Canonical Monte Carlo determination of the connective constant of self-avoiding walks

We define a statistic an(w), the size of the atmosphere of a self-avoiding walk, w, of length n, with the property that 〈an(w)〉 → μ as n → ∞, where μ is the growth constant of lattice self-avoiding walks. Both μ and the entropic exponent γ may be estimated to high precision from 〈a(w)〉 using canonical Monte Carlo simulations of self-avoiding walks. Previous Monte Carlo measurements of μ and γ have used grand canonical Monte Carlo simulations. Our simulations indicate that μ = 2.638 16 ± 0.000 06 and γ = 1.345 ± 0.002. These results, based on a modest computer run, are comparable to the best estimates for μ and γ from (grand canonical) Monte Carlo simulations, and are at most two digits of the best series estimates of μ for self-avoiding walks available in the literature. PACS number: 05.50.+q The lattice self-avoiding walk is a hard combinatorial model of polymers with self-excluded volume [3, 4, 19]. The most important quantity in this model is cn, which is the number of distinct self-avoiding walks of length n steps, starting from the origin in (say) the square lattice. There is overwhelming analytic and numerical evidence that cn = Aμnnγ−1(1 + o(1)) (1) where A is an amplitude, μ = e is the growth constant of self-avoiding walks while κ is called the connective constant [2], and γ is the entropic exponent3. The exponential growth of cn with n was established decades ago in [2, 7, 8] but the power law correction to the exponential growth is a conjecture in low dimensions (see, for example, [19] and 3 This terminology is frequently abused in the literature; μ is often called the connective constant of selfavoiding walks. However, the connective constant is κ = logμ, as it was originally defined by Broadbent and Hammersley [2]. 0305-4470/02/420605+08$30.00 © 2002 IOP Publishing Ltd Printed in the UK L605 L606 Letter to the Editor references therein). This asymptotic expression for cn has been proven for dimensions d 5 [11] , but remains a conjecture if d < 4. In d = 4 dimensions the conjecture is modified by a logarithmic correction (see [17]). In this letter we focus on d = 2 dimensions, and we show that μ and γ can be estimated using a canonical Monte Carlo algorithm. The motivation for this is to verify the digits of μ determined by exact enumeration studies independently, using a completely different method, which will also provide statistical confidence intervals on our estimates. It is known that the connective constant can be defined via the limit [10] κ = lim n→∞ cn)/n (2) and one would ideally like to strengthen this result to μ = lim n→∞ cn+1 cn (3) but this remains an open question [8] and no one has proven the existence or non-existence of this limit in the square lattice (however, the limit exists in non-bipartite lattices such as the triangular lattice (see [19]). It is known that μ = lim n→∞ cn+2 cn (4) a result due to Kesten [15, 16] (and also see [18]). Showing that cn+2 cn is not difficult, but the monotonicity of cn (i.e. cn+1 cn for all n) is far more challenging—this result has been proven by O’Brien [23]. The numerical value of μ has been estimated for the square lattice using a variety of techniques, including series analysis and grand canonical Monte Carlo simulations. However, the best estimates for μ have been obtained from series analysis of lattice polygons (rather than self-avoiding walks), which are known to have the same connective constant [9]. Using series analysis, μ for polygons (together with their entropic exponent, α) has been determined [13, 14] to an amazing number of digits: μ = 2.638 158 529 27± 0.000 000 000 01 (lattice polygons) (5) α = 0.500 000 5± 0.000 001 0. (6) Determining μ from self-avoiding walk data is not nearly this successful. The best estimate for μ and the entropic exponent γ obtained from self-avoiding walk data are μ = 2.638 158 7± 0.000 000 7 (self-avoiding walks) (7) γ = 1.343 72± 0.000 10 (8) as determined, for example, in [6]. In two dimensions, Monte Carlo determination of the connective constant of self-avoiding walks does not even approach the precision obtained in the series enumeration above. In higher dimensions, where the finite lattice method [14] is less efficient, the precision of Monte Carlo estimates is more competitive [24]. Nevertheless, Monte Carlo simulations have the added benefit that statistical error bars can be obtained, and provide an independent means of confirming the digits found by series analysis. It is therefore of interest that the numerical determination of connective constants and critical exponents be pursued by Monte Carlo techniques, and that more efficient algorithms be developed for estimating these quantities. Letter to the Editor L607 20 0 40 60 80 120 140 160 180 100 200 length 2.84