gases when, as timentends to infinity, the scatterer size\documentclass[12pt]{minimal}
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				\begin{document}$$\rho $$\end{document}ρmay 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\documentclass[12pt]{minimal}
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				\begin{document}$$\sqrt{n\log n}$$\end{document}nlognscaling (i) for fixed infinite horizon configurations—letting first\documentclass[12pt]{minimal}
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				\begin{document}$$n\rightarrow \infty $$\end{document}n→∞and then\documentclass[12pt]{minimal}
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				\begin{document}$$\rho \rightarrow 0$$\end{document}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\documentclass[12pt]{minimal}
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				\begin{document}$$\rho \rightarrow 0$$\end{document}ρ→0and then\documentclass[12pt]{minimal}
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				\begin{document}$$n\rightarrow \infty $$\end{document}n→∞—studied by Marklof and Tóth (Commun Math Phys 347(3):933–981, 2016) .