Free boundary dimers: random walk representation and scaling limit

Area of Science:

• Statistical Mechanics
• Mathematical Physics
• Combinatorics

Background:

• The study focuses on the dimer model, a statistical mechanics model on a lattice.
• It considers variations with unmatched vertices (monomers) on a free boundary, introducing a weight parameter.
• A known bijection connects this to a dimer model on a non-bipartite graph.

Purpose of the Study:

• To establish a random walk representation for the inverse Kasteleyn matrix of the modified dimer model.
• To determine the continuum scaling limit of the centered height function under specific assumptions.
• To investigate the emergence of Neumann boundary conditions in the scaling limit.

Main Methods:

• Utilizing a bijection to a dimer model on a non-bipartite graph.
• Analyzing the Kasteleyn matrix and its properties, including negative transition weights.
• Proving an effective random walk representation for the inverse Kasteleyn matrix.
• Investigating the scaling limit of the height function in the infinite volume limit.

Main Results:

• An effective, true random walk representation for the inverse Kasteleyn matrix is proven.
• The scaling limit of the centered height function is shown to be the Gaussian free field.
• Neumann (free) boundary conditions are demonstrated to arise in the continuum scaling limit, independent of the monomer weight.

Conclusions:

• This work provides the first discrete model example where Neumann boundary conditions emerge in the continuum scaling limit.
• The findings offer insights into the behavior of dimer models with boundary defects and their connection to continuous fields.
• The established random walk representation is a key tool for further analysis of such systems.