A Periodic Hexagon Tiling Model and Non-Hermitian Orthogonal Polynomials

KTH, Matematik (Avd.) Springer 2020

We study a one-parameter family of probability measures on lozenge tilings of large regular hexagons that interpolates between the uniform measure on all possible tilings and a particular fully frozen tiling. The description of the asymptotic behavior can be separated into two regimes: the low and the high temperature regime. Our main results are the computations of the disordered regions in both regimes and the limiting densities of the different lozenges there. For low temperatures, the disordered region consists of two disjoint ellipses. In the high temperature regime the two ellipses merge into a single simply connected region. At the transition from the low to the high temperature a tacnode appears. The key to our asymptotic study is a recent approach introduced by Duits and Kuijlaars providing a double integral representation for the correlation kernel. One of the factors in the integrand is the Christoffel-Darboux kernel associated to polynomials that satisfy non-Hermitian orthogonality relations with respect to a complex-valued weight on a contour in the complex plane. We compute the asymptotic behavior of these orthogonal polynomials and the Christoffel-Darboux kernel by means of a Riemann-Hilbert analysis. After substituting the resulting asymptotic formulas into the double integral we prove our main results by classical steepest descent arguments.
QC 20201214

Mathematical Analysis; Matematisk analys; Article in journal; info:eu-repo/semantics/article; text

10.1007.s00220-020-03779-0

Open access content. Open access content
info:eu-repo/semantics/restrictedAccess

English

UPE oai:DiVA.org:kth-286534
0000-0001-6890-344x
0000-0002-7598-4521
0000-0001-6191-7769
doi:10.1007/s00220-020-03779-0
PMID 32704184
ISI:000534992400003
Scopus 2-s2.0-85085339742
1235095586

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