## Counting in Lattices: Combinatorial Problems from Statistical Mechanics (1994)

Citations: | 2 - 1 self |

### BibTeX

@TECHREPORT{Randall94countingin,

author = {Dana Randall},

title = {Counting in Lattices: Combinatorial Problems from Statistical Mechanics},

institution = {},

year = {1994}

}

### OpenURL

### Abstract

In this thesis we consider two classical combinatorial problems arising in statistical mechanics: counting matchings and self-avoiding walks in lattice graphs. The first problem arises in the study of the thermodynamical properties of monomers and dimers (diatomic molecules) in crystals. Fisher, Kasteleyn and Temperley discovered an elegant technique to exactly count the number of perfect matchings in two dimensional lattices, but it is not applicable for matchings of arbitrary size, or in higher dimensional lattices. We present the first efficient approximation algorithm for computing the number of matchings of any size in any periodic lattice in arbitrary dimension. The algorithm is based on Monte Carlo simulation of a suitable Markov chain and has rigorously derived performance guarantees that do not rely on any assumptions. In addition, we show that these results generalize to counting matchings in any graph which is the Cayley graph of a finite group. The second problem is countin...

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101 |
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Citation Context ... algorithm fails to generalize to all non-planar graphs. Furthermore, Jerrum showed that counting the total number of matchings (of all sizes) in a graph is #P-complete, even when the graph is planar =-=[32]-=-. This gives evidence for why the Fisher, Kasteleyn, Temperley algorithm does not easily generalize to counting matchings of arbitrary size. Similarly, consider the problem #SAW(G;sN \Gamma 1; x 0 ) w... |

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Citation Context ...ar lattice (or chessboard). Example 1: The Ising Model The Ising model was introduced in the 1920's to study ferromagnetism and is one of the most famous models from statistical mechanics (see, e.g., =-=[8]-=- for a history and review). The system is modeled by a finite lattice where the vertices represent atoms of a ferromagnetic 7 material. Each configuration oe consists of an assignment of a +1 or-1 spi... |

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Citation Context ...cal mechanics was achieved in 1961 when Fisher, Kasteleyn and Temperley independently discovered a polynomial-time algorithm for a special case of the monomer-dimer problem known as the dimer problem =-=[15,40, 63]-=- (see section 2.1). The dimer problem asks for the number of perfect matchings, or coverings of the lattice by N=2 dimers (and no monomers). (More generally, a perfect matching in a graph is a matchin... |

37 |
Statistical mechanics of dimers on a plane lattice
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- 1961
(Show Context)
Citation Context ...cal mechanics was achieved in 1961 when Fisher, Kasteleyn and Temperley independently discovered a polynomial-time algorithm for a special case of the monomer-dimer problem known as the dimer problem =-=[15,40, 63]-=- (see section 2.1). The dimer problem asks for the number of perfect matchings, or coverings of the lattice by N=2 dimers (and no monomers). (More generally, a perfect matching in a graph is a matchin... |

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Citation Context ...ff(G) is large can be efficiently decomposed in such a way that the resulting components have a small value of ff , and hence fall within the scope of the Monte Carlo algorithm; this idea was used in =-=[37]-=- to obtain an approximation scheme for general graphs whose running time, though still exponential, improves substantially on naive deterministic methods. 33 The question of whether ff is polynomially... |

28 |
Poor man’s Monte Carlo
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Citation Context ...r arbitrary dimensions d , they are of greatest interest in the case of low-dimensional lattices with 2sds4 . One key fact that holds in all dimensions was discovered in 1954 by Hammersley and Morton =-=[27]-=-; they observed that lim n!1 c 1=n n =sexists, and thatsnsc n =sn f(n) , where lim n!1 f(n) 1=n = 1 . This is a straightforward consequence of the obvious fact that 35 the sequence ` n = log c n is su... |

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Citation Context ...ial time algorithms where the statistical errors are rigorously controlled. Our algorithms are based on modifications and extensions of a Monte Carlo approach studied originally by Berretti and Sokal =-=[3]-=-. In the next subsection we sketch this approach and point out its limitations. Then, in section 3.1.3, we summarize our algorithms and explain how they overcome these problems. 36 3.1.2 Monte Carlo s... |

25 |
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Citation Context ... 1, Sokal and Thomas [62] argue that with such values of fi the Markov chain is rapidly mixing, i.e., it gets very close to stationarity after a number of steps that is only polynomial in n (see also =-=[46]-=-). In order to appreciate the role of fi here, consider a truncated version of this Markov chain in which the length of a walk is never allowed to exceed n , so that the stationary distribution is alw... |

20 | The Computational Complexity of Some Classical Problems from Statistical
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(Show Context)
Citation Context ...ested in counting the number of monomer-dimer arrangements with any fixed density. In this section we present some of the highlights in the history of this problem. For further information see, e.g., =-=[30, 42, 66]-=- and the references given there. The monomer-dimer problem gained prominence in 1937 through the early paper of Fowler and Rushbrooke [17]. A breakthrough was achieved in 1961, when, independently, Fi... |

18 | Testable algorithms for self–avoiding walks
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(Show Context)
Citation Context ...ations as well. These are the first algorithms for counting and generating self-avoiding walks where the error bounds are rigorously controlled. These results are based on work with Alistair Sinclair =-=[54]-=-. An optimized version of the algorithm was implemented by Rob Pike yielding numerical estimates for the number of self-avoiding walks in 2 and 3 dimensions; these estimates are presented in section 3... |

