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Asymptotic geometry of discrete interlaced patterns: Part II
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study the boundary of the liquid region L in large random lozenge tiling models defined by uniform random interlacing particle systems with general initial configuration, which lies on the line (x,1), x∈R≡∂H. We assume that the initial particle configuration converges weakly to a limiting density ϕ(x), 0≤ϕ≤1. The liquid region is given by a homeomorphism WL:L→H, the upper half plane, and we consider the extension of W−1L to H¯¯¯. Part of ∂L is given by a curve, the edge E, parametrized by intervals in ∂H, and this corresponds to points where ϕ is identical to 0 or 1. If 0<ϕ<1, the non-trivial support, there are two cases. Either W−1L(w) has the limit (x,1) as w→x non-tangentially and we have a \emph{regular point}, or we have what we call a singular point. In this case W−1L does not extend continuously to H¯¯¯. Singular points give rise to parts of ∂L not given by E and which can border a frozen region, or be "inside" the liquid region. This shows that in general the boundary of ∂L can be very complicated. We expect that on the singular parts of ∂L we do not get a universal point process like the Airy or the extended Sine kernel point processes. Furthermore, E and the singular parts of ∂L are shocks of the complex Burgers equation.

National Category
Natural Sciences
Research subject
URN: urn:nbn:se:kth:diva-176649OAI: diva2:868186
Knut and Alice Wallenberg Foundation, 2010.0063

QC 20151110

Available from: 2015-11-09 Created: 2015-11-09 Last updated: 2015-11-10Bibliographically approved
In thesis
1. On Uniformly Random Discrete Interlacing Systems
Open this publication in new window or tab >>On Uniformly Random Discrete Interlacing Systems
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns uniformly random discrete interlacing particle sys-tems and their connections to certain random lozenge tiling models. In par-ticular it contains the first derivation of a relatively unknown universal scalinglimit, which we call the Cusp-Airy process, of certain lozenge tiling modelsat a cusp point. In addition it contains a characterization of the geometryof the macroscopic behavior of uniformly random discrete interlaced parti-cle systems that, although not complete, shows many new and interestingfeatures.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2015. vii, 56 p.
TRITA-MAT-A, 2015:12
National Category
Research subject
urn:nbn:se:kth:diva-176652 (URN)978-91-7595-732-6 (ISBN)
Public defence
2015-12-04, Sal D3, Lindstedtsvägen 5, KTH, Stockholm, 13:00 (English)

QC 20151110

Available from: 2015-11-10 Created: 2015-11-09 Last updated: 2015-11-10Bibliographically approved

Open Access in DiVA

fulltext(828 kB)