[en] The numerical analysis of combinatorial problems with non-standard scaling is an important testing ground for the limits of current techniques. One problem that has proven especially difficult to analyse with all available numerical techniques, including various Monte Carlo simulation methods and careful series analysis, is anisotropic spiral walks in two dimensions. Here we revisit this problem discussing various non-standard scaling hypotheses and showing how these best fit the available data. This highlights the difficulties with the analysis of data when the standard scaling forms may not hold true and also provides a testing ground for improved techniques

[en] A self-avoiding walk adsorbing on a line in the square lattice, and on a plane in the cubic lattice, is studied numerically as a model of an adsorbing polymer in dilute solution. The walk is simulated by a multiple Markov chain Monte Carlo implementation of the pivot algorithm for self-avoiding walks. Vertices in the walk that are visits in the adsorbing line or plane are weighted by e^β. The critical value of β, where the walk adsorbs on the adsorbing line or adsorbing plane, is determined by considering energy ratios and approximations to the free energy. We determine that the critical values of β are β_c = 0.565±0.010 in the square lattice and β_c = 0.288±0.020 in the cubic lattice. In addition, the value of the crossover exponent is determined: Φ = 0.501±0.015 in the square lattice and Φ = 0.5005±0.0036 in the cubic lattice. Metric quantities, including the mean square radius of gyration, are also considered, as well as rescaling of the specific heat and free energy, as the critical point is approached

[en] We show that the classical Rosenbluth method for sampling self-avoiding walks (Hammersley and Morton 1954 J. R. Stat. Soc. B 16 23, Rosenbluth and Rosenbluth 1955 J. Chem. Phys. 23 356) can be extended to a general algorithm for sampling many families of objects, including self-avoiding polygons. The implementation relies on an elementary move which is a generalization of kinetic growth; rather than only appending edges to the endpoint, edges may be inserted at any vertex provided the resulting objects still lie within the same family. This gives, for the first time, a kinetic growth algorithm for sampling self-avoiding polygons. We implement this method using pruning and enrichment (Grassberger 1997 Phys. Rev. E 56 3682) to sample self-avoiding walks and polygons. The algorithm can be further extended by mixing it with length-preserving moves, such as pivots and crank-shaft moves. (fast track communication)

[en] We analyse directed walk models of random copolymer adsorption and localization. Ideally we would like to solve the quenched problem, but it appears to be intractable even for simple directed models. The annealed approximation is solvable, but is inadequate in the strong interaction regime for the adsorption problem and gives a qualitatively incorrect phase diagram for the localization problem. In this paper, we treat these directed models using an approximation suggested by Morita (1964 J. Math. Phys. 5 1401-5) in which the proportion of each comonomer is fixed. We find that the Morita approximation leads to behaviour that is closer to that of the quenched average model and this is particularly interesting in the localization problem where the phase diagram is (at least qualitatively) very similar to that of the quenched average problem. We also show that the phase boundaries in the Morita approximation are bounds on the locations of the phase boundaries of the quenched model. (author)

[en] In this paper, we introduce a new Monte Carlo method for sampling lattice self-avoiding walks. The method, which we call 'GAS' (generalized atmospheric sampling), samples walks along weighted sequences by implementing elementary moves generated by the positive, negative and neutral atmospheric statistics of the walks. A realized sequence is weighted such that the average weight of states of length n is proportional to the number of self-avoiding walks from the origin c_n. In addition, the method also self-tunes to sample from uniform distributions over walks of lengths in an interval [0, n_max]. We show how to implement GAS using both generalized and endpoint atmospheres of walks and analyse our data to obtain estimates of the growth constant and entropic exponent of self-avoiding walks in the square and cubic lattices.

