Asymptotics of dimer coverings on free boundary rail-yard graphs (2304.00650v1)

Abstract: Rail-yard graphs are a general class of graphs introduced in \cite{bbccr} on which the random dimer coverings form Schur processes. We study asymptotic limits of random dimer coverings on rail yard graphs with free boundary conditions on both the left boundary and the right boundary (double-sided free boundary) when the mesh sizes of the graphs go to 0. Each dimer covering corresponds to a sequence of interlacing partitions starting with an arbitrary partition and ending in an arbitrary partition. Under the assumption that the probability of each dimer covering is proportional to the product of weights of present edges, we obtain the moment formula for the height function which includes an infinite product. By passing down to the scaling limit, we compute the limit shape (law of large numbers) of the rescaled height functions and prove the convergence of unrescaled height fluctuations to a diffeomorphic image of the restriction of the 0-boundary Gaussian free field (central limit theorem) on the upper half plane to a subset. Applications include the limit shape and height fluctuations for free boundary steep tilings as proposed in \cite{BCC17}. The technique to obtain these results is to analyze a class of Macdonald processes with dual specializations, subject to further complexities arising from the infinite product in the moment formula. We also obtain a new algorithm to sample double-sided free boundary dimer coverings on rail-yard graphs, which fulfills an open problem in \cite{bbbccv14}.