NI 92016, OUTP 92-54S, UCSBTH-92-50

Universal Amplitude Combinations for Self-Avoiding Walks, Polygons and Trails John L. Cardy^{∗} Isaac Newton Institute for Mathematical Sciences 20 Clarkson Road Cambridge CB3 0EH, UK Anthony J. Guttmann† Department of Physics Theoretical Physics University of Oxford 1 Keble Road Oxford OX1 3NP, UK

We give exact relations for a number of amplitude combinations that occur in the study of self-avoiding walks, polygons and lattice trails. In particular, we elucidate the lattice-dependent factors which occur in those combinations which are otherwise universal, show how these are modified for oriented lattices, and give new results for amplitude ratios involving even moments of the area of polygons. We also survey numerical results for a wide range of amplitudes on a number of oriented and regular lattices, and provide some new ones.

∗ Permanent address: Dept. of Physics, University of California, Santa Barbara CA 93106, USA † Permanent address: Dept. of Mathematics, University of Melbourne, Parkville, Vic 3052, Australia 1. Introduction.

In this paper we consider a number of amplitude combinations that are universal, up to explicit lattice-dependent factors, in the context of the N → 0 limit of the O(N) model, which describes self-avoiding walks and polygons. We correct and extend a table of such values given previously by one of us [1], and correct and generalise an exact expression for a particular amplitude combination given by the other [2]. We also point out an apparent error in another amplitude combination given previously [3]. We make clear the role that lattice-dependent factors (such as the number of sites per unit area) play in some of these otherwise universal combinations. In particular, we point out for the first time a distinction between oriented and unoriented lattices. In order to confirm these results, we have generated short series for the radius of gyration of self-avoiding polygons on the triangular and honeycomb lattices. We also give arguments for the universality and lattice independence of amplitude ratios involving even moments of the area of self-avoiding polygons.

The functions we are considering are: (i) the chain generating function for SAWs, C(x) = Σc_{n}x^{n}; (ii) the corresponding polygon generating function, P (x) = Σp_{n}x^{n}; (iii) the generating function for lattice trails, T (x) = Σt_{n}x^{n}; (iv) the generating function for dumb-bell graphs Δ(x) = Σd_{n}x^{n}; (v) the mean-square end-to-end distance of n-step selfavoiding walks (SAWs) ⟨R_{e}^{2}⟩_{n}; (vi) the mean-square radius of gyration of n-step polygons ⟨R^{2}⟩_{n}; (vii) the mean-square radius of gyration of n-step SAWs ⟨R_{g}^{2}⟩_{n}; (viii) moments of the area of polygons of perimeter n, ⟨a^{p}⟩_{n}; and (ix) the mean-square distance of a monomer from the origin of n-step SAWs ⟨R_{m}^{2}⟩_{n}. In the above generating functions, c_{n}, p_{n}, t_{n} and d_{n} denote the total number of n-step SAWs, polygons, trails and dumb-bells. For SAWs and trails, an origin is chosen on an infinite lattice, and all distinct SAWs and trails are enumerated. For dumb-bells and polygons, we adopt the normal convention in which the total number per lattice site is given. As we discuss below, this is different from the number of distinct unrooted polygons (up to translations) on certain lattices, such as the honeycomb lattice. This distinction is one source of error in Ref. 1.

In terms of the quantities defined above, we denote the relevant amplitudes as follows:

c_{n} = Aμ^{n}n^{γ−1}[1 + o(1)]
p_{n} = Bμ^{n}n^{α−3}[1 + o(1)]
t_{n} = Hμ^{n}n^{γ−1}[1 + o(1)]
d_{n} = Jμ^{n}n^{γ−1}[1 + o(1)] ;

(where the growth exponent μ for trails applies only to lattices of coordination number three [4]), and

