On the asymptotics of dimers on tori

• Published in: Probability theory and related fields 2016-12, Vol.166 (3-4), p.971-1023
• Main Authors: Kenyon, Richard W., Sun, Nike, Wilson, David B.
• Format: Article
• Language: English
• Subjects: Asymptotic methods; Asymptotic properties; Asymptotic series; Bulk density; Dimers; Economics; Energy; Finance; Flux density; Fourier transforms; Free energy; Graph theory; Graphs; Insurance; Management; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Partitions (mathematics); Polynomials; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Quotients; Statistics for Business; Studies; Texts; Theoretical; Toruses
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-015-0687-8

We study asymptotics of the dimer model on large toric graphs. Let L be a weighted Z 2 -periodic planar graph, and let Z 2 E be a large-index sublattice of  Z 2 . For L bipartite we show that the dimer partition function Z E on the quotient L / ( Z 2 E ) has the asymptotic expansion Z = exp { A f 0 + fsc + o ( 1 ) } where A is the area of L / ( Z 2 E ) , f 0 is the free energy density in the bulk, and fsc 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 L 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.