Correlations in totally symmetric self-complementary plane partitions

TL;DR

This paper models totally symmetric self-complementary plane partitions as a Pfaffian point process using dimer models, providing explicit formulas for correlations and conjecturing their limit shape in the scaling limit.

Contribution

It introduces a Pfaffian point process framework for TSSCPPs and derives explicit inverse Kasteleyn matrices, advancing understanding of their correlation structure.

Findings

Edges form a Pfaffian point process

Explicit formulas for inverse Kasteleyn matrix

Conjecture for the limit shape of TSSCPPs

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.