Renormalization of self-avoiding walks on the square lattice

The complications encountered in direct renormalization approaches for the self-avoiding walk problem are discussed. Using a decimation transformation on the square lattice, sequences of approximants to the critical exponent ν and to the inverse connective constant $\text{K}{}_{\mathrm{<mtext>c</mtext>}}\text{(}\text{1}\text{K}{}_{\mathrm{<mtext>c</mtext>}}\text{= μ)}$ are obtained.     

True self-avoiding walk on critical percolation clusters and lattice animal

The statistics of true-self-avoiding walk model on two dimensional critical percolation clusters and lattice animals are studied using real-space renormalization group method. The correlation length exponents 's are found to be _TSAW ^pc 0.576 and _TSAW ^LA 0.623 respectively for the critical percolation clusters and lattice animals.     

Random walk on a self-avoiding walk with superconducting local bridges

Random walk on a self-avoiding walk with superconducting local bridges is studied by Real Space renormalization group technique. We enumerate SAWs in two dimensions for a square lattice by using corner rule and equal averaging method. For a SAW network with superconducting bridges we estimate the exponents for end to end resistance and linear part as 0.8625 and 0.81907 respectively. We also obtain the shortest path exponent =0.9782 by equal averaging technique.     

Ising model on self-avoiding walk chains

The phase transitions of nearest-neighbour interacting Ising models on self-avoiding walk (SAW) chains on square and triangular lattices have been studied using Monte Carlo technique. To estimate the transition temperature (T _c) bounds, the average number of nearest-neighbours (Z _eff) of spins on SAWs have been determined using the computer simulation results, and the percolation thresholds (p _c) for site dilution on SAWs have been determined using Monte Carlo methods. We find, for SAWs on square and triangular lattices respectively,Z _eff=2.330 and 3.005 (which compare very well with our previous theoretically estimated values) andp _c=0.022±0.003 and 0.045±0.005. When put in Bethe-Peierls approximations, the above values ofZ _eff givekT _c/J<1.06 and 1.65 for Ising models on SAWs on square and triangular lattices respectively, while, using the semi-empirical relation connecting the Ising modelT _c's andp _c's for the same lattices, we findkT _c/J0.57 and 0.78 for the respective models. Using the computer simulation results for the shortest connecting path lengths in SAWs on both kinds of lattices, and integrating the spin correlations on them, we find the susceptibility exponent =1.024±0.007, for the model on SAWs on two dimensional lattices.     

Peculiar scaling of self-avoiding walk contacts

The nearest neighbor contacts between the two halves of an N-site lattice self-avoiding walk offer an unusual example of scaling random geometry: for N-->infinity they are strictly finite in number but their radius of gyration R(c) is power law distributed proportional to R(-tau)(c), where tau>1 is a novel exponent characterizing universal behavior. A continuum of diverging length scales is associated with the R(c) distribution. A possibly superuniversal tau = 2 is also expected for the contacts of a self-avoiding or random walk with a confining wall.     

Renormalization of self-avoiding walks by non-linear transformations

Renormalizations of self-avoiding lattice walks by non-linear transformations are discussed. A procedure to obtain approximate renormalization group equations, together with the value of the critical index ν, is carried through for the triangular and the square lattice.     

The XY-model and the self-avoiding walk approximation

The SAW-approximation is calculated for the quantum-mechanical XY-model for spin $\text{1}\text{2}$ in a parallel magnetic field. For this purpose we derive an exact expression for the internal weight factors of the polygon graphs.In this approximation we obtain an analytic expression for the phase-separation line in the (H, T)-phase. The critical exponents are the same as for classical models.     

The statistics of self-avoiding walks on a disordered lattice

The relevance of lattice disorder on the critical behaviour of self-avoiding walks is discussed. A crossover from nonclassical to classical behaviour seems to take place.Supported by Special Research Area SFB 125Supported by German Academic Exchange Service (DAAD).     

