Properties Of Self-avoiding Walks Research Articles

Fractals meet fractals: self-avoiding random walks on percolation clusters

The scaling behavior of linear polymers in disordered media, modelled by self-avoiding walks (SAWs) on the backbone of percolation clusters in two, three and four dimensions is studied by numerical simulations. We apply the pruned-enriched Rosenbluth chain-growth method (PERM). Our numerical results yield estimates of critical exponents, governing the scaling laws of disorder averages of the configurational properties of SAWs, and clearly indicate a multifractal spectrum which emerges when two fractals meet each other.

Walking on fractals: diffusion and self-avoiding walks on percolation clusters

We consider random walks (RWs) and self-avoiding walks (SAWs) on disordered lattices directly at the percolation threshold. Applying numerical simulations, we study the scaling behavior of the models on the incipient percolation cluster in space dimensions d = 2, 3, 4. Our analysis yields estimates of universal exponents, governing the scaling laws for configurational properties of RWs and SAWs.

Polymers in media with long-range-correlated quenched disorder

Abstract We study the scaling properties of self-avoiding walks (SAWs) on a d -dimensional disordered lattice with quenched defects obeying a power law correlation ∼ r − a for large distances r . Such type of disorder is known to be relevant for magnetic phase transitions. We apply the field-theoretical renormalization group approach and perform calculations in a double expansion in e = 4 − d , δ = 4 − a . The asymptotic behaviour of SAWs on a lattice with long-range-correlated disorder is found to be governed by a new exponent ν long = 1/2 + δ/8, (e/2 pure = 1/2 + e/16, (e > 0).

SELF-AVOIDING WALKS ON QUASILATTICES

I consider the asymptotic properties of self-avoiding walks on a variety of quasilattices. The critical points and exponents are estimated. Walks on several of the patterns appear to belong to the same universality class as walks on the hexagonal periodic lattice.

Universality of surface exponents of self-avoiding walks on a Manhattan lattice

The authors use phenomenological renormalization group techniques to study the bulk and surface properties of self-avoiding walks on a Manhattan lattice. They find that, as is the case for the bulk exponents, the underlying bond directionality does not change the universality class of the surface exponents relative to undirected lattices. They also obtain an estimate for the position of the binding transition.

Self-avoiding walks on random lattices

The properties of self-avoiding walks on dilute lattices are studied, both directly and using the replica formalism. It is shown that dilution does not affect the exponents and careful use of the Haris criterion also leads to this conclusion.

Configurational studies of the Potts models

Eigenvalues and eigenvectors are derived for the q-state two-level model introduced by Potts in 1952, and it is shown that for any net the partition function depends only on the topology and not on any two-degree vertices. This enables a simple method to be used calculating the partition functions of standard star topologies. The second q-orientation model introduced by Potts (termed the planar Potts model) is discussed, and it is shown that the same property holds. Partition functions for certain star topologies are derived for this model. By considering the asymptotic form of the coefficients in high-temperature expansions of the partition function, estimates are obtained of the critical temperatures of these models in terms of the geometrical properties of self-avoiding walks.