q-Distributions on boxed plane partitions

Abstract

We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon's product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of these weights that are related to orthogonal polynomials; they form three 2-D families. For distributions from these families, we prove two types of results. First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm. Second, we consider a limit when all dimensions of the box grow and plane partitions become large and prove that the local correlations converge to those of ergodic translation invariant Gibbs measures. For fixed proportions of the box, the slopes of the limiting Gibbs measures (that can also

Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions

“…The same phenomenon is present in the case of discrete and continuous log-gases. Subsequently, we apply our general results to a class of tiling models that was introduced in [16] and obtain explicit formulas for their limit shape and global fluctuations. The tiling model we investigate corresponds to a special case of (1.3) when θ = 1, and we remark that for general θ > 0 the interaction term H q,v θ ( i , j ) can be linked to Macdonald-Koorwinder polynomials [49] similarly to how H θ (λ i , λ j ) in (1.2) is linked to Jack symmetric polynomials, see also Remark 2.1.2.…”

“…As discussed in Section 1 our main motivation for studying discrete log-gases on shifted quadratic lattices comes from the q-Racah tiling model that was introduced in [16]. In Section 7.1 we give a formal definition of the model and in Section 7.2 we state the main results we prove about it in Theorems 7.2.2 and 7.2.4.…”

Arctic boundaries of the ice model on three-bundle domains

“…This phenomenon was soon observed to be ubiquitous within the context of highly correlated statistical mechanical systems; see, for instance, [1,2,5,6,7,10,12,13,15,16,17,18,19,20,25,28,29,30,32,33,34,35,42,43,44,45,51,54,57,61]. In particular, Cohn-Kenyon-Propp developed a variational principle [12] that prescribes a law of large numbers for random domino tilings on almost arbitrary domains, which was used effectively by to explicitly determine the arctic boundaries of uniformly random lozenge tilings on polygonal domains.…”

Lozenge tilings of hexagons with arbitrary dents

“…Assume therefore that n, a, b and k are non-negative integers with a even, and define S n,a,b,k to be the region obtained from the hexagon H n,n+a+3b of side-lengths n, n + a + 3b, n, n + a + 3b, n, n + a + 3b (clockwise from top) by removing a triangle of side a from its center and three satellite triangular holes, each of side b, as indicated on the left in Figure 1 (we emphasize that k is the length of a chain of lozenges that would bridge the gap between each satellite and the core; there are 2k lattice spacings between a satellite and the core). For non-negative integers n, a, b and k with b even, 3 Throughout this paper, with the exception of Section 3, we draw the triangular lattice so that one family of lattice lines is horizontal. define S ′ n,a,b,k to be the region obtained from the same hexagon H n,n+a+3b by removing a triangle of side a from its center and three satellite triangular holes of side b as indicated on the right in Figure 1 (k has the same significance as in the picture on the left in that figure).…”