Thermodynamics of a dense self-avoiding walk with contact interactions

A lattice model is used to study the properties of an infinite self-avoiding linear polymer chain that occupies a fraction, 01, of sites on ad-dimensional hypercubic lattice. The model introduces an (attractive or repulsive) interaction energy between nonbonded monomers that are nearest neighbors on the lattice. The lattice cluster theory enables us to derive a double series expansion in and d^–1 for the chain free energy per segment while retaining the full dependence. Thermodynamic quantities, such as the entropy, energy, and mean number of contacts per segment, are evaluated, and their dependences on, , andd are discussed. The results are in good accordance with known limiting cases.     

The broken supersymmetry phase of a self-avoiding random walk

We consider a weakly self-avoiding random walk on a hierarchical lattice ind=4 dimensions. We show that for choices of the killing ratea less than the critical valuea _cthe dominant walks fill space, which corresponds to a spontaneously broken supersymmetry phase. We identify the asymptotic density to which walks fill space, (a), to be a supersymmetric order parameter for this transition. We prove that (a)(a _c–a) (–log(a _c–a))^1/2 asaa _c, which is mean-field behavior with logarithmic corrections, as expected for a system in its upper critical dimension.Research partially supported by NSF Grants DMS 91-2096 and DMS 91-96161.     

Coherent-anomaly method in self-avoiding walk problems

Self-avoiding walk (SAW), being a nonequilibrium cooperative phenomenon, is investigated with a finite-order-restricted-walk (finite-ORW or FORW) coherent-anomaly method (CAM). The coefficient β_1r in the asymptotic form C_nr≅ β_1r λⁿ_1r for the total number C_nr of r- ORW's with respect to the step number n is investigated for the first time. An asymptotic form for SAW's is thus obtained form the series of FORW approximants, C_nr≅ br^gμⁿ(1 + a/r)ⁿ, as the envelope curve C_n≅b(ae/g)^gμⁿn^g. Numerical results are given by C_n≅1.424n^0.27884.1507ⁿ and C_n≅1.179n^0.158710.005ⁿ for the plane triangular lattice and f.c.c. lattice, respectively. A good coincidence of the total numbers estimated from the above simple formulae with exact enumerations for finite-step SAW's implies that the essential nature of SAW (non-Markov process) can be understood from FORW (Markov process) in the CAM framework.     

New Monte Carlo method for the self-avoiding walk

We introduce a new Monte Carlo algorithm for the self-avoiding walk (SAW), and show that it is particularly efficient in the critical region (long chains). We also introduce new and more efficient statistical techniques. We employ these methods to extract numerical estimates for the critical parameters of the SAW on the square lattice. We find=2.63820 ± 0.00004 ± 0.00030=1.352 ± 0.006 ± 0.025v=0.7590 ± 0.0062 ± 0.0042 where the first error bar represents systematic error due to corrections to scaling (subjective 95% confidence limits) and the second bar represents statistical error (classical 95% confidence limits). These results are based on SAWs of average length 166, using 340 hours CPU time on a CDC Cyber 170–730. We compare our results to previous work and indicate some directions for future research.     

Semi-flexible interacting self-avoiding trails on the square lattice

Self-avoiding walks self-interacting via nearest neighbours (ISAW) and self-avoiding trails interacting via multiply-visited sites (ISAT) are two models of the polymer collapse transition of a polymer in a dilute solution. On the square lattice it has been established numerically that the collapse transition of each model lies in a different universality class.     

Relation between the two-dimensional self-avoiding walk and the Eden model

When the interface of a two-dimensional lattice gas is grown by an algorithm producing self-avoiding walks, some features of the Eden model appear. The scaling behavior of the widths and lengths of these interfaces and of their accessible perimeters are examined in several regions of the phase diagram. It is found that both the accessible perimeter and the interface behave like the Eden model below a critical point. Above this point, only the accessible perimeter behaves like the Eden model. The behavior in the critical region is suprisingly the same for the interface and its external perimeter but is different from the Eden model.     

Renormalization group for a critical lattice model

New methods are developed for the study of the Kadanoff-Wilson renormalization group for critical lattice systems of unbounded spins. The methods are based on a combination of expansion and analyticity techniques and are applied to a nonlocal hierarchical model of the dipole gas. They remove the main obstacle against the use of the block spin strategy in more realistic models such as φ _d ^/4 , the dipole gas and the anharmonic crystal.