On the asymptotics of dimers on tori

• 成果类型: Article
• 署名作者: Kenyon, Richard W.; Sun, Nike; Wilson, David B.
• 署名单位: Brown University; Stanford University; Microsoft
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0687-8
• 发表日期: 2016
• 页码: 971-1023
• 关键词: finite-size corrections; ising-model; statistical-mechanics; conformal-invariance; lattice; graphs

摘要

We study asymptotics of the dimer model on large toric graphs. Let be a weighted -periodic planar graph, and let be a large-index sublattice of . For bipartite we show that the dimer partition function on the quotient has the asymptotic expansion where A is the area of , is the free energy density in the bulk, and is a finite-size correction term depending only on the conformal shape of the domain together with some parity-type information. Assuming a conjectural condition on the zero locus of the dimer characteristic polynomial, we show that an analogous expansion holds for non-bipartite. The functional form of the finite-size correction differs between the two classes, but is universal within each class. Our calculations yield new information concerning the distribution of the number of loops winding around the torus in the associated double-dimer models.