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On the Shuffling Algorithm for Domino Tilings
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). (Analys)
2010 (English)In: Electronic Journal of Probability, ISSN 1083-6489, E-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.

##### Place, publisher, year, edition, pages
2010. Vol. 15, 75-95 p.
##### Keyword [en]
Aztec diamond, domino tilings, interlaced particles, GUE
Mathematics
##### Identifiers
ISI: 000273780800002Scopus ID: 2-s2.0-77952804389OAI: oai:DiVA.org:kth-9829DiVA: diva2:133475
##### 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
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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