Annals of Combinatorics 3 (1999) 191-203

Planar Lattice Gases with Nearest-Neighbor Exclusion

R.J. Baxter

Theoretical Physics, I.A.S. and School of Mathematical Sciences, The Australian National University, Canberra, ACT 0200, Australia

Received November 27, 1998

AMS Subject Classification: 05B20, 82B20, 82B80

Abstract. We discuss the hard-hexagon and hard-square problems, as well as the corresponding problem on the honeycomb lattice. The case when the activity is unity is of interest to combinatorialists, being the problem of counting binary matrices with no two adjacent 1's. For this case, we use the powerful corner transfer matrix method to numerically evaluate the partition function per site, density and some near-neighbor correlations to high accuracy. In particular, for the square lattice, we obtain the partition function per site to 43 decimal places.

Keywords: combinatorics, statistical mechanics, legal matrices, lattice models, nearest-neighbor exclusion, hard squares, hard hexagons


1.  R.J. Baxter, Variational approximations for square lattice models in statistical mechanics, J. Stat. Phys. 19 (1978) 461–478.

2.  R.J. Baxter, Hard hexagons: Exact solution, J. Phys. A 13 (1980) L61–L70.

3.  R.J. Baxter, Corner transfer matrices, Physica A 106 (1981) 18–27.

4.  R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic, London, 1982.

5.  R.J. Baxter, Numerics, conjectures and exact results, Phys. Chem. and Earth Science 30 (1990) 24.

6.  R.J. Baxter and I.G. Enting, Series expansions from corner transfer matrices: the square lattice ising model, J. Stat. Phys. 21 (1979) 103–123.

7.  R.J. Baxter, I.G. Enting, and S.K. Tsang, Hard square lattice gas, J. Stat. Phys. 22 (1980) 465–489.

8.  R.J. Baxter and S.K. Tsang, Entropy of hard hexagons, J. Phys. A: Math. Gen. 13 (1980) 1023–1030.

9.  N.J. Calkin and H.S. Wilf, The number of independent sets in a grid graph, SIAM J. Discrete Math. 11 (1998) 54–60.

10.  S. Finch, private communication; Hard square entropy constant, MathSoft. Inc.,

11.  S. Finch, Several constants arising in statistical mechanics,

12.  K. Engel, On the Fibonacci number of an m by n lattice, Fibonacci Quarterly 28 (1990) 72–78.

13.  I.G. Enting, Triplet order parameters in triangular and honeycomb Ising models, J. Phys. A: Math. Gen. 10 (1977) 1737–1743.

14.  M.E. Fisher, Transformations of Ising models, Phys. Rev. 113 (1959) 969–981.

15.  M.E. Fisher, Perpendicular susceptibility of the Ising model, J. Math. Phys. 4 (1963) 124–135.

16.  M.E. Fisher, Lattice statistics --- a review and an exact isotherm for a plane lattice gas, J. Math. Phys. 4 (1963) 278–286.

17.  D.S. Gaunt, Hard-sphere lattice gases, II. Plane-triangular and three- dimensional lattices, J. Chem. Phys. 46 (1967) 3237–3259.

18.  D.S. Gaunt and M.E. Fisher, Hard-sphere lattice gases, I. Plane-square lattice, J. Chem. Phys. 43 (1965) 2840–2863.

19.  G.S. Joyce, On the hard-hexagon model and the theory of modular functions, Phil. Trans. Roy. Soc. London A 325 (1988) 643–702.

20.  H.A. Kramers and G.H. Wannier, Statistics of the two-dimensional ferromaget. Part I, Phys. Rev. 60 (1941) 252–262.

21.  N.G. Markley and M.E. Paul, Maximal measures and entropy for Zn subshifts of finite type, In: Classical Mechanics and Dynamical Systems, Dekker Lecture Notes, Vol. 70, R.L. Devaney and Z.H. Nitecki, Eds., 1981, pp. 154–155.

22.  B.D. McKay, private communication (1996)

23.  B.D. Metcalf and C.P. Yang, Degeneracy of anti-ferromagnetic Ising lattices at critical magnetic field and zero temperature, Phys. Rev. B 18 (1978) 2304–2307.

24.  S. Milosevic, B. Stosic, and T. Stosic, Towards finding exact residual entropies of the Ising ferromagnets, Physica A 157 (1989) 899–906.

25.  P.A. Pearce and K.A. Seaton, A classical theory of hard squares, J. Stat. Phys. 53 (1988) 1061–1072.

26.  L.K. Runnels and L.L. Coombs, Exact finite method of lattice statistics. I. Square and triangular lattice gases of hard molecules, J. Chem. Phys. 45 (1966) 2482–2492.

27.  L.K. Runnels, L.L. Coombs, and J.P. Salvant, Exact finite method of lattice statistics, II. Honeycomb lattice gas of hard molecules, J. Chem. Phys. 47 (1967) 4015–4020.

28.  M.F. Sykes, J.W. Essam, and D.S.Gaunt, Derivation of low-temperature expansions for the Ising model of a ferromagnet and an anti-ferromagnet, J. Math. Phys. 6 (1965) 283–298.

Get the | DVI| PS | PDF file of this abstract.