### 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^{-ε} when the dimension n goes to infinity.

Original language | English (US) |
---|---|

Pages (from-to) | 4101-4134 |

Number of pages | 34 |

Journal | International Mathematics Research Notices |

Volume | 2013 |

Issue number | 18 |

DOIs | |

State | Published - Aug 2013 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*International Mathematics Research Notices*,

*2013*(18), 4101-4134. https://doi.org/10.1093/imrn/rns167

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

Research output: Contribution to journal › Article

*International Mathematics Research Notices*, vol. 2013, no. 18, pp. 4101-4134. https://doi.org/10.1093/imrn/rns167

}

TY - JOUR

T1 - A unitary extension of virtual permutations

AU - Bourgade, Paul

AU - Najnudel, Joseph

AU - Nikeghbali, Ashkan

PY - 2013/8

Y1 - 2013/8

N2 - 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-ε when the dimension n goes to infinity.

AB - 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-ε when the dimension n goes to infinity.

UR - http://www.scopus.com/inward/record.url?scp=84890448282&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84890448282&partnerID=8YFLogxK

U2 - 10.1093/imrn/rns167

DO - 10.1093/imrn/rns167

M3 - Article

VL - 2013

SP - 4101

EP - 4134

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 18

ER -