We prove a central limit theorem for the mesoscopic linear statistics of N×N\documentclass[12pt]{minimal}
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\begin{document}$$N\times N$$\end{document} Wigner matrices H satisfying E|Hij|2=1/N\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {E}}|H_{ij}|^2=1/N$$\end{document} and EHij2=σ/N\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {E}} H_{ij}^2= \sigma /N$$\end{document}, where σ∈[-1,1]\documentclass[12pt]{minimal}
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\begin{document}$$\sigma \in [-1,1]$$\end{document}. We show that on all mesoscopic scales η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document} (1/N≪η≪1\documentclass[12pt]{minimal}
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\begin{document}$$1/N \ll \eta \ll 1$$\end{document}), the linear statistics of H have a sharp transition at 1-σ∼η\documentclass[12pt]{minimal}
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\begin{document}$$1-\sigma \sim \eta $$\end{document}. As an application, we identify the mesoscopic linear statistics of Dyson’s Brownian motion Ht\documentclass[12pt]{minimal}
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\begin{document}$$H_t$$\end{document} started from a real symmetric Wigner matrix H0\documentclass[12pt]{minimal}
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\begin{document}$$H_0$$\end{document} at any nonnegative time t∈[0,∞]\documentclass[12pt]{minimal}
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\begin{document}$$t \in [0,\infty ]$$\end{document}. In particular, we obtain the transition from the central limit theorem for GOE to the one for GUE at time t∼η\documentclass[12pt]{minimal}
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\begin{document}$$t \sim \eta $$\end{document}.
