# Asymptotics of Random Lozenge Tilings via Gelfand-Tsetlin Schemes

### 2012/02/16

Leonid Petrov

*Probability Theory and Related Fields, 160 (2014), no. 3, 429-487* •

arXiv: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