Theory of computation of multidimensional entropy with an application to the monomer-dimer problem

Advances in Applied Mathematics

Volumn 34, Issue 3, 2005, Pages 486-522

  Friedland, Shmuel ^a,b   Peled, Uri N ^a  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^b UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Monomer dimer; Subshifts of finite type; Topological entropy; Transfer matrix

Indexed keywords

GRAPH THEORY; HEURISTIC METHODS; MATHEMATICAL MODELS; MATRIX ALGEBRA; NUMERICAL METHODS; SET THEORY; STATISTICAL METHODS; TOPOLOGY;

MONOMER-DIMER PROBLEM; SUBSHIFTS-OF-FINITE TYPE; TOPOLOGICAL ENTROPY; TRANSFER MATRIX;

COMPUTATION THEORY;

EID: 14844318558     PISSN: 01968858     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.aam.2004.08.005     Document Type: Article

References (35)

1. R.J. Baxter Dimers on a rectangular lattice J. Math. Phys. 9 1968 650-654
2. R.J. Baxter Hard hexagons: exact solution J. Phys. A 13 1980 L61-L70
3. I. Beichl F. Sullivan Approximating the permanent via importance sampling with application to the dimer cover problem J. Comput. Phys. 149 1999 128-147
4. R. Berger The undecidability of the domino problem Mem. Amer. Math. Soc. 66 1966
5. J.A. Bondy D.J.A. Welsh A note on the monomer dimer problem Proc. Cambridge Philos. Soc. 62 1966 503-505
6. N.J. Calkin H.S. Wilf The number of independent sets in a grid graph SIAM J. Discrete Math. 11 1998 54-60
7. M. Ciucu An improved upper bound for the 3-dimensional dimer problem Duke Math. J. 94 1998 1-11
8. R. Diestel Graph Theory second ed. 2000 Springer New York
9. S. Even Graph Algorithms 1979 Computer Science Press Potomac, MD
10. M.E. Fisher Statistical mechanics of dimers on a plane lattice Phys. Rev. 124 1961 1664-1672
11. S. Forschhammer J. Justesen Entropy bounds for constrained two-dimensional random fileds IEEE Trans. Inform. Theory 45 1999 118-127
12. R.H. Fowler G.S. Rushbrooke Statistical theory of perfect solutions Trans. Faraday Soc. 33 1937 1272-1294
13. S. Friedland A proof of a generalized van der Waerden conjecture on permanents Linear Multilin. Algebra 11 1982 107-120
14. d subshifts of finite type Linear Algebra Appl. 252 1997 199-220
15. S. Friedland Multi-dimensional capacity, pressure and Hausdorff dimension in: J. Rosenthal D. Gilliam (Eds.) Mathematical System Theory in Biology, Communication, Computation and Finance in: IMA Vol. Ser. vol. 134 2003 Springer New York 183-222
16. O.J. Heilman E.H. Lieb Theory of monomer-dimer systems Comm. Math. Phys. 25 1972 190-232 Errata: 27 (1972) 166
17. D.S. Gaunt Exact series-expansion study of the monomer-dimer problem Phys. Rev. 179 1969 174-186
18. J.M. Hammersley Existence theorems and Monte Carlo methods for the monomer-dimer problem in: F.N. David (Ed.) Research Papers in Statistics: Festschrift for J. Neyman 1966 Wiley London 125-146
19. J.M. Hammersley An improved lower bound for the multidimensional dimer problem Proc. Cambridge Philos. Soc. 64 1966 455-463
20. J.M. Hammersley Calculations of lattice statistics in: Proc. Comput. Physics Conf. 1970 Inst. of Phys. & Phys. Soc. London pp. 1-8
21. J. Hammersley V. Menon A lower bound for the monomer-dimer problem J. Inst. Math. Appl. 6 1970 341-364
22. L.P. Hurd J. Kari K. Culik The topological entropy of cellular automata is uncomputable Ergodic Theory Dynam. Systems 12 1992 255-265
23. P.W. Kasteleyn The statistics of dimers on a lattice Physica 27 1961 1209-1225
24. C. Kenyon D. Randall A. Sinclair Approximating the number of monomer-dimer coverings of a lattice J. Statist. Phys. 83 1996 637-659
25. M. Jerrum Two-dimensional monomer-dimer systems are computationally intractible J. Statist. Phys. 48 1987 121-134
26. E.H. Lieb The solution of the dimer problem by the transfer matrix method J. Math. Phys. 8 1967 2339-2341
27. E.H. Lieb Residual entropy of square ice Phys. Rev. 162 1967 162-172
28. P.H. Lundow Compression of transfer matrices Discrete Math. 231 2001 321-329
29. J.F. Nagle New series expansion method for the dimer problem Phys. Rev. 152 1966 190-197
30. Z. Nagy K. Zeger Capacity bounds for the 3-dimensional (0, 1) run length limited channel IEEE Trans. Inform. Theory 46 2000 1030-1033
31. L. Pauling J. Amer. Chem. Soc. 57 1935 2680
32. K. Schmidt Algebraic Ideas in Ergodic Theory 1990 Amer. Math. Soc.
33. A. Schrijver Counting 1-factors in regular bipartite graphs J. Combin. Theory B 72 1998 122-135
34. k, Linear Algebra Appl. 373 2003 153-167
35. W. Weeks R.E. Blahut The capacity and coding gain of certain checkerboard codes IEEE Trans. Inform. Theory 44 1998 1193-1203