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Eigenvalues of GUE minors
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). (Analys)
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). (Analys)ORCID iD: 0000-0003-2943-7006
2006 (English)In: Electronic Journal of Probability, ISSN 1083-6489, Vol. 11, no 50, 1342-1371 p.Article in journal (Refereed) Published
##### Abstract [en]

Consider an infinite random matrix H = (hij)o<i,j picked from the Gaussian Unitary Ensemble (GUE). Denote its main minors by Hi = (hrs)1≤r,s≤i and let the j:th largest eigenvalue of Hi be μji. We show that the configuration of all these eigenvalues (i, μji) form a determinantal point process on ℕ × ℝ. Furthermore we show that this process can be obtained as the scaling limit in random tilings of the Aztec diamond close to the boundary. We also discuss the corresponding limit for random lozenge tilings of a hexagon.

##### Place, publisher, year, edition, pages
2006. Vol. 11, no 50, 1342-1371 p.
##### Keyword [en]
GUE, Aztec diamond, domino tilings, lozenge tilings, interlaced particles
Mathematics
##### Identifiers
ISI: 000243135900001ScopusID: 2-s2.0-33845777983OAI: oai:DiVA.org:kth-9830DiVA: diva2:133476
##### Note
This version also contains the corrections from the erratum published in the same journal. QC 20100804Available from: 2009-01-19 Created: 2009-01-12 Last updated: 2010-12-06Bibliographically 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
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)
##### Note
QC 20100804Available from: 2009-01-19 Created: 2009-01-12 Last updated: 2010-08-04Bibliographically approved

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Nordenstam, EricJohansson, Kurt
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Electronic Journal of Probability
Mathematics