Asymptotics of pure dimer coverings on rail-yard graphs

TL;DR

This paper investigates the asymptotic behavior of random pure dimer coverings on rail-yard graphs, establishing a limit shape and Gaussian Free Field fluctuations, with applications to steep tilings and pyramid partitions.

Contribution

It introduces a method to analyze the asymptotics of dimer models on rail-yard graphs using Mcdonald processes, answering a previously open question.

Findings

Derives the limit shape of the height function.

Shows height fluctuations converge to Gaussian Free Field.

Applies results to steep tilings and pyramid partitions.

Abstract

We study asymptotic limit of random pure dimer coverings on rail yardgraphs when the mesh sizes of the graphs go to 0. Each pure dimer covering correspondsto a sequence of interlacing partitions starting with an empty partition and ending inan empty partition. Under the assumption that the probability of each dimer covering isproportional to the product of weights of present edges, we obtain the limit shape (Law ofLarge Numbers) of the rescaled height function and the convergence of unrescaled heightfluctuation to a diffeomorphic image of Gaussian Free Field (Central Limit Theorem); an-swering a question in [6]. Applications include the limit shape and height fluctuations forpure steep tilings ([8]) and pyramid partitions ([20, 35, 36, 37]). The technique to obtainthese results is to analyze a class of Mcdonald processes which involve dual partitions as well.