Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes

  title={Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes},
  author={Leonid A. Petrov},
  journal={Probability Theory and Related Fields},
  • L. Petrov
  • Published 17 February 2012
  • Mathematics, Physics
  • Probability Theory and Related Fields
A Gelfand–Tsetlin scheme of depth $$N$$N is a triangular array with $$m$$m integers at level $$m$$m, $$m=1,\ldots ,N$$m=1,…,N, subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand–Tsetlin schemes with arbitrary fixed $$N$$Nth row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its $$q$$q-deformation). This provides new tools for asymptotic analysis of uniformly random… Expand
GUE corners limit of q-distributed lozenge tilings
We study asymptotics of $q$-distributed random lozenge tilings of sawtooth domains (equivalently, of random interlacing integer arrays with fixed top row). Under the distribution we consider eachExpand
Universal edge fluctuations of discrete interlaced particle systems
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 equivalentlyExpand
Lozenge Tilings, Glauber Dynamics and Macroscopic Shape
We study the Glauber dynamics on the set of tilings of a finite domain of the plane with lozenges of side 1/L. Under the invariant measure of the process (the uniform measure over all tilings), it isExpand
Fourier transform on high-dimensional unitary groups with applications to random tilings
A combination of direct and inverse Fourier transforms on the unitary group $U(N)$ identifies normalized characters with probability measures on $N$-tuples of integers. We develop the $N\to\infty$Expand
Universality for Lozenge Tiling Local Statistics
In this paper we consider uniformly random lozenge tilings of arbitrary domains approximating (after suitable normalization) a closed, simply-connected subset of $\mathbb{R}^2$ with piecewise smooth,Expand
The local limit of random sorting networks
A sorting network is a geodesic path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of $S_n$ generated by adjacent transpositions. For a uniformly random sorting network, we establish theExpand
Universality of local statistics for noncolliding random walks
We consider the $N$-particle noncolliding Bernoulli random walk --- a discrete time Markov process in $\mathbb{Z}^{N}$ obtained from a collection of $N$ independent simple random walks with stepsExpand
Rigidity and Edge Universality of Discreteβ‐Ensembles
We study discrete $\beta$-ensembles as introduced in [17]. We obtain rigidity estimates on the particle locations, i.e. with high probability, the particles are close to their classical locationsExpand
Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field
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 allowsExpand
Asymptotic Geometry of Discrete Interlaced Patterns: Part II
We study the boundary of the liquid region $\mathcal{L}$ in large random lozenge tiling models defined by uniform random interlacing particle systems with general initial configuration, which lies onExpand


Universality properties of Gelfand–Tsetlin patterns
A standard Gelfand–Tsetlin pattern of depth n is a configuration of particles in $${\{1,\ldots,n\} \times \mathbb{R}}$$ . For each $${r \in \{1, \ldots, n\}, \{r\} \times \mathbb{R}}$$ is referred toExpand
The boundary of the Gelfand–Tsetlin graph: A new approach
The Gelfand–Tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups. The boundary of the Gelfand–Tsetlin graph has at least threeExpand
Eigenvalues of GUE Minors
Consider an infinite random matrix $H=(h_{ij})_{0 < i,j}$ picked from the Gaussian Unitary Ensemble (GUE). Denote its main minors by $H_i=(h_{rs})_{1\leq r,s\leq i}$ and let the $j$:th largestExpand
Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions
We construct a family of stochastic growth models in 2 + 1 dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield 1 + 1 dimensional growth models in theExpand
Asymptotics of Plancherel measures for symmetric groups
1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G∧ of irreducible representations of G which assigns to aExpand
Nonintersecting paths with a staircase initial condition
We consider an ensemble of $N$ discrete nonintersecting paths starting from equidistant points and ending at consecutive integers. Our first result is an explicit formula for the correlation kernelExpand
q-Distributions on boxed plane partitions
We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon’s product formula for the number of plane partitions in a box. We then focus onExpand
Correlations for the Novak process
We study random lozenge tilings of a certain shape in the plane called the Novak half-hexagon, and compute the correlation functions for this process. This model was introduced by Nordenstam andExpand
Local statistics of lattice dimers
Abstract We show how to compute the probability of any given local configuration in a random tiling of the plane with dominos. That is, we explicitly compute the measures of cylinder sets for theExpand
The Shape of a Typical Boxed Plane Partition
Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on allExpand