Abstract
In this note, we consider the fluctuation theorem for \(X_{f_n}^{(n)}:=\sum _{i=1}^n f(\lambda _i)I(\lambda _i\ge \theta _n)\), where \(\lambda _i, i=1,\ldots ,n\) are eigenvalues from a Wigner matrix and \(\theta _n\rightarrow 2^-\). We prove that in the edge case \(X_{f_n}^{(n)}\) behaves like the counting function of Wigner matrix. Our results can be viewed as a complement of Bao et al. (J Stat Phys 150(1):88–129, 2013).
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Acknowledgments
The authors are grateful to Han, X. for useful discussions and also express their gratitude to two anonymous referees for useful comments which improve the presentation of the paper. G.M. Pan was partially supported by a MOE Tier 2 grant 2014-T2-2-060 and by a MOE Tier 1 Grant RG25/14 at the Nanyang Technological University, Singapore; S.C. Wang was supported by the Project Funded by China Postdoctoral Science Foundation (Grant No. 2015M580713); W. Zhou was partially supported by a grant R-155-000-165-112 at the National University of Singapore.
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Pan, G., Wang, S. & Zhou, W. Fluctuations of Linear Eigenvalues Statistics for Wigner Matrices: Edge Case. J Stat Phys 165, 507–520 (2016). https://doi.org/10.1007/s10955-016-1618-5
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DOI: https://doi.org/10.1007/s10955-016-1618-5