⟨R_{e}^{2}⟩_{n} = Cn^{2ν}[1 + o(1)]
⟨R^{2}⟩_{n} = Dn^{2ν}[1 + o(1)]  (1)
⟨a^{p}⟩_{n} = E^{(p)}n^{2pν}[1 + o(1)]
⟨R_{g}^{2}⟩_{n} = F n^{2ν}[1 + o(1)]
⟨R_{m}^{2}⟩_{n} = Gn^{2ν}[1 + o(1)] ;

where γ = 43/32, α = \frac{1}{2}, ν = \frac{3}{4} and μ = x_{c}^{−1}, the reciprocal of the critical point. Amplitudes B, D and E^{(p)}, which relate to polygons, will be zero for odd-order terms on loose packed lattices, and will be non-zero only for every fourth term on certain lattices such as the L-lattice (an oriented square lattice in which every step must be followed by a step perpendicular to the preceding step) and the Manhattan lattice. We should remind the reader that the above asymptotic forms, and the exact values of the critical exponents, are assumptions in the sense that they have not been proven rigorously. Nevertheless they all follow from the central assumption that these problems have a continuum limit which corresponds to a particular exactly soluble field theory, and it is this correspondence on which shall base our analytic results. The appearance of the same growth exponent μ in each of the above has been rigorously demonstrated, however.

In the next section we generalise and correct the first amplitude relation, which gives the value of BD. We subsequently discuss the other known relations, and correct the relation for the combination BC. We then use these to provide amplitude estimates for some cases in which they have not been directly estimated.

2. The combination BD.

Rather than giving the modifications to the argument in Ref. 2 (which in fact contains several errors of factors of 2), it is clearer to repeat the whole argument for a general lattice. The universality of the combination BD follows from the integral form [5] of Zamolodchikov’s c-theorem [6], which reads

c = 3πt^{2}ν^{−2} ∫ r^{2}⟨E(r)E(0)⟩_{c}d^{2}r ,  (2)

where E(r) is the energy density, which enters the Hamiltonian in the form t ∫ E(r)d^{2}r, and ν is the usual critical exponent governing the divergence of the correlation length. We wish to apply this to the O(N) model whose lattice Hamiltonian is H = −x ∑_{links} E_{lat}(r) where r now labels links, and E_{lat}(r) = s⃗(r_{<})·s⃗(r_{>}). Here s⃗(r_{<}) and s⃗(r_{>}) are O(N) spins located at the sites r_{<} and r_{>} at the ends of the link at r, ordered in some standard fashion. In order to apply Eq. (2) correctly, we must relate the continuum energy density E(r) to its lattice counterpart. This is done by equating the continuum and lattice Hamiltonians, so that

t ∫ E(r)d^{2}r ⟶ (x_{c} − x) ∑_{r} E_{lat}(r) .  (3)

It is convenient to work temporarily on a very large but finite lattice of total area A, and to rewrite Eq. (2), using translational invariance, as

c = 3πt^{2}ν^{−2}A^{−1} ∫ ∫ (r − r′)^{2}⟨E(r)E(r′)⟩_{c} d^{2}rd^{2}r′ .

It is then apparent from Eq. (3) that we may simply replace this by its lattice version

c = 3π(x_{c} − x)^{2}ν^{−2}A^{−1} ∑_{r,r′}(r − r′)^{2}⟨E_{lat}(r)E_{lat}(r′)⟩_{c} .  (4)

This should be valid in the scaling limit as x → x_{c}, when the integral will be dominated by values of |r − r′| on the scale of the correlation length ξ, which is much larger than the lattice spacing, so that the continuum approximation becomes arbitrarily accurate.

The next step is to evaluate the right hand side of Eq. (4) in the limit N → 0 as a sum over pairs of mutually self-avoiding walks connecting (r_{<}, r_{>}) and (r′_{<}, r′_{>}). These pairs of walks are then identified with self-avoiding polygons where the links at r and r′ are marked. The contribution of a given polygon of length n, whose links are at (r_{1}, r_{2}, …, r_{n}), to the sum in Eq. (4) is then

Nx^{n−2}∑_{k,l}′(r_{k} − r_{l})^{2} ,  (5)

