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Bridges and random truncations of random matrices

Abstract : Let U be a Haar distributed matrix in U(n) or O(n). In a previous paper, we proved that after centering, the two-parameter process T (n) (s, t) = i≤⌊ns⌋,j≤⌊nt⌋ |Uij | 2 , s, t ∈ [0, 1] converges in distribution to the bivariate tied-down Brownian bridge. In the present paper, we replace the deterministic truncation of U by a random one, in which each row (resp. column) is chosen with probability s (resp. t) independently. We prove that the corresponding two-parameter process, after centering and normalization by n −1/2 converges to a Gaussian process. On the way we meet other interesting conver-gences.
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Vincent Beffara, Catherine Donati-Martin, Alain Rouault. Bridges and random truncations of random matrices. Random Matrices: Theory and Applications, World Scientific, 2014, 03 (02), ⟨10.1142/S2010326314500063⟩. ⟨hal-01693144⟩

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