Abstract
The theory of renewal sequences is used to obtain some exact results for the number of dimer and monomer-dimer configurations on blocks and rods in the hypercubical lattice and these results, in turn, are applied to give lower bounds for the dimer problem on the unrestricted lattice.
Keywords
- Transfer Matrix Method
- Hypercubical Lattice
- Critical Configuration
- Dime Configuration
- Renewal Sequence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
N.G. de Bruijn and P. Erdös, ‘Some linear and some quadratic recursion formulas', Indagationes Math. 13 (1951), 374–382.
C. Domb and M.F. Sykes, ‘Use of series expansions for the Ising model susceptibility and excluded volume problem', J. Math. Phys. 2 (1961), 63–67.
R.M. Fowler and G.S. Rushbrook, ‘Statistical theory of perfect solutions', Trans. Faraday Soc. 33 (1937), 1272–1294.
R.C. Grimson, ‘Exact formulas for 2×n arrays of dumbbells', J. Math. Phys. 15 (1974), 214–216.
R.C. Grimson, ‘Enumeration of dimer (domino) configurations', Discrete Math. 18 (1977), 167–177.
J.M. Hammersley, ‘Existence theorems and Monte Carlo methods for the monomer dimer problem', in Research papers in statistics: Festschrift für J. Neyman, ed. F.N. David, J. Wiley, New York, 1966.
J.M. Hammersley, ‘An improved lower bound for the multidimensional dimer problem', Proc. Camb. Phil. Soc. 64 (1968), 455–463.
J.M. Hammersley and V.V. Menon, ‘A lower bound for the monomer-dimer problem', J. Inst. Math. Applics. 6 (1970), 341–364.
O.J. Heilmann and E.H. Lieb, ‘Theory of monomer-dimer systems', Comm. Math. Phys. 25 (1972), 190–232.
H. Kesten, ‘On the number of self avoiding walks', J. Math. Phys. 4 (1963), 960–969.
J.F.C. Kingman, Regenerative Phenomena, J. Wiley, London, 1972.
E.H. Lieb, ‘Solution of the dimer problem by the transfer matrix method', J. Math. Phys. 8(1967), 2339–2341.
R.B. McQuistan and S.J. Lichtman, ‘Exact recursion relation for 2×n arrays of dumbbells', J. Math. Phys. (1970), 3095–3097.
R.B. McQuistan, S.J. Lichtman, and L.P. Levine, ‘Occupation statistics for parallel dumbbells on a 2×N lattice space', J. Math. Phys. 13 (1972), 242–248.
H. Minc, ‘An upper bound for the multidimensional dimer problem', Math. Proc. Camb. Phil. Soc. 83 (1978), 461–462.
E.W. Montroll, ‘Lattice statistics', in Applied Combinatorial Mathematics, ed. F. Beckenback, J. Wiley, New York, 1964.
J.K. Percus, Combinatorial Methods, Springer-Verlag, Berlin, 1971.
M.A. Rashid, ‘Occupation statistics from exact recursion relations for occupation by dumbbells of a 2×n array', J. Math. Phys. 15 (1974), 474–476.
J. Riordan, Introduction to Combinatorial Analysis, J. Wiley, New York, 1958.
D.G. Rogers, ‘Pascal triangles, Catalan numbers, and renewal arrays', Discrete Math. 22(1978), 301–311.
D.G. Rogers, ‘Lattice paths with diagonal steps.’
D.G. Rogers, 'similarity relations and semiorders.’
D.N. Shanbhag, ‘On renewal sequences', Bull. London Math. Soc. 9 (1977), 79–80.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1979 Springer-Verlag
About this paper
Cite this paper
Rogers, D.G. (1979). An application of renewal sequences to the dimer problem. In: Horadam, A.F., Wallis, W.D. (eds) Combinatorial Mathematics VI. Lecture Notes in Mathematics, vol 748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0102693
Download citation
DOI: https://doi.org/10.1007/BFb0102693
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-09555-2
Online ISBN: 978-3-540-34857-3
eBook Packages: Springer Book Archive
