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

@inproceedings{Aggarwal2021EdgeSF, title={Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble}, author={Amol Aggarwal and Jiaoyang Huang}, year={2021} }

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…

## 3 Citations

### Gaussian unitary ensemble in random lozenge tilings

- MathematicsProbability Theory and Related Fields
- 2022

This paper establishes a universality result for scaling limits of uniformly random lozenge tilings of large domains. We prove that whenever the boundary of the domain has three adjacent straight…

### KPZ fluctuations in the planar stochastic heat equation

- Mathematics
- 2022

We use a version of the Skorokhod integral to give a simple and rigorous formulation of the Wick-ordered (stochastic) heat equation with planar white noise, representing the free energy of an…

### Lyapunov exponents for truncated unitary and Ginibre matrices

- Mathematics
- 2021

In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced ‘picket-fence’ statistics.…

## References

SHOWING 1-10 OF 50 REFERENCES

### Edge Statistics for Lozenge Tilings of Polygons, I: Concentration of Height Function on Strip Domains

- Mathematics
- 2021

In this paper we study uniformly random lozenge tilings of strip domains. Under the assumption that the limiting arctic boundary has at most one cusp, we prove a nearly optimal concentration estimate…

### Tilings of Non-convex Polygons, Skew-Young Tableaux and Determinantal Processes

- MathematicsCommunications in Mathematical Physics
- 2018

This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to new…

### Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field

- Mathematics
- 2012

We study large-scale height fluctuations of random stepped surfaces corresponding to uniformly random lozenge tilings of polygons on the triangular lattice. For a class of polygons (which allows…

### Probability distributions related to tilings of non-convex polygons

- MathematicsJournal of Mathematical Physics
- 2018

This paper is based on the study of random lozenge tilings of non-convex polygonal regions with interacting non-convexities (cuts) and the corresponding asymptotic kernel as in [3] and [4] (discrete…

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

- Mathematics
- 2017

This paper investigates lozenge tilings of non-convex hexagonal regions and more specifically the asymptotic fluctuations of the tilings within and near the strip formed by opposite cuts in the…

### Local statistics for random domino tilings of the Aztec diamond

- Mathematics
- 1996

We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, the tiling will…

### Non-intersecting paths, random tilings and random matrices

- Mathematics
- 2000

Abstract. We investigate certain measures induced by families of non-intersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abc-hexagon, a dimer model on a cylindrical brick…

### Bulk Universality for Random Lozenge Tilings Near Straight Boundaries and for Tensor Products

- Mathematics
- 2016

We prove that the asymptotic of the bulk local statistics in models of random lozenge tilings is universal in the vicinity of straight boundaries of the tiled domains. The result applies to uniformly…

### Local limits of lozenge tilings are stable under bounded boundary height perturbations

- MathematicsProbability Theory and Related Fields
- 2018

We show that bounded changes to the boundary of a lozenge tilings do not affect the local behaviour inside the domain. As a consequence we prove the existence of a local limit in all domains with…

### Universal edge fluctuations of discrete interlaced particle systems

- Mathematics
- 2017

We impose the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of depth $n$ with the particles on row $n$ in deterministic positions. These systems equivalently…