Non-intersecting Paths, Random Tilings and Random Matrices

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 lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the same structure as the eigenvalue measures in random matrix theory like GUE, which can in fact be obtained from non-intersecting Brownian motions. The derivations of the measures are based on the Karlin-McGregor or Lindström-Gessel-Viennot method. We use the measure to show some asymptotic results for the models.

48 pages

References in corpus (7)

• Level-Spacing Distributions and the Airy Kernel
• Universality of the local spacing distribution in certain ensembles of hermitian Wigner matrices
• Local statistics of lattice dimers
• Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions
• Statistical Self-Similarity of One-Dimensional Growth Processes
• Z-Measures on partitions, Robinson-Schensted-Knuth correspondence, and beta=2 random matrix ensembles
• Random walks and random permutations

Cited by in corpus (7)

• Universality of the local spacing distribution in certain ensembles of hermitian Wigner matrices
• Noncommutative integrability, paths and quasi-determinants
• Duality of orthogonal polynomials on a finite set
• Shuffling algorithm for boxed plane partitions
• Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram
• Asymptotics of Random Lozenge Tilings via Gelfand-Tsetlin Schemes
• Pfaffian Stochastic Dynamics of Strict Partitions