Pearcey universality at cusps of polygonal lozenge tiling

Abstract

We study uniformly random lozenge tilings of general simply connected polygons. Under a technical assumption that is presumably generic with respect to polygon shapes, we show that the local statistics around a cusp point of the arctic curve converge to the Pearcey process. This verifies the widely predicted universality of edge statistics in the cusp case. Together with the smooth and tangent cases proved in Aggarwal-Huang and Aggarwal-Gorin, these are believed to be the three types of edge statistics that can arise in a generic polygon. Our proof is via a local coupling of the random tiling with non-intersecting Bernoulli random walks (NBRW). To leverage this coupling, we establish an optimal concentration estimate for the tiling height function around the cusp. As another step and also a result of potential independent interest, we show that the local statistics of NBRW around a cusp converge to the Pearcey process when the initial configuration consists of two parts with proper density growth, via careful asymptotic analysis of the determinantal formula.

arXiv commentMinor change; 59 pages, 9 figures

References in corpus (21)

• A variational principle for domino tilings
• The shape of a typical boxed plane partition
• Universal singularity at the closure of a gap in a random matrix theory
• Local statistics for random domino tilings of the Aztec diamond
• Level Spacing of Random Matrices in an External Source
• The Pearcey Process
• Airy processes with wanderers and new universality classes
• Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge
• Uniform convergence to the Airy line ensemble
• Universality of the Pearcey process
• Minimizers of Convex Functionals Arising in Random Surfaces
• Random tilings and Markov chains for interlacing particles
• Cusp Universality for Random Matrices II: The Real Symmetric Case
• Bulk universality for random lozenge tilings near straight boundaries and for tensor products
• From the Pearcey to the Airy process
• A periodic hexagon tiling model and non-Hermitian orthogonal polynomials
• Local limits of lozenge tilings are stable under bounded boundary height perturbations
• Tilings of non-convex Polygons, skew-Young Tableaux and determinantal Processes
• Lozenge tilings of hexagons with cuts and asymptotic fluctuations: a new universality class
• Double Interlacing in Random Tiling Models
• Edge Statistics for Lozenge Tilings of Polygons, I: Concentration of Height Function on Strip Domains

Cited by in corpus (3)

• Tail estimates for the stationary stochastic six vertex model and ASEP
• A convergence framework for Airy${}_{\beta }$ line ensemble via pole evolution
• Dimer Models and Conformal Structures