where the prime on the sum indicates that adjacent links are to be excluded. This excludes contributions where, for example, r_{>} coincides with r′_{<}. Such contributions are certainly included in the right hand side of Eq. (4), but they are more complicated to evaluate. Fortunately they are expected to be negligible in the scaling limit. It is worth checking the normalisation in Eq. (5) for small polygons. For example, for n = 4 on a square lattice, the sum on the right hand side of Eq. (4) gives 2NN_{b}x^{2}, where N_{b} is the total number of links, while Eq. (5) gives 4Nx^{2} for each elementary square, of which there are N_{b}/2. It is straightforward to check this also for larger polygons. Now, if the sum in Eq. (5) were unrestricted, it would be equal to 2n^{2}R_{b}^{2}, where R_{b}^{2} is the link-weighted squared radius of gyration of the polygon (which differs from the corresponding site-weighted quantity by terms of O(1) as n → ∞.) However, the effect of the restriction is only at the level of subleading terms down by one power of n, and is negligible in the scaling limit. If now we denote the mean square radius of gyration over all polygons of perimeter n by ⟨R^{2}⟩_{n}, and the total number of such polygons by N_{s}p_{n}, where N_{s} is the total number of sites on our large lattice, then, in the limits N → 0 and x → x_{c},

c(N) ∼ 6πN(x_{c} − x)^{2}ν^{−2} (\frac{N_{s}}{A} ) ∑_{n} n^{2}p_{n}⟨R^{2}⟩_{n}x^{n−2} .  (6)

This implies that the sum on the right hand side has a singularity of the form (x_{c} − x)^{−2}. Since each term in these series is non-negative, this singularity on the positive real axis gives the radius of convergence, and, for close-packed lattices, we assume that there are no other singularities on |x| = x_{c}. Thus this singularity solely dictates the behaviour of the terms as n → ∞. For loose-packed and oriented lattices, however, p_{n} is non-zero only if n is divisible by an integer σ. In this case the generating function is invariant under rotations x → xe^{2πi/σ}, so that there must exist singularities of equal strengths at x = x_{c}e^{2πir/σ} where r = 0, 1, …, σ − 1. Thus, the large n behaviour of p_{n}, for n divisible by σ, is greater by a factor σ than that expected on the basis of the singularity described by Eq. (6).

We conclude that

lim_{n→∞} np_{n}⟨R^{2}⟩_{n}x_{c}^{n} = σa_{0} \frac{c′(0)ν^{2}}{6π} ,

where we have introduced the area per site a_{0} = A/N_{s}. From the known values [7] c = 1 − 6/m(m + 1) and ν = (m + 1)/4, where N = 2 cos(π/m), for the O(N) model, the second numerical factor is 5/32π^{2}, so that, in terms of the amplitudes defined previously, we have

BD = \frac{5}{32π^{2}} σa_{0} .  (7)

For the square lattice, σ = 2 and the area per site (measured in units of square lattice spacings) is unity. This then gives the result 5/16π^{2} found previously [2], which agrees with results from enumerations given in Ref. 8, as well as later data summarised in the final section.

For a general periodic lattice, we may write the factor a_{0} = A/N_{s} as v/κ, where v is the area of the unit cell, and κ counts the the number of sites per unit cell. In the above, we have defined p_{n} in the conventional way as the number of n-step polygons per site. If instead, as in Ref. 3, it is defined as the number of distinct n-step unrooted polygons (where two such polygons are regarded as being equivalent if they are related by translation by a lattice vector) then we may associate with each such polygon a unique site. This may be done, for example, by ordering the sites in a systematic manner, and associating with a given polygon the lowest site according to this ordering. In this way of counting, N_{s} should be replaced by the total number of sites which are equivalent to each other by lattice translations. These lattice translations divide the sites into equivalence classes, and there is just one representative of each equivalence class corresponding to each unit cell. Thus, with this definition of p_{n}, the factor κ does not enter the formula for BD. This is most easily seen by considering polygons on the honeycomb lattice. The first term in the generating function is p_{6} = \frac{1}{2} if the normalisation is per site, and then κ = 2. Alternatively, p_{6} = 1 gives the number of 6-step unrooted polygons.

This result may easily be generalised to polygons on oriented lattices. In this case each link r has a specified orientation, say from r_{<} to r_{>}. Oriented walks on such a lattice are described by the N → 0 limit of a complex O(N) spin model, where now the link energy is E^{lat}(r) = s⃗(r_{>})^{∗} · s⃗(r_{<}). Proceeding as before, the sum on the right hand side of Eq. (4) is now equal to

