Abstract
We investigate the solvability of a variety of well-known problems in lattice statistical mechanics. We provide a new numerical procedure which enables one to conjecture whether the solution falls into a class of functions calleddifferentiably finite functions. Almost all solved problems fall into this class. The fact that one can conjecture whether a given problem is or is not D-finite then informs one as to whether the solution is likely to be tractable or not. We also show how, for certain problems, it is possible to prove that the solutions are notD-finite, based on the work of Rechnitzer [1–3].
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References
A Rechnitzer,Adv. Appl. Math. 30, 228 (2003)
A Rechnitzer, Haruspicy 2: The self-avoiding polygon generating function is not Definite, accepted for publication inJ. Combinatorial Theory-Series A
A Rechnitzer, Haruspicy 3: The directed bond-animal generating function is not D-finite, to appear inJ. Combinatorial Theory-Series A
L Onsager,Phys. Rev. 65, 117 (1944)
C N Yang,Phys. Rev. 85, 808 (1952)
P W Kasteleyn,Physica 27, 1209 (1961)
M E Fisher,Phys. Rev. 124, 1644 (1961)
E H Lieb,Phys. Rev. Lett. 18, 1046 (1967)
R J Baxter,Phys. Rev. Lett. 26, 834 (1971)
R J Baxter,J. Phys. A13, L61 (1980)
GFUN, a program developed by B Salvy, P Zimmerman, F Chyzak and colleagues at INRIA, France. Available from http://pauillac.inria.fr/algo
A J Guttmann, Asymptotic Analysis of Power Series Expansions, inPhase transitions and critical phenomena, edited C Domb and J Lebowitz (Academic Press, 1989) vol. 13, pp. 1–234
T de Neef and I G Enting,J. Phys. A10, 801 (1977)
I G Enting,Nucl. Phys. B47, 180 (1996)
I Jensen and A J Guttmann,J. Phys. A32, 4867 (1999)
I Jensen,J. Phys. A37, 5503 (2004)
W P Orrick, B Nickel, A J Guttmann and J H H Perk,Phys. Rev. Lett. 86, 4120 (2001)
W P Orrick, B Nickel, A J Guttmann and J H H Perk,J. Stat. Phys. 102, 795 (2001)
R J Baxter,Exactly solved models in statistical mechanics (Academic Press, London, 1982)
J M Hammersley,Proc. Camb. Phil. Soc. 57, 516 (1961)
J M Hammersley and D J A Welsh,Q. J. Math. 2nd series (Oxford, 1962) vol.13, pp. 108–10
R P Stanley,Enumerative combinatorics Vol. 2, volume 62 ofCambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 1999)
A J Guttmann and I Jensen (in preparation)
R J Baxter, inFundamental Problems in Statistical Mechanics edited by E G D Cohen (North Holland, Amsterdam 1981) vol. 5, pp. 109–41
R J Baxter,J. Stat. Phys. 28, 1 (1982)
A J Guttmann and I G Enting,Phys. Rev. Letts 76, 344 (1996)
H N V Temperley,Phys. Rev. 103, 1 (1956)
M Bousquet-Mélou,Disc. Math. 154, 1 (1996)
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Guttmann, A.J. The analytic structure of lattice models — why can’t we solve most models?. Pramana - J Phys 64, 829–846 (2005). https://doi.org/10.1007/BF02704146
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DOI: https://doi.org/10.1007/BF02704146
Keywords
- Solvability
- differentiably finite
- bond animal
- Ising model
- susceptibility
- self-avoiding walks
- self-avoiding polygons