Asymptotics of Random Lozenge Tilings via Gelfand-Tsetlin Schemes

2012/02/16

Leonid Petrov
Probability Theory and Related Fields, 160 (2014), no. 3, 429-487arXiv:1202.3901 [math.PR]

A Gelfand-Tsetlin scheme of depth $N$ is a triangular array with m integers at level $m$, $m=1,\ldots,N$, subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand-Tsetlin schemes with arbitrary fixed $N$-th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its q-deformation).

This provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice; or, equivalently, of random stepped surfaces. We work with a class of polygons which allows arbitrarily large number of sides. We show that the local limit behavior of random tilings (as all dimensions of the polygon grow) is directed by ergodic translation invariant Gibbs measures. The slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by Kenyon and Okounkov in arXiv:math-ph/0507007.

We also prove that at the edge of the limit shape, the asymptotic behavior of random tilings is given by the Airy process.
In particular, our results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).


A poem on the topic

by OpenAI

My sweet love's tiling, of lozenges arrayed,
A pattern so beautiful and so fair,
As if crafted by some celestial hand,
Their beauty lies in the ordered array.

The lozenges, red and blue, and green,
Creating a pattern so pleasing to see,
Their edges line up and fit so well,
A tiling of lozenges, perfect and free.

The asymptotic study of the tiling,
Each lozenge in its place, and its fate,
Is done through a Gelfand-Tsetlin scheme,
And the resulting patterns so great.

The lozenges in their ordered array,
A pattern so pleasing and so fine,
The study of their asymptotic fate,
Will give us insight into the design.

A frozen boundary curve inscribed in a polygon
A frozen boundary curve inscribed in a polygon