N ∑_{n} x^{n−2} ∑_{all n−step oriented polygons} ∑_{k,l}′(r_{k} − r_{l})^{2} ,

so that the appropriate lattice-dependent factor is the total number of n-step oriented polygons divided by the total area. If we define p_{n} as the number of n-step oriented polygons per site, then this factor is equal to p_{n}/a_{0}, as before. However, the complex O(N) model is equivalent to a real O(2N) model, and therefore the appropriate value of the central charge on the left hand side of Eq. (6) is c(2N). This leads to an extra factor of 2 on differentiating with respect to N in the N → 0 limit, so the final result for oriented lattices is BD/σa_{0} = 5/16π^{2}, as compared with Eq. (7) for the unoriented case. If we define p_{n} as the number of distinct unrooted oriented polygons, as in Ref. 3, then a_{0} should be replaced by v, the unit cell size, in the above formula.

The results for unoriented and oriented lattices may be unified by supposing that in the unoriented case each link may be oriented in either direction, and by defining p_{n} in both cases in terms of the number of oriented polygons. The values of p_{n} for unoriented lattices will then be doubled, since each polygon on such a lattice has precisely two orientations. The result above will then apply to both types of lattice, and, presumably, to partially oriented lattices also. It makes physical sense, because if one looks at polygons with a fixed R^{2}, then the number of oriented polygons on an oriented lattice should be asymptotically the same as the number of such polygons on an unoriented, or partially oriented, lattice.

3. Polygon area amplitudes.

Next we consider the universality of amplitudes involving the area of polygons. Our approach is as follows: let \overrightarrow{dl} be an infinitesimal element of the curve formed by the polygon. Then, by Stokes’ theorem, the area within the polygon is proportional to ∮ r⃗ × \overrightarrow{dl} We can write this equivalently as ∮ (r⃗ × J⃗)dl, where J⃗ is a unit current flowing through the links of the polygon. Thus, moments of the area distribution are related to integrals over correlation functions of J⃗. Such a current will be conserved in the ensemble of polygons. In the continuum field theory, it turns out that it is possible to identify the corresponding current. Conserved currents in field theory enjoy the special property that their correlation functions scale according to naive dimensional counting arguments. Moreover, the normalization of this particular current is fixed by considering its behavior in the larger ensemble including self-avoiding walks with ends. By definition, each end will be a unit source or sink for this current. Thus the functional form, and overall normalization, of these correlation functions are in principle completely determined in the field theory.

We now put some flesh on these remarks. Consider a given oriented polygon, and let J⃗^{lat} = (J_{1}^{lat}, J_{2}^{lat}) be a unit current flowing along the links, each of length b and labelled by r⃗, in the direction of their orientation. Then the sum

a = \frac{1}{2}b ∑_{r⃗} ϵ_{ij}r_{i}J_{j}^{lat} ,

where ϵ_{ij} is the totally antisymmetric symbol, gives the signed area of the polygon (that is, the area is positive for anticlockwise orientation, negative for clockwise orientation.) The mean signed area over all polygons with n steps is of course zero, since opposite orientations of a given polymer give an equal and opposite contribution. However, the mean square area, and, indeed, all the higher even moments, are non-zero and the same as for unoriented polygons. The squared area of a given polygon is

a^{2} = \frac{1}{4}b^{2} ∑_{r,r′} (r_{i}r′_{i}J_{j}^{lat}(r)J_{j}^{lat}(r′) − r_{i}r′_{j}J_{i}^{lat}(r)J_{j}^{lat}(r′)) .

Using ∑_{r} J_{i}^{lat}(r) = 0 and ∑_{r} r_{i}J_{i}^{lat}(r) = 0, the right hand side may be rewritten as −\frac{1}{8}b^{2} ∑_{r,r′}(r − r′)^{2}J_{j}^{lat}(r)J_{j}^{lat}(r′).

Oriented self-avoiding polygons and walks are described by a complex O(N) spin model in the limit N → 0. The lattice degrees of freedom are complex O(N) spins, and the Hamiltonian is H = −x ∑_{r} s⃗(r_{>})^{∗} · s⃗(r_{<}) + c.c.. Within this model, we may identify

J⃗^{lat}(r) = xb^{−1}(r⃗_{>} − r⃗_{<})(s⃗(r_{>})^{∗} · s⃗(r_{<}) − s⃗(r_{<})^{∗} · s⃗(r_{>})) .

