Lozenge tilings of a hexagon and random stepped surfaces

The talk is about a probability model of discrete random stepped surfaces that was studied a lot in the last 10 years.
There are several equivalent ways to introduce the model. Here are some of them. Consider 3d Young diagrams in $A\times B\times C$ box (i.e. piles of unit cubes flushed to the corner of the box). A boundary of such diagram is a stepped surface. If we equip the set of all 3d Young diagrams in a given box with some probability measure (the simplest one being uniform), then we get a model of random stepped surfaces. Another equivalent description is the following one: Consider an equiangular hexagon with side lengths A, B, C, A, B, C and tile it with rhombi of 3 different types (“lozenges”). This tilings are in one-to-one correspondence with stepped surfaces again. Some pictures the objects under consideration are available at http://www.mccme.ru/~vadicgor/Random_tilings.html.
In 1998 Cohn, Larsen and Propp proved that for big A, B and C random surfaces degenerate in some sense: An analogue of the law of big numbers holds and a typical random surfaces is very close to a certain non-random limit shape. Limit shapes in this and related models were further studied in papers by Kenyon and Okounkov. In the first part of talk I will speak about the most important results obtained since 1998 and some problems in the field that remain unsolved.
In the second part of the talk some recent results about this model obtained by the speaker and coauthors will be presented. Two aspects will be highlighted in more details. First, we will speak about local probability characteristics of random stepped surfaces in a big box. The simplest example of such characteristic is an average slope of the surfaces, but much more general ones will be also described. Also we will speak about a Markov chain which relates stepped surfaces in boxes of various sizes. The possibility of a simple construction of transition probability matrices relating random 3d diagrams in $A\times B\times C$ and $A\times (B-1)\times (C+1)$ boxes will be explained. The existence of such Markov chain provides, in particular, an efficient algorithm to sample random stepped surfaces in computer experiments.