17 |
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Citation Context ...h have also been employed to obtain numerical estimates for the constantssand fl . Elementary considerations show thats2 (d; 2d \Gamma 1) . For d = 2 , it has actually been proven thats2 (2:62; 2:70) =-=[2, 10]-=-. (See also [29] for similar bounds in higher dimensions.) However, these rigorous bounds are much weaker than the non-rigorous estimates obtained by empirical methods, which are typically quoted to f... |

17 |
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(Show Context)
Citation Context ...) , where lim n!1 f(n) 1=n = 1 . This is a straightforward consequence of the obvious fact that 35 the sequence ` n = log c n is subadditive, i.e., ` n+ms` n + ` m for all n; m . Hammersley and Welsh =-=[28]-=- later showed that f(n) = O(a n 1=2 ) for some constant a . It is a celebrated and long-standing conjecture that f(n) is in fact polynomially bounded. More precisely, we have: Conjecture 1: c n =sn e ... |

17 | On the number of Eulerian orientations of a graph - Mihail, Winkler - 1992 |

16 |
Solution of the Dimer Problem on an Hexagonal Lattice with Boundary
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- 1984
(Show Context)
Citation Context ...e dimer covering problem for this planar lattice has been known for some time [65, 41]. In contrast to the rectangular lattice, the assumption of periodic boundary conditions is important here: Elser =-=[14]-=- has solved the dimer covering problem on a hexagonal lattice with fixed boundaries, and shown that the result depends significantly on the shape of the boundary. 21 has at most jEj neighbors, where j... |

15 |
Existence theorems and Monte Carlo methods for the monomer-dimer problem
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(Show Context)
Citation Context ...rectangular lattice with dimers, is one of the classical unsolved problems of solid-state chemistry. A few facts are known: for example, ln(f(n))=n 3 tends 19 to a finite limitsas n tends to infinity =-=[21]-=-. Hammersley [22] proved the lower bounds0:418347 , while the early paper by Fowler and Rushbrooke [17] showed the upper bounds0:54931 . It has been conjectured thatslies between 0:43 and 0:45 . In ot... |

13 |
Statistical mechanics of dimers on a plane lattice. II. Dimer correlations and monomers,” Phys
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(Show Context)
Citation Context ...y shed more light on other quantities related to monomerdimer systems, and in particular the correlation between monomers at two specified sites, as studied in two dimensions by Fisher and Stephenson =-=[16]-=-. The remainder of the chapter is organized as follows. In the next section, we prove bounds of the above form for rectangular lattices with periodic boundary conditions in any dimension, and with fix... |

13 |
Statistical theory of perfect solutions
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(Show Context)
Citation Context ...y of this problem. For further information see, e.g., [30, 42, 66] and the references given there. The monomer-dimer problem gained prominence in 1937 through the early paper of Fowler and Rushbrooke =-=[17]-=-. A breakthrough was achieved in 1961, when, independently, Fisher, Kasteleyn and Temperley provided an analytic solution for the case of dimer coveringss(i.e., arrangements with dimer density 1) on a... |

12 |
A Monte Carlo algorithm for estimating the permanent
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(Show Context)
Citation Context ...y other currently known algorithm, so it is of interest to establish precisely which graphs are amenable to it. (For other simpler, but apparently less widely applicable approximation algorithms, see =-=[38, 33]-=- and [55].) Moreover, it is conceivable that any graph G for which ff(G) is large can be efficiently decomposed in such a way that the resulting components have a small value of ff , and hence fall wi... |

12 |
Approximating the Permanent: A Simple Approach ,Random Structures and Algorithms 5
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Citation Context ...ntly known algorithm, so it is of interest to establish precisely which graphs are amenable to it. (For other simpler, but apparently less widely applicable approximation algorithms, see [38, 33] and =-=[55]-=-.) Moreover, it is conceivable that any graph G for which ff(G) is large can be efficiently decomposed in such a way that the resulting components have a small value of ff , and hence fall within the ... |

11 | Algorithmic research problems in molecular bioinformatics - Lengauer - 1993 |

9 | The number of polygons on a lattice - Hammersley - 1961 |

9 |
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Citation Context ...rigorous results holds for the problem at dimer densities less than 1, even in two dimensions. Notable exceptions are series expansions valid at low densities [19] and lower bounds on the free energy =-=[6, 26]-=-. 2.1.2 Results We make progress on the monomer-dimer problem in cases where the technique of Fisher, Kasteleyn and Temperley fails. Specifically, we give a polynomial-time algorithm for computing, to... |

8 |
An analysis of a Monte Carlo algorithm for estimating the permanent
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(Show Context)
Citation Context ...y other currently known algorithm, so it is of interest to establish precisely which graphs are amenable to it. (For other simpler, but apparently less widely applicable approximation algorithms, see =-=[38, 33]-=- and [55].) Moreover, it is conceivable that any graph G for which ff(G) is large can be efficiently decomposed in such a way that the resulting components have a small value of ff , and hence fall wi... |

7 |
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(Show Context)
Citation Context ...f(n) to good accuracy. A similar lack of rigorous results holds for the problem at dimer densities less than 1, even in two dimensions. Notable exceptions are series expansions valid at low densities =-=[19]-=- and lower bounds on the free energy [6, 26]. 2.1.2 Results We make progress on the monomer-dimer problem in cases where the technique of Fisher, Kasteleyn and Temperley fails. Specifically, we give a... |

7 |
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(Show Context)
Citation Context ...nomially with n for any fixed dimension d . A similar bound holds for arbitrary Cayley graphs. We stress that our Monte Carlo algorithm differs from earlier ones for monomer-dimer systems (see, e.g., =-=[23]-=-) in that it is guaranteed (independent of any heuristic arguments) to provide statistically reliable estimates in a running time that grows only polynomially with the number of lattice sites. Our pro... |