Thus, if ⟨a^{2}⟩_{n} denotes the mean square area of oriented polygons of n steps, and, as before, 2N_{s}p_{n} is the number of such polygons (the factor of 2 is correct on an unoriented lattice, if p_{n} counts the number of unoriented polygons per site),

2N_{s} ∑_{n} p_{n}⟨a^{2}⟩_{n}x^{n} = −\frac{1}{8}b^{2} ∑_{r,r′}(r − r′)^{2}⟨J⃗^{lat}(r) · J⃗^{lat}(r′)⟩ .  (8)

In the continuum limit, J⃗^{lat} is replaced by a conserved current J⃗, which is the Noether current for the U(1) transformations in the complex O(N) field theory which commute with the O(N) rotations [9]. The connection between them is simply

∑_{r} J⃗^{lat}(r) ⟶ ∫ J⃗(r)d^{2}r ,

whereby the double sum over r and r′ in Eq. (8) becomes a double integral, which, by translational invariance, equals

−\frac{1}{8}Ab^{2} ∫ r^{2}⟨J⃗(r) · J⃗(0)⟩d^{2}r .  (9)

Now the main point is that the normalisation of J⃗ in the continuum limit is fixed by the requirement that the unit source for this current is the free end of an oriented SAW [9].

This means that if ϕ(r) is the continuum field corresponding to s(r) (suppressing O(N) indices), representing the magnetisation density of the O(N) model, then

\begin{aligned}∫ J⃗(r′) · \overrightarrow{dS}_{r′} ϕ(r) & = ϕ(r) , \\ ∫ J⃗(r′) · \overrightarrow{dS}_{r′} ϕ(r)^{∗} & = −ϕ(r)^{∗} , \end{aligned}  (10)

where the integral is over the boundary of a small neighbourhood of r, and \overrightarrow{dS}_{r′} is the outward pointing normal. Eq. (10) is supposed to make sense when inserted into correlation functions with other operators. It specifies [9] the normalisation of the pole term in the short-distance expansion of J⃗ with ϕ and ϕ^{∗}.

We see from Eq. (10) that J⃗ has unit scaling dimension (since J⃗ is a conserved current which generates a symmetry of the theory, it has no anomalous dimension.) The fact that the normalisation of J⃗ is fixed, and therefore does not contain any metric factors, implies that its two-point function has the universal form

⟨J⃗(r) · J⃗(0)⟩ = ξ^{−2}f(r/ξ) ,  (11)

where ξ is the correlation length. Therefore the integral in Eq. (9) is of the form Uξ^{2} ∼ Uξ_{0}^{2}(1 − x/x_{c})^{−2ν}, where U is a universal number, in principle calculable from the underlying continuum field theory, and ξ_{0} is the non-universal metric factor in the correlation length.

It then follows from Eq. (8,11) that

p_{n}⟨a^{2}⟩_{n} ∼ σa_{0}U′ξ_{0}^{2} n^{2ν−1}x_{c}^{−n} ,

where U′ is universal. Now we know that ⟨R^{2}⟩_{n} is related to the ratio of the second to the zeroth moments of the energy-energy correlation function in the O(N) model, which is of the form U′′ξ^{2}, where again U′′ is universal. Therefore the amplitude D defined in Eq. (1) has the form U′′ξ_{0}^{2}. Also p_{n} ∼ Bn^{−2ν−1}x_{c}^{−n}. Combining these, we find

\frac{⟨a^{2}⟩_{n}}{⟨R^{2}⟩_{n}^{2}} ∼ b^{2}U′′′ \frac{σa_{0}}{BD} .

But we argued above that the combination BD/σa_{0} should be universal and lattice independent. This establishes the universality and lattice independence of the ratio on the left hand side, provided the area is measured in square lattice spacings.

If we now consider the higher even moments of the area, the argument generalises straightforwardly. Schematically,

N_{s} ∑_{n} p_{n}⟨a^{p}⟩_{n}x^{n} ∼ b^{p} ∫ … ∫ r_{1} … r_{p}⟨J(r_{1}) … J(r_{p})⟩d^{2}r_{1} … d^{2}r_{p} ,

where all indices have been suppressed. The integral on the the right hand side is equal to AU^{(p)}ξ^{2p−2}, where U^{(p)} is universal. Following the argument through, we then discover that ⟨a^{p}⟩_{n}/⟨R^{2}⟩_{n}^{p} should be universal, with no lattice dependent factors.

