A unitary extension of virtual permutations

Paul Bourgade, Joseph Najnudel, Ashkan Nikeghbali

Research output: Contribution to journalArticle

Abstract

Analogously to the space of virtual permutations [5], we define projective limits of isometries: these sequences of unitary operators are natural in the sense that they minimize the rank norm between successive matrices of increasing sizes. The space of virtual isometries we construct this way may be viewed as a natural extension of the space of virtual permutations of Kerov et al. [5] as well as an extension of the space of virtual isometries of Neretin [9]. We then derive with purely probabilistic methods an almost sure convergence for these random matrices under the Haar measure: for a coherent Haar measure on virtual isometries, the smallest normalized eigenangles converge almost surely to a point process whose correlation function is given by the sine kernel. This almost sure convergence actually holds for a larger class of measures as is proved by Borodin and Olshanski [1]. We give a different proof, probabilistic in the sense that it makes use of martingale arguments and shows how the eigenangles interlace when going from dimension n to n+1. Our method also proves that for some universal constant ε>0, the rate of convergence is almost surely dominated by n when the dimension n goes to infinity.

Original languageEnglish (US)
Pages (from-to)4101-4134
Number of pages34
JournalInternational Mathematics Research Notices
Volume2013
Issue number18
DOIs
StatePublished - Aug 2013

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Permutation
Isometry
Almost Sure Convergence
Haar Measure
Projective Limit
Unitary Operator
Probabilistic Methods
Point Process
Natural Extension
Random Matrices
Martingale
Correlation Function
Rate of Convergence
Infinity
kernel
Converge
Minimise
Norm

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

A unitary extension of virtual permutations. / Bourgade, Paul; Najnudel, Joseph; Nikeghbali, Ashkan.

In: International Mathematics Research Notices, Vol. 2013, No. 18, 08.2013, p. 4101-4134.

Research output: Contribution to journalArticle

Bourgade, Paul ; Najnudel, Joseph ; Nikeghbali, Ashkan. / A unitary extension of virtual permutations. In: International Mathematics Research Notices. 2013 ; Vol. 2013, No. 18. pp. 4101-4134.
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