[en] In two dimensions the universality classes of self-avoiding walks on the square lattice, restricted by allowing only certain two-step configurations to occur within each walk, has been argued to be determined primarily by the symmetry of the set of allowed two-step configurations. In a recent paper (Rechnitzer A. and Owczarek A.L. 2000 On three-dimensional self-avoiding walk symmetry classes J. Phys. A : Math. Gen. 33 2685-723), primarily tackling the three-dimensional analogues of these models, a novel two-dimensional model was discovered that seemed either to break the classification of the models into universality classes according to microscopic symmetry or was itself a member of a novel universality class. This was supported by series analysis of exact enumeration data. Here we provide conclusive evidence that this model, known as 'anti-spiral walks', is in the directed walk universality class. We arrive at these conclusions from Monte Carlo simulations of these walks using a PERM algorithm modified for this problem. We point out that the behaviour of this model is unusual in that other models in the directed walk universality class remain directed when the self-avoidance condition is removed, whereas the behaviour of anti-spiral walks becomes that of a isotropic simple random walk. We also remark that the symmetry classification of walk models can be kept by adding a natural condition to the scheme that disallows models, all of whose configurations avoid some infinite region of the plane by virtue of their microscopic constraints. (author)

[en] In this paper we examine numerically the properties of minimal length knotted lattice polygons in the simple cubic, face-centered cubic, and body-centered cubic lattices by sieving minimal length polygons from a data stream of a Monte Carlo algorithm, implemented as described in Aragão de Carvalho and Caracciolo (1983 Phys. Rev. B 27 1635), Aragão de Carvalho et al (1983 Nucl. Phys. B 215 209) and Berg and Foester (1981 Phys. Lett. B 106 323). The entropy, mean writhe, and mean curvature of minimal length polygons are computed (in some cases exactly). While the minimal length and mean curvature are found to be lattice dependent, the mean writhe is found to be only weakly dependent on the lattice type. Comparison of our results to numerical results for the writhe obtained elsewhere (see Janse van Rensburg et al 1999 Contributed to Ideal Knots (Series on Knots and Everything vol 19) ed Stasiak, Katritch and Kauffman (Singapore: World Scientific), Portillo et al 2011 J. Phys. A: Math. Theor. 44 275004) shows that the mean writhe is also insensitive to the length of a knotted polygon. Thus, while these results for the mean writhe and mean absolute writhe at minimal length are not universal, our results demonstrate that these values are quite close the those of long polygons regardless of the underlying lattice and length

[en] In this paper, the elementary moves of the BFACF-algorithm (Aragao de Carvalho and Caracciolo 1983 Phys. Rev. B 27 1635-45, Aragao de Carvalho and Caracciolo 1983 Nucl. Phys. B 215 209-48, Berg and Foester 1981 Phys. Lett. B 106 323-6) for lattice polygons are generalized to elementary moves of BFACF-style algorithms for lattice polygons in the body-centered (BCC) and face-centered (FCC) cubic lattices. We prove that the ergodicity classes of these new elementary moves coincide with the knot types of unrooted polygons in the BCC and FCC lattices and so expand a similar result for the cubic lattice (see Janse van Rensburg and Whittington (1991 J. Phys. A: Math. Gen. 24 5553-67)). Implementations of these algorithms for knotted polygons using the GAS algorithm produce estimates of the minimal length of knotted polygons in the BCC and FCC lattices.

[en] We discuss and analyse the Morita approximation for a number of different models of quenched random copolymer localization at the interface between two immiscible liquids. We focus on two directed models, bilateral Dyck paths and bilateral Motzkin paths, for which this approximation can be carried through analytically. We study the form of the phase diagram and find that the Morita approximation gives phase boundaries which are qualitatively correct. This is also true when a monomer-interface interaction is included in the model. When this interaction is attractive it can lead to separation of the phase boundaries, which is also a feature of the quenched problem. We note the existence of non-analytic points on the phase boundaries which may reflect tricritical points on the phase boundaries of the full quenched average problem. In certain regions of the phase plane this approximation can be extended to the self-avoiding walk model

[en] In this paper the number and lengths of minimal length lattice knots confined to slabs of width L are determined. Our data on minimal length verify the recent results by Ishihara et al for the similar problem, except in a single case, where an improvement is found. From our data we construct two models of grafted knotted ring polymers squeezed between hard walls, or by an external force. In each model, we determine the entropic forces arising when the lattice polygon is squeezed by externally applied forces. The profile of forces and compressibility of several knot types are presented and compared, and in addition, the total work done on the lattice knots when they are squeezed to a minimal state is determined. (paper)