Dimer model: full asymptotic expansion of the partition function. (English) Zbl 1406.82005
The authors consider the dimer model on a square lattice \(\Gamma_{m,n}=(V_{m,n}, E_{m,n})\) on the torus \(\mathbb{Z}_m\times\mathbb{Z}_n=\mathbb{Z}^2/(m\mathbb{Z}\times n\mathbb{Z})\) (periodic boundary conditions), where \(V_{m,n}\) and \(E_{m,n}\) are the sets of vertices and edges of \(\Gamma_{m,n}\), respectively. A complete rigorous proof of the full asymptotic expansion of the partition function of this dimer model is given.
Reviewer: Nasir N. Ganikhodjaev (Kuantan)
MSC:
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B30 | Statistical thermodynamics |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
Citations:
Zbl 1043.82009References:
| [1] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (1972), Dover Publications: Dover Publications, New York · Zbl 0543.33001 |
| [2] | Bleher, P.; Elwood, B.; Petrović, D., The Pfaffian sign theorem for the dimer model on a triangular lattice, J. Stat. Phys., 171, 3, 400-426, (2018) · Zbl 1395.82037 · doi:10.1007/s10955-018-2007-z |
| [3] | Cimasoni, D.; Reshetikhin, N., Dimers on surface graphs and spin structures. I, Commun. Math. Phys., 275, 187-208, (2007) · Zbl 1135.82006 · doi:10.1007/s00220-007-0302-7 |
| [4] | Ferdinand, A. E., Statistical mechanics of dimers on a quadratic lattice, J. Math. Phys., 8, 2332-2339, (1967) · doi:10.1063/1.1705162 |
| [5] | Galluccio, A.; Loebl, M., On the theory of Pfaffian orientations. I. Perfect matchings and permanents, Electron. J. Combin., 6, 18, (1999) · Zbl 0909.05005 |
| [6] | Ivashkevich, E. V.; Izmailian, N. Sh.; Hu, C.-K., Kronecker’s double series and exact asymptotic expansions for free models of statistical mechanics on torus, J. Phys. A: Math. Gen., 35, 5543-5561, (2002) · Zbl 1043.82009 · doi:10.1088/0305-4470/35/27/302 |
| [7] | Izmailian, N. Sh.; Oganesyan, K. B.; Hu, C.-K., Exact finite-size corrections of the free energy for the square lattice dimer model under different boundary conditions, Phys. Rev. E, 67, 066114, (2003) · doi:10.1103/physreve.67.066114 |
| [8] | Kasteleyn, P. W., The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice, Physica, 27, 1209-1225, (1961) · Zbl 1244.82014 · doi:10.1016/0031-8914(61)90063-5 |
| [9] | Kasteleyn, P. W., Dimer statistics and phase transitions, J. Math. Phys., 4, 287-293, (1963) · doi:10.1063/1.1703953 |
| [10] | Kasteleyn, P. W., Graph theory and crystal physics, Graph Theory and Theoretical Physics, (1967), Academic Press: Academic Press, London · Zbl 0205.28402 |
| [11] | Kenyon, R. W.; Sun, N.; Wilson, D. B., On the asymptotics of dimers on tori, Probab. Theory Relat. Fields, 166, 3, 971-1023, (2016) · Zbl 1354.82007 · doi:10.1007/s00440-015-0687-8 |
| [12] | McCoy, B. M., Advanced Statistical Mechanics, (2010), Oxford University Press: Oxford University Press, Oxford · Zbl 1198.82001 |
| [13] | McCoy, B. M.; Wu, T. T., The Two-Dimensional Ising Model, (2014), Dover Publications, Inc.: Dover Publications, Inc., Mineola, New York · Zbl 1409.82001 |
| [14] | Temperley, H.; Fisher, M. E., The dimer problem in statistical mechanics-an exact result, Philos. Mag., 6, 1061-1063, (1961) · Zbl 0126.25102 · doi:10.1080/14786436108243366 |
| [15] | Tesler, G., Matchings in graphs on non-orientable surfaces, J. Comb. Theory, Ser. B, 78, 198-231, (2000) · Zbl 1025.05052 · doi:10.1006/jctb.1999.1941 |
| [16] | Weber, H. M., Lehrbuch Der Algebra, (1961), Chelsea Publishing Company: Chelsea Publishing Company, New York |
| [17] | Weil, A., Elliptic Functions According to Eisenstein and Kronecker, (2015), Springer-Verlag: Springer-Verlag, New York |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
