Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble

TL;DR

This work proves that edge fluctuations for uniformly random lozenge tilings of simply connected polygons converge, away from cusps and tangency points, to the Airy line ensemble under a precise rescaling. The authors develop a two-pronged strategy: (i) establish a near-optimal rigidity (height-function concentration) via a Markov alternating-dynamics scheme that preserves concentration while decomposing the domain into tractable pieces, and (ii) perform a local comparison to hexagonal tilings whose edge statistics are governed by the Airy line ensemble. A key innovation is the tilted-height-function framework, which produces barrier profiles that enable robust control of the height function and the arctic boundary under dynamics and tilt perturbations. The results yield a universal edge-statistics phenomenon for polygonal lozenge tilings, highlighting the robustness of the Airy line ensemble as a universal edge limit in the KPZ universality class and extending known hexagonal-domain universality to broad polygonal geometries.

Abstract

We consider uniformly random lozenge tilings of simply connected polygons subject to a technical assumption on their limit shape. We show that the edge statistics around any point on the arctic boundary, that is not a cusp or tangency location, converge to the Airy line ensemble. Our proof proceeds by locally comparing these edge statistics with those for a random tiling of a hexagon, which are well understood. To realize this comparison, we require a nearly optimal concentration estimate for the tiling height function, which we establish by exhibiting a certain Markov chain on the set of all tilings that preserves such concentration estimates under its dynamics.

Figures (13)

• Figure 1: Depicted above are the four scenarios for arctic curve $\mathfrak{A}$ forbidden by \ref{['pa']}.
• Figure 2: Shown to the left and middle are arctic boundaries exhibiting a single cusp. Shown to the right is an arctic boundary exhibiting two cusps that point in opposite directions, and a decomposition of that strip into overlapping regions that each have (at most) one cusp.
• Figure 3: Depicted to the right are the three types of lozenges. Depicted in the middle is a lozenge tiling of a hexagon. One may view this tiling as a packing of boxes (of the type depicted on the left) into a large corner, which gives rise to a height function (shown in the middle).
• Figure 4: Depicted to the left is an ensemble $\mathsf{Q} = ( \mathsf{q}_{-2}, \mathsf{q}_{-1}, \mathsf{q}_0, \mathsf{q}_1, \mathsf{q}_2, \mathsf{q}_3 )$ consisting of six non-intersecting Bernoulli walks. Depicted to the right is an associated lozenge tiling.
• Figure 5: Shown to the left is the arctic boundary of an octagon, and shown to the right is the arctic boundary of a $12$-gon. Both examples satisfy the constraints listed in \ref{['pa']}.

Theorems & Definitions (88)

• Theorem : \ref{['walkspconverge']} below
• Lemma 2.1: VPT
• Definition 2.2
• Lemma 2.3: LSCEDMCS
• Proposition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Remark 2.9
• Theorem 2.10