Vacancy localization in the square dimer model

Abstract

We study the classical dimer model on a square lattice with a single vacancy by developing a graph-theoretic classification of the set of all configurations which extends the spanning tree formulation of close-packed dimers. With this formalism, we can address the question of the possible motion of the vacancy induced by dimer slidings. We find a probability 57/4-10Sqrt[2] for the vacancy to be strictly jammed in an infinite system. More generally, the size distribution of the domain accessible to the vacancy is characterized by a power law decay with exponent 9/8. On a finite system, the probability that a vacancy in the bulk can reach the boundary falls off as a power law of the system size with exponent 1/4. The resultant weak localization of vacancies still allows for unbounded diffusion, characterized by a diffusion exponent that we relate to that of diffusion on spanning trees. We also implement numerical simulations of the model with both free and periodic boundary conditions.

35 pages, 24 figures. Improved version with one added figure (figure 9), a shift s->s+1 in the definition of the tree size, and minor corrections

References in corpus (4)

• Classical dimers on the triangular lattice
• Pocket Monte Carlo algorithm for classical doped dimer models
• The Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary
• Non-Local Finite-Size Effects in the Dimer Model

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