On the critical properties of the Edwards and the self-avoiding walk model of polymer chains

We study the two- and three-dimensional, superrenormalizable Edwards model and the self-avoiding walk model of polymers. Using a Schwinger-Dyson equation and upper and lower bounds on correlations in terms of “skeleton diagrams” [6] we establish the existence of a non-trivial continuum limit in the two- and three-dimensional, superrenormalizable Edwards model. We also prove that perturbation theory is asymptotic for the continuum correlations of these models.A fairly detailed analysis of the approach to the critical point in the self-avoiding walk model is presented. In particular, we show that η<1. In dimension d?4, we discuss rigorous consequences of the conjecture that η is non-negative: among other implications, we derive that the continuum limit is trivial and that γ=1, in d?5 dimensions, and that corrections to mean-field scaling laws are at most logarithmic in four dimensions.     

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.     

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.     

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 the self-avoiding walk on a lattice

It is shown that the Niemeijer-Van Leeuwen renormalization method can be extended, via a general n-vector model, to the case n = 0, which describes the self-avoiding walk. Results of a second cumulant approximation are given.     

Monte carlo study of the interacting self-avoiding walk model in three dimensions

We consider self-avoiding walks on the simple cubic lattice in which neighboring pairs of vertices of the walk (not connected by an edge) have an associated pair-wise additive energy. If the associated force is attractive, then the walk can collapse from a coil to a compact ball. We describe two Monte Carlo algorithms which we used to investigate this collapse process, and the properties of the walk as a function of the energy or temperature. We report results about the thermodynamic and configurational properties of the walks and estimate the location of the collapse transition.     

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.     

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.     

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.     

A note on mean-field behavior for self-avoiding walk on branching planes

We consider the critical behavior of the susceptibility of the self-avoiding walk on the graphT×Z, whereT is a Bethe lattice with degreek andZ is the one dimensional lattice. By directly estimating the two-point function using a method of Grimmett and Newman, we show that the bubble condition is satisfied whenk>2, and therefore the critical exponent associated with the susceptibility equals 1.     

Loop-erased self-avoiding random walk in two and three dimensions

If(n) is the position of the self-avoiding random walk in ^d obtained by erasing loops from simple random walk, then it is proved that the mean square displacementE(¦(n)¦²) grows at least as fast as the Flory predictions for the usual SAW, i.e., at least as fast asn ^3/2 ford=2 andn ^6/5 ford=3. In particular, if the mean square displacement of the usual SAW grows liken ^1.18... ind=3, as expected, then the loop-erased process is in a different universality class.     

Compressible Ising double chains

We obtain exact expressions for the free energy and the magnetic susceptibility in zero field of a compressible double Ising chain with first and second neighbour interactions. The chain is supposed to be made of rigid rods which move like dumbbells in an elastic harmonic potential. The exchange interactions along the direction of the chain are linear functions of the spacing between rods. The effective spin hamiltonian of the double chain involves short-range two and four-spin interactions. Due to the existence of compensation points, we obtain regions of peculiar thermodynamic properties in the pressure-temperature phase diagram.     

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.     

The diffusion of self-avoiding random walk in high dimensions

We use the Brydges-Spencer lace expansion to prove that the mean square displacement of aT step strictly self-avoiding random walk in thed dimensional square lattice is asymptotically of the formDT asT approaches infinity, ifd is sufficiently large. The diffusion constantD is greater than one.     

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.     

Critical exponents for the self-avoiding random walk in three dimensions

We compute by direct Monte Carlo simulation the main critical exponents, , ₄, andv and the effective coordination number for the self-avoiding random walk in three dimensions on a cubic lattice. We find both hyperscaling relationsdv=2– anddv– 2 ₄+=0 satisfied ind = 3.     

Confined space and effective interactions of multiple self-avoiding chains

We study the link between three seeming-disparate cases of self-avoiding polymers: strongly overlapping multiple chains in dilute solution, chains under spherical confinement, and the onset of semidilute solutions. Our main result is that the free energy for overlapping n chains is independent of chain length and scales as n9/4, slowly crossing over to n3, as n increases. For strongly confined polymers inside a spherical cavity, we show that rearranging the chains does not cost an additional free energy. Our results imply that, during cell cycle, global reorganization of eukaryotic chromosomes in a large cell nucleus could be readily achieved.