Spectrum of random permutation matrices

Abstract : In this thesis, our goal is to study random matrices related to permutations. We tackle the study of their spectra in various ways, and at different scales. First, we extend the work of Wieand about the numbers of eigenvalues lying in some fixed arcs of the unit circle. We take advantage of the results of Ben Arous and Dang on the linear statistics of the spectrum of permutation matrices for a one-parameter family of deformations of the uniform law on the symmetric group, called Ewens' measures. One of the most innovative parts of our work is the generalization to non-fixed arcs. Indeed we get similar results when we let the lengths of the arcs decrease to zero slower than 1/n. Then, we look at the spectrum at microscopic scale. Inspired by the work of Najnudel and Nikeghbali about the convergence of empirical measures of rescaled eigenangles, we give a meaning to the convergence in terms of indicator functions of intervals. From the limiting point process, we show that the number of points in any interval is asymptotically normal as the length of the interval goes to infinity. Finally, we adapt some results of Chhaibi, Najnudel and Nikeghbali on the characteristic polynomial of the CUE at microscopic scale, and develop them in our framework. Analogously but with different techniques of proof, we get that the characteristic polynomials converge to entire functions, and this for a large family of laws including the Ewens' measures.
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Valentin Bahier. Spectrum of random permutation matrices. Combinatorics [math.CO]. Université Paul Sabatier - Toulouse III, 2018. English. ⟨NNT : 2018TOU30069⟩. ⟨tel-02084015⟩

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