In this Letter we present a flat histogram algorithm based on the pruned and enriched Rosenbluth method. This algorithm incorporates in a straightforward manner microcanonical reweighting techniques, leading to "flat histogram" sampling in the chosen parameter space. As an additional benefit, our algorithm is completely parameter free and, hence, easy to implement. We apply this algorithm to interacting self-avoiding walks, the generic lattice model of polymer collapse.

Owczarek, Prellberg, and Brak Reply: In our recent Letter [1] we made a conjecture on the scaling form of the partition function at low temperatures of a single polymer (poor solvent). The canonical model of interest is that of self-avoiding walks (SAW) with nearest-neighbor attraction. We supported this conjecture with some long series for partially directed SAW in two dimensions and have subsequently calculated this form exactly [2]. As has been pointed out in our Letter, and in the preceding Comment [3], this scaling form is mathematically similar to forms found elsewhere in the literature including the theory of dense polymers [4]. To set this in context we mention that high temperature polymers are believed to be described by the critical Oin) model, in the limit Az-• 0, while low temperature polymers are related to the first order line of the tricritical 0(0) model. In contrast, dense polymers are believed to be described by the low temperature phase of the critical 0(0) model (here the temperature is associated with a different coupling than in the tricritical model). The scaling form conjectured arises as a generic form for low temperatures (first order or condensationlike behavior) [5] and hence the values of 7 involved in each case are not a priori identical. Moreover, universality cannot be invoked since it has not been shown that there even exists a field theory equivalent in the continuum limit for collapsed, rather than dense, polymers. In the preceding Comment [3] the scaling form of a dense polymer system, which has no interactions, and a single interacting polymer are related in two dimensions. This is certainly intriguing and warrants further study.The argument connecting dense polymers to collapsed ones is based on the idea that collapsed polymers are internally dense and on the assumption that the surface of the collapsed polymer is smooth, which induces the same value of cr== y. However, even if on average the surface is smooth, it is not clear to us that the sum over all configurations will lead to a value of 7 at low temperature which is the same as in dense polymers where the surface is constrained to be smooth by the boundary conditions imposed. We point out also that the work on dense polymers was accomplished on the Manhattan lattice. However, it is believed that at the B point this lattice constraint changes the partition function scaling. For example, the ratio of the open to closed walk partition functions will be different to that of walks on an isotropic lattice (that is, it is believed that 7^= f on the square lattice while 7^ = 1 on the Manhattan lattice). At low tem-peratures, it might be expected that lattice effects would be even stronger. Recent work [6] on the compact subset of self-avoiding walk configurations I7] also indicates behavior inconsistent with a conjecture identifying dense, compact, and collapsed polymers.To shed some light on the conjecture we have extended and examined the available series for interacting walks and polygons. We have focused on the ratio of th...

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We present a method for the derivation of the generating function and computation of critical exponents for several cluster models (staircase, bar-graph, and directed columnconvex polygons, as well as partially directed self-avoiding walks), starting with nonlinear functional equations for the generating function. By linearising these equations, we first give a derivation of the generating functions. The non-linear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz. Alternatively, taking the continuum limit leads to non-linear differential equations, from which one can extract the scaling function. We find that all the above models are in the same universality class with exponents: γ u = −1/2, γ t = −1/3 and φ = 2/3. All models have as their scaling function the logarithmic derivative of the Airy function.

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