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The Anti-Symmetric GUE Minor Process
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). (Analys)
(Discrete Mathematics & Algebraic Combinatorics)
2009 (English)In: Moscow Mathematical Journal, ISSN 1609-3321, E-ISSN 1609-4514, Vol. 9, no 4, 749-774 p.Article in journal (Refereed) Published
Abstract [en]

Our study is initiated by a multi-component particle system underlyingthe tiling of a half hexagon by three species of rhombi. In this particlesystem species $j$ consists of $\lfloor j/2 \rfloor$ particles which areinterlaced with neigbouring species. The joint probability densityfunction (PDF) for this particle system is obtained, and is shown in asuitable scaling limit to coincide with the joint eigenvalue PDFfor the process formed by the successive minors of anti-symmetric GUEmatrices, which in turn we compute from first principles. The correlationsfor this process are determinantal and we give an explicit formula for thecorresponding correlation kernel in terms of Hermite polynomials.Scaling limits of the latter are computed, giving rise to theAiry kernel, extended Airy kernel and bead kernel at the soft edge and inthe bulk, as well as a new kernel at the hard edge.

Place, publisher, year, edition, pages
2009. Vol. 9, no 4, 749-774 p.
Keyword [en]
Anti-symmetric GUE, lozenge tilings, interlaced particles
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-9831ISI: 000273089600002OAI: oai:DiVA.org:kth-9831DiVA: diva2:133479
Note
QC 20100804Available from: 2009-01-19 Created: 2009-01-12 Last updated: 2017-12-14Bibliographically approved
In thesis
1. Interlaced particles in tilings and random matrices
Open this publication in new window or tab >>Interlaced particles in tilings and random matrices
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of three articles all relatedin some way to eigenvalues of random matrices and theirprincipal minors and also to tilings of various planar regions with dominoes or rhombuses.Consider an $N\times N$ matrix $H_N=[h_{ij}]_{i,j=1}^N$ from the Gaussian unitary ensemble (GUE). Denote its principal minors (submatrices in the upper left corner) by $H_n=[h_{ij}]_{i,j=1}^n$ for  $n=1$, \dots, $N$. We show in paper A that  all the $N(N+1)/2$ eigenvaluesof $H_1$, \dots, $H_N$ form a determinantal process on $N$ copies of the real line $\mathbb{R}$. We also show that this distribution arises as a scaling limit in tilings of an Aztec diamond with dominoes.We discuss a corresponding result for rhombus tilings of a hexagonwhich was later proved by Okounkov and Reshtikhin. We give a new proof of that statement in the introductionto this thesis.In paper B we perform a similar analysis for the Anti-symmetric Gaussian unitary ensemble (A-GUE). We show that the positive eigenvalues of an $N\times N$ A-GUE matrix andits principal minors form a determinantal processon $N$ copies of the positive real line $\mathbb{R}^+$.We also show that this distribution of all these eigenvalues appears as a scaling limit of tilings of half a hexagon with rhombuses. In paper C we study the shuffling algorithm for tilings of an Aztec diamond. This leads to the study of an interacting set of interlacedparticles that evolve in time. We conjecture that the diffusion limit of thisprocess is a process studied by Warrenand establish some results in this direction.

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. vii, 23 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 08:14
Keyword
Interlaced particles, GUE, Anti-symmetric GUE, domino tilings, lozenge tilings
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-9834 (URN)978-91-7415-212-8 (ISBN)
Public defence
2009-02-06, Kollegiesalen, F3, KTH, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20100804Available from: 2009-01-19 Created: 2009-01-12 Last updated: 2010-08-04Bibliographically approved

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