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Abstract: We outline the most recent theory for the computation of the exponential growth rate of the number of configurations on a multi-dimensional grid. As an application we compute the monomer-dimer constant for the 2-dimensional grid to 8 decimal digits, agreeing with the heuristic computations of Baxter, and for the 3-dimensional grid with an error smaller than 1.35%. (Update)
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BibTeX entry: (Update)
@misc{ friedland-theory,
author = "Shmuel Friedland and Uri N. Peled",
title = "Theory of Computation of Multidimensional Entropy with an Application to
the Monomer-Dimer Problem",
url = "citeseer.ist.psu.edu/friedland04theory.html" }
Citations (may not include all citations):58 The undecidability of the domino problem (context) - Berger - 1966
30 The statistics of dimers on a lattice (context) - Kasteleyn - 1961
28 The number of independent sets in a grid graph (context) - Calkin, Wilf - 1998
26 Theory of monomer-dimer systems (context) - Heilman, Lieb
21 Algebraic Ideas in Ergodic Theory (context) - Schmidt - 1990
21 The capacity and coding gain of certain checkerboard codes (context) - Weeks, Blahut - 1998
18 Statistical mechanics of dimers on a plane lattice (context) - Fisher - 1961
13 Residual entropy of square ice (context) - Lieb - 1967
8 Hard hexagons: exact solution (context) - Baxter - 1980
7 Two-dimensional monomer-dimer systems are computationally in.. (context) - Jerrum - 1987
6 Approximating the number of monomer-dimer coverings of a lat.. - Kenyon, Randall et al. - 1996
5 Existence theorems and Monte Carlo methods for the monomer-d.. (context) - Hammersley - 1966
5 Statistical theory of perfect solutions (context) - Fowler, Rushbrooke - 1937
5 A note on the monomer dimer problem (context) - Bondy, Welsh - 1966
4 Exact series-expansion study of the monomer-dimer problem (context) - Gaunt - 1969
4 The topological entropy of cellular automata is uncomputable (context) - Hurd, Kari et al. - 1992
4 Calculations of lattice statistics (context) - Hammersley - 1970
4 A lower bound for the monomer-dimer problem (context) - Hammersley, Menon - 1970
3 Capacity bounds for the 3-dimensional (context) - Nagy, Zeger - 2000
2 A proof of a generalized van der Waerden conjecture on perma.. (context) - Friedland - 1982
2 New series expansion method for the dimer problem (context) - Nagle - 1966
1 A lower bound on the maximum permanent in (context) - Wanless - 2003
1 Dimers on a rectangular lattice (context) - Baxter - 1968
1 Approximating the permanent via importance sampling with app.. (context) - Beichl, Sullivan - 1999
1 pressure and Hausdor# dimension (context) - Friedland, capacity - 2003
1 An improved lower bound for the multidimesional dimer proble.. (context) - Hammersley - 1966
1 Compression of transfer matrices (context) - Lundow - 2001
1 subshifts of finite type (context) - Friedland, entropy et al. - 1997
1 An improved upper bound for the 3-dimensional dimer problem (context) - Ciucu - 1998
1 Counting 1-factors in regular bipartite graphs - Schrijver - 1998
1 Entropy bounds for constrained twodimensional random fileds (context) - Forschhammer, Justesen - 1999
1 The solution of the dimer problem by the transfer matrix met.. (context) - Lieb - 1967
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