Shuffling algorithm for boxed plane partitions

We introduce discrete time Markov chains that preserve uniform measures on boxed plane partitions. Elementary Markov steps change the size of the box from a × b × c to (a − 1)× (b + 1)× c or (a + 1)× (b − 1)× c. Algorithmic realization of each step involves O((a+ b)c) operations. One application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions. Trajectories of our Markov chains can be viewed as random point configurations in the three-dimensional lattice. We compute the bulk limits of the correlation functions of the resulting random point process on suitable two-dimensional sections. The limiting correlation functions define a two-dimensional determinantal point processes with certain Gibbs properties. Intoduction For any integers a, b, c ≥ 1 consider a hexagon with sides a, b, c, a, b, c drawn on a regular triangular lattice. Denote by Ωa×b×c the set of all tilings of this hexagon by rhombi obtained by gluing two of the neighboring elementary triangles together (such rhombi are called lozenges). Equivalently, Ωa×b×c is the set of all dimers on the part of the dual hexagonal lattice cut out by our (a, b, c)-hexagon. An element of Ω4×5×5 is shown on Figure 1. Elements of Ωa×b×c have a number of different interpretations, see e.g. Section 1 below. In particular, they can be viewed as plane partitions or stepped surfaces inside a three-dimensional box of size a×b×c. A uniformly distributed element of Ωa×b×c then provides a basic model of a random surface. This model and its generalizations have been thoroughly studied, see e.g. [CLP], [CKP], [DMB], [Des], [J1] [J2], [JN], [Ke1], [KO], [Kr], [P1], [P2], [W1], [W2]. California Institute of Technology and Institute for Information Transmission Problems, Moscow Moscow State University