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\(q\)-distributions on boxed plane partitions. (English) Zbl 1205.82122

The uniform distribution on boxed plane partitions (equivalently, lozenge tilings of a hexagon) is one of the most studied model of random surfaces. In this paper, measures on boxed plane partitions that generalize the uniform distribution are studied. The weight of a tiling is defined as the product of certain sample factors over all lozenges of a fixed type. One special case is the weight \(q^{volume}\), where \(volume\) is the volume of the corresponding plane partition, and \(q\) is an arbitrary positive number. In the most general case the weight of a lozenge is considered as elliptic which leads to a generalization of the MacMahon formula for the total number of plane partition in a given box. A perfect sampling algorithm is obtained from a more general construction of relatively simple Markov chains that change the size of the box and that map the measures from the class which are considered to be similar. Asymptotic results show that near any point of the limit shape the measure on tilings converges to a translation-invariant ergodic Gibbs measure on the lozenge tiling of the plane of a given slope and the slope coincides with the slope of the tangent plane to the limit shape at the chosen point.

MSC:

82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
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