Abstract
Totally symmetric self-complementary plane partitions (TSSCPPs) are boxed plane partitions with the maximum possible symmetry. We use the well-known representation of TSSCPPs as a dimer model on a honeycomb graph enclosed in one-twelfth of a hexagon with free boundary to express them as perfect matchings of a family of non-bipartite planar graphs. Our main result is that the edges of the TSSCPPs form a Pfaffian point process, for which we give explicit formulas for the inverse Kasteleyn matrix. Preliminary analysis of these correlations are then used to give a precise conjecture for the limit shape of TSSCPPs in the scaling limit.
Highlights
Symmetric self-complementary plane partitions (TSSCPPs) of order n are the subset of plane partitions in a (2n) × (2n) × (2n) box with the maximum possible symmetry
There are still microscopic fluctuations which are believed to be governed by universal probability distributions originating in both statistical mechanics and random matrix theory
The correlations of the associated particle system to the tiling model are given by a Pfaffian point process; see [3] for an example where the authors give a formula for the correlation kernel for both symmetric plane partitions and plane overpartitions
Summary
Symmetric self-complementary plane partitions (TSSCPPs) of order n are the subset of plane partitions in a (2n) × (2n) × (2n) box with the maximum possible symmetry. For a specific class of tiling models, interesting probabilistic features are observed when the system size becomes large, such as a macroscopic limit shape, which is a type of law of large numbers result Around this limit shape, there are still microscopic fluctuations which are believed to be governed by universal probability distributions originating in both statistical mechanics and random matrix theory. The Eynard–Mehta theorem has a Pfaffian analog where the final positions of the nonintersecting lattice paths are free In this case, the correlations of the associated particle system to the tiling model are given by a Pfaffian point process; see [3] for an example where the authors give a formula for the correlation kernel for both symmetric plane partitions and plane overpartitions.
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