Double-dimer pairings and skew Young diagrams

TL;DR

This paper explores the enumeration of skew Young diagram tilings with ribbon tiles and applies these results to compute connection probabilities in the double-dimer model on planar graphs, especially in a scaling limit.

Contribution

It introduces a novel combinatorial approach to counting ribbon tilings of skew Young diagrams and links these counts to probabilistic properties of the double-dimer model.

Findings

Derived formulas for tilings of skew Young diagrams with Dyck path-shaped tiles.

Connected combinatorial counts to double-dimer pairing probabilities.

Analyzed the scaling limit for evenly spaced boundary nodes on a planar graph.

Abstract

We study the number of tilings of skew Young diagrams by ribbon tiles shaped like Dyck paths, in which the tiles are "vertically decreasing". We use these quantities to compute pairing probabilities in the double-dimer model: Given a planar bipartite graph $G$ with special vertices, called nodes, on the outer face, the double-dimer model is formed by the superposition of a uniformly random dimer configuration (perfect matching) of $G$ together with a random dimer configuration of the graph formed from $G$ by deleting the nodes. The double-dimer configuration consists of loops, doubled edges, and chains that start and end at the boundary nodes. We are interested in how the chains connect the nodes. An interesting special case is when the graph is $\epsilon(\Z\times\N)$ and the nodes are at evenly spaced locations on the boundary $\R$ as the grid spacing $\epsilon\to 0$.