However, it appears difficult to extend this result to odd moments of the unsigned area. This is because these quantities appear not to have any simple expression in terms of correlation functions in the continuum field theory. Nevertheless, in Ref. 10 it was argued that the corresponding ratio involving the first moment of the area ⟨a^{1}⟩_{n}/⟨R^{2}⟩_{n} = E^{(1)}/D should be universal, with no lattice dependent factors. This is to be expected on physical grounds [11], since if lattice polygons are regarded as a model for 2-dimensional vesicles, the area couples to the pressure p. Since p and ⟨R^{2}⟩ are macroscopically measurable quantities, one might expect a universal relationship between them independent of the particular lattice model. Nevertheless, a theoretical demonstration of this is so far lacking. In Ref. 12, an argument was made for the finiteness of ratio E^{(1)}/D, but this does not prove its universality. Note that [11] the fact that moments of the area satisfy the inequalities

⟨a^{1}⟩ ≤ ⟨a^{2}⟩^{\frac{1}{2}} ≤ ⟨a^{3}⟩^{\frac{1}{3}} ≤ …,

together with our results for the even moments, does imply that the higher odd moments ⟨a^{p}⟩ do scale as expected for p ≥ 3, but it does not imply that their ratios are universal or lattice independent. As pointed out [11], it is essential that area be measured in Euclidean units such that the lattice spacing is the same in calculating the radius of gyration and the mean area, nd its moments.

4. Numerical results.

Another universal amplitude ratio, first given in Ref. 13 and corrected in Ref. 14 is

(2 + \frac{y_{t}}{y_{h}})\frac{F}{C} − \frac{2G}{C} + \frac{1}{2} = 0 .

The individual quotients F/C and G/C are also universal [13]. The best estimates of these quantities is given in Ref. 14, as F/C = 0.14029 ± 0.00012 and G/C = 0.43962 ± 0.00033, obtained by Monte Carlo calculations.

The quantity BC/σa_{0} has been shown in Ref. 3 to be universal. These authors expressed their result in terms of BC/vσ, but they used a definition of p_{n} as the number of distinct unrooted polygons. As discussed earlier, these results are completely equivalent. It is possible to show directly from the scaling forms of the relevant correlation functions in the O(N) model that C/D is universal. This also follows from Ref. 13 and Ref. 15, where it was argued that F/C and F/D, respectively, are universal. Hence, the universality of BC/σa_{0} follows from our result for BD/σa_{0}.

The value of C/D should also be the same for walks on oriented lattices. Since we have argued that in this case BD/σa_{0} gains an additional factor of two, this should carry over to BC/σa_{0}.

The amplitude for dumb-bells, J, is related to the amplitude for SAWs A by

J = A(1 − 2τ/μ + τ^{2}/μ^{2})/8 ,  (12)

as shown in Ref. 16, where τ = q − 1 and q is the lattice coordination number. It was also shown there that the amplitude for lattice trails, H, on the honeycomb lattice only, is related to the SAW amplitude by

H = \frac{4A}{2 + √2} .  (13)

Details of the various series and their analysis to estimate critical amplitudes have already been discussed in Ref. 1. Since then, extension of the square lattice SAW series [17] and the square lattice trail series [18] have allowed more accurate estimates of these amplitudes. These are given in Table I below. Corrected amplitudes for the honeycomb lattice are also given in Table I. To ensure that the various normalisation factors were correct, we generated and analysed the series for the radius of gyratio for triangular lattice and honeycomb lattice polygons, and estimated the radius of gyration amplitude directly. These new series are given in Table II, and the amplitude estimates in Table I. The quantity E(triangular) was given in Ref. 1 in units of unit triangles. It is more correctly given in Table I in units of area, assuming unit lattice spacing, as also used in the estimate of other metrically dependent amplitudes. Direct estimates of the amplitudes A, B and C for the L and Manhattan lattices are also given. These have been obtained from the series in Ref. 19 and Ref. 20. The polygon counts for Manhattan lattice polygons quoted in Ref. 19 should be divided by 2 in order that the correct normalisation per-site be retained.

