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Interlaced particles in tilings and random matrices
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). (Analys)
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 [en]
Interlaced particles, GUE, Anti-symmetric GUE, domino tilings, lozenge tilings
Mathematics
##### Identifiers
ISBN: 978-91-7415-212-8OAI: oai:DiVA.org:kth-9834DiVA: diva2:133494
##### 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
##### List of papers
1. Eigenvalues of GUE minors
Open this publication in new window or tab >>Eigenvalues of GUE minors
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.

##### Keyword
GUE, Aztec diamond, domino tilings, lozenge tilings, interlaced particles
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-9830 (URN)000243135900001 ()2-s2.0-33845777983 (ScopusID)
##### 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
2. The Anti-Symmetric GUE Minor Process
Open this publication in new window or tab >>The Anti-Symmetric GUE Minor Process
2009 (English)In: Moscow Mathematical Journal, ISSN 1609-3321, 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.

##### Keyword
Anti-symmetric GUE, lozenge tilings, interlaced particles
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-9831 (URN)000273089600002 ()
##### Note
QC 20100804Available from: 2009-01-19 Created: 2009-01-12 Last updated: 2011-07-06Bibliographically approved
3. On the Shuffling Algorithm for Domino Tilings
Open this publication in new window or tab >>On the Shuffling Algorithm for Domino Tilings
2010 (English)In: Electronic Journal of Probability, ISSN 1083-6489, Vol. 15, 75-95 p.Article in journal (Refereed) Published
##### Abstract [en]

We study the dynamics of a certain discrete model of interacting interlaced particles that comes from the so called shuffling algorithm for sampling arandom tiling of an Aztec diamond.  It turns out that the transition probabilitieshave a particularly convenient determinantal form. An analogous formula in a continuous setting has recently been obtained by Jon Warren studying certain model of  interlacing Brownian motions which can be used to construct Dyson's non-intersecting Brownian motion.We conjecture that Warren's model can be recovered as a scaling limit of our discrete model and prove some partial results in this direction. As an application to one of these results  we use it to  rederive the known result that random tilings of an Aztec diamond, suitably rescaled near a turning point, converge to the GUE minor process.

##### Keyword
Aztec diamond, domino tilings, interlaced particles, GUE
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-9829 (URN)000273780800002 ()2-s2.0-77952804389 (ScopusID)
##### Note
QC 20100804Available from: 2009-01-19 Created: 2009-01-12 Last updated: 2010-12-08Bibliographically approved

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