On Smooth Mesoscopic Linear Statistics of the Eigenvalues of Random Permutation Matrices

Valentin Bahier and Joseph Najnudel ()
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Valentin Bahier: University of Bristol School of Mathematics
Joseph Najnudel: University of Bristol School of Mathematics

Journal of Theoretical Probability, 2022, vol. 35, issue 3, 1640-1661

Abstract: Abstract We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough test function f to all the determinations of the eigenangles of the permutations, we get a convergence in distribution when the order of the permutation tends to infinity. Two distinct kinds of limit appear: if $$f(0)\ne 0$$ f ( 0 ) ≠ 0 , we have a central limit theorem with a logarithmic variance; and if $$f(0) = 0$$ f ( 0 ) = 0 , the convergence holds without normalization and the limit involves a scale-invariant Poisson point process.

Keywords: Random permutation matrices; Linear statistics of eigenvalues; Mesoscopic scale; 15B52; 60F05 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01106-4

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