The square lattice amplitudes are generally the most accurate. We see that 32π^{2}BD/5σa_{0} = 1.0000 for the square lattice, 1.020 for the honeycomb lattice and 0.997 for the triangular lattice. This is perfect agreement, given the accuracy of the amplitude estimates, and allows D(L, Man.) to be estimated. Similarly, the invariant BC/σa_{0} = 0.2168 (square), 0.2167 (triangular) and this value is used to predict C(honeycomb). The ratio F/C = 0.1403(square), 0.1402 (triangular), in good agreement with the Monte Carlo estimate cited above. This then permits F (honeycomb, L, Man.) to be estimated. We see that this quantity is a factor 2 greater for the oriented L and Manhattan lattices, as predicted. This factor was not found by Privman and Redner [3], who found that (with their convention for p_{n} as the number of distinct unrooted oriented polygons) the same result for BC/vσ as for unoriented lattices, within their numerical errors. It seems likely that these authors miscounted by a factor of 2. For example, the number of distinct oriented 4-step polygons on the L lattice is 2, not 1.

Similarly, G/C = 0.4397(square), and 0.4402(triangular), again in agreement with the precise Monte Carlo estimate. The predicted values of G(honeycomb, L, Man) are given in Table I. The ratio E/D (where E is E^{(1)} in our previous notation) was found to be 2.515(square) and 2.529(triangular), where we believe the square lattice estimate to be more precise, as the series from which E was estimated is far longer for the square lattice. This value then permits E(honeycomb, L, Man.) to be estimated. The amplitudes J follow from the amplitudes for A and Eq. (12). The amplitude H(honeycomb) follows from A(honeycomb) and Eq. (13). All amplitudes quoted are expected to have an associated error confined to the last decimal place quoted. Apart from some minor gaps for the triangular, Manhattan and L lattice amplitude, Table I now gives a complete and corrected tabulation of critical amplitudes. Acknowledgments. We would like to thank Prof. Michael E. Fisher for valuable comments and correspondence, and Mr. Andrew Conway for programming assistance in calculating the radii of gyration. This work was supported in part by the Isaac Newton Institute, the UK Science and Engineering Research Council, the US National Science Foundation Grant PHY 91-16964, and by the Australian Research Council.

Tables

Table I. Estimates of critical amplitudes A … J defined in the text, for the honeycomb, square, triangular, L and Manhattan lattices. Quantities in parentheses are estimates from the amplitude relations discussed in the text. The remainder are estimates from the corresponding series expansions.

Amplitude Honeycomb Square Triangular  L  Manhattan
A  1.145  1.1771  1.186  1.05  0.89
B  0.6358  0.5623  0.2640  2.47  2.5
C  (0.889)  0.771  0.711  0.67  0.73
D  0.0660  0.05631  0.0518 (0.049) (0.053)
E  (0.166)  0.1416  0.131  (0.12)  (0.13)
F  (0.125)  0.1082  0.0997 (0.095) (0.10)
G  (0.389)  0.339  0.313  (0.30)  (0.32)
H  (1.341)  1.272
J  (0.000972) (0.002768) (0.006205)
μ  1.8477591 2.6381585 4.1507951 1.5657  1.733
σ  2  2  1  4  4
a_{0}  3√3/4  1  √3/2  1  1
32π^{2}BD/5σa_{0}  1.020  1.0000  0.997
BC/σa_{0}  0.2168  0.2167  0.41  0.46

Table II. Number of polygons and the sum of the squares of the radii of gyration of n-step polygons on the triangular and honeycomb lattices.

Triangular  Honeycomb
n  p_{n}  4n^{2}p_{n}⟨R^{2}⟩_{n}  n 2p_{n} 8n^{2}p_{n}⟨R^{2}⟩_{n}
3  2  24  6  1  144
4  3  96  8  0  0
5  6  408  10  3  2460
6  15  1872  12  2  3168
7  42  8688  14 12  32052
8  123  39912  16 18  77976
9  380  183264  18 65  420444
10  1212  834744  20 138  1310088
11  3966  3779064  22 432  5655204
12 13265  17013936  24 1074 19291968
13 45144  76186320  26 3231 76066992
14 155955  339566616 28 8718 268063080
15 545690 1507025568 30 25999 1011675420
16 1930635 6662739096
17 6897210 29355291552
18 24852576 128932421592

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