Structural properties of self-attracting walks.

Self-attracting walks (SATW) with attractive interaction u>0 display a swelling-collapse transition at a critical u(c) for dimensions d>or=2, analogous to the Theta transition of polymers. We are interested in the structure of the clusters generated by SATW below u(c) (swollen walk), above u(c) (collapsed walk), and at u(c), which can be characterized by the fractal dimensions of the clusters d(f) and their interface d(I). Using scaling arguments and Monte Carlo simulations, we find that for uu(c), the clusters are compact, i.e., d(f)=d and d(I)=d-1. At u(c), the SATW is in a new universality class. The clusters are compact in both d=2 and d=3, but their interface is fractal: d(I)=1.50+/-0.01 and 2.73+/-0.03 in d=2 and d=3, respectively. In d=1, where the walk is collapsed for all u and no swelling-collapse transition exists, we derive analytical expressions for the average number of visited sites and the mean time to visit S sites.

"Generalized des Cloizeaux" exponent for self-avoiding walks on the incipient percolation cluster.

We study the asymptotic shape of self-avoiding random walks (SAW) on the backbone of the incipient percolation cluster in d-dimensional lattices analytically. It is generally accepted that the configurational averaged probability distribution function for the end-to-end distance r of an N step SAW behaves as a power law for r-->0. In this work, we determine the corresponding exponent using scaling arguments, and show that our suggested "generalized des Cloizeaux" expression for the exponent is in excellent agreement with exact enumeration results in two and three dimensions.

Multifractal behavior of linear polymers in disordered media.

The scaling behavior of linear polymers in disordered media modeled by self-avoiding random walks (SAWs) on the backbone of two- and three-dimensional percolation clusters at their critical concentrations p(c) is studied. All possible SAW configurations of N steps on a single backbone configuration are enumerated exactly. We find that the moments of order q of the total number of SAWs obtained by averaging over many backbone configurations display multifractal behavior; i.e., different moments are dominated by different subsets of the backbone. This leads to generalized coordination numbers mu(q) and enhancement exponents gamma(q), which depend on q. Our numerical results suggest that the relation mu(1)=p(c)mu between the first moment mu(1) and its regular lattice counterpart mu is valid.

Swelling-collapse transition of self-attracting walks

We study the structural properties of self-attracting walks in d dimensions using scaling arguments and Monte Carlo simulations. We find evidence of a transition analogous to the Theta transition of polymers. Above a critical attractive interaction u(c), the walk collapses and the exponents nu and k, characterizing the scaling with time t of the mean square end-to-end distance approximately t(2nu) and the average number of visited sites approximately t(k), are universal and given by nu=1/(d+1) and k=d/(d+1). Below u(c), the walk swells and the exponents are as with no interaction, i.e., nu=1/2 for all d, k=1/2 for d=1 and k=1 for d>/=2. At u(c), the exponents are found to be in a different universality class.