Asymptotic geometry of discrete interlaced patterns: Part I

Abstract

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 px, 1q, x P R " BH. We assume that the initial particle configuration converges weakly to a limiting density φpxq, 0 ď φ ď 1. The liquid region is given by a homeomorphism WL : L Ñ H, the upper half plane, and we consider the extension of W´1 L to H. Part of BL is given by a curve, the edge E , parametrized by intervals in BH, and this corresponds to points where φ is identical to 0 or 1. If 0 ă φ ă 1, the non-trivial support, there are two cases. Either W´1 L pwq has the limit px, 1q as w Ñ x non-tangentially and we have a regular point, or we have what we call a singular point. In this case W´1 L does not extend continuously to H. Singular points give rise to parts of BL 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 BL can be very complicated. We expect that on the singular parts of BL 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 BL are shocks of the complex Burgers equation.

GUE corners limit of $q$-distributed lozenge tilings

“…1 In fact, this paper considers a more general weighting of lozenge tilings related to the q-Racah univariate orthogonal polynomials. In the uniform (q = 1) case the frozen boundary is known explicitly for a much wider family of boundary conditions, e.g., see [Pet14b], [BG15], [DM15].…”

Concavity analysis of the tangent method

“…Arctic phenomena have been observed in many other models, including lozenge tilings of hexagons [3], the 6-vertex model with domain wall boundary conditions [4,5], alternating sign matrices [6], lozenge tilings of polygonal domains [7,8] and the 2-periodic Aztec diamond [9]. For dimer models on bipartite planar graphs, rather general results have been obtained in [10,11].…”

Lozenge Tilings of Hexagons with Cuts and Asymptotic Fluctuations: a New Universality Class

“…with two cuts of sizes d and d , as in Fig. 4, satisfying m 1 + m 2 = n 1 + n 2 and N = b L + c L = b R + c R and c L + d = c R + d , the same limit (14) holds. Here also the limiting kernel L dTac , as in (9), only depends on the width ρ of the oblique strip {ρ } formed by the two cuts, the number r of dots on each oblique line in the strip {ρ }, and β as in (13),where…”

“…with r = r + δ and ρ = n 1 − m 1 + b − d = ρ. Substituting the formulas (15) in the scaling (11) gives the correct scaling for the model of Theorem 1.3. Given the kernel (9) it is very natural to ask for the (joint) density of blue dots along oblique levels τ = τ 1 , τ 2 .…”