True Self-avoiding Walk Research Articles

Monte Carlo renormalisation group for the true self-avoiding walk

Following the general ideas by Swendsen (1982) a Monte Carlo renormalisation group (MCRG) estimation of the end-to-end exponent nu for the one-dimensional true self-avoiding walk (TSAW) has been carried out. Because the TSAW is known to reach its asymptotic behaviour only very slowly an approximate expression for the fixed point has been determined by Monte Carlo techniques. This approximate fixed point has been used as a starting point for the MCRG. The final result for the exponent nu is 0.665+or-0.004.

Kinetic growth of polyelectrolytes: Long-range true self-avoiding walks.

A long-range interacting 1/\ensuremath{\Vert}r${\ensuremath{\Vert}}^{\ensuremath{\omega}}$ true self-avoiding walk is introduced. A Flory argument gives the scaling exponent \ensuremath{\nu}=2/(\ensuremath{\omega}+2) for the end-to-end distance. For a Coulomb interaction this yields \ensuremath{\nu}=2/d, which implies that four is the upper critical dimension for growing polyelectrolytes. A Monte Carlo simulation is consistent with this conclusion.

The generalised true self-avoiding walk-a model with continuously variable exponent ν

The true self-avoiding walk is generalised by admitting that the self-avoidance parameter depends on the past history of the walk. For the generalised model the end-to-end distance exponent nu is calculated by a method that gives the same result as a Flory-type theory for the usual true self-avoiding walk. The exponent nu is predicted to be continuously variable with one of the parameters of the generalised model. The resulting expression for the exponent nu is confirmed by Monte Carlo simulations.

A short note on the true self-avoiding walk

The scaling laws for the true self-avoiding walk and a Fokker-Planck equation for the end-to-end probability distribution are combined in order to calculate the scaling functions and the end-to-end distance in one dimension explicitly.

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 ν's are found to beν TSAW pc ≈0.576 andν TSAW LA ≈0.623 respectively for the critical percolation clusters and lattice animals.

"True" self-avoiding walk in one dimension

The true self-avoiding walk in one dimension is studied via extensive Monte Carlo simulations. For any finite and nonzero value of the repulsion parameter $g$, the asymptotic behavior of the end-to-end distance is characterized by a universal exponent $\ensuremath{\nu}=0.67\ifmmode\pm\else\textpm\fi{}0.01$, in close agreement with the value $\ensuremath{\nu}=\frac{2}{3}$ recently predicted by one of us (L.P.).

Experimental realization of true self-avoiding walks

We show that the statistics of a linear polymer in an extremely polydispersed solution with a broad distribution of chain sizes is the same as that of a true self-avoiding walk. We also develop a Flory theory for a true self-avoiding walk and determine the upper critical dimension ${d}_{c}$ and the correlation-length exponent $\ensuremath{\nu}$ We find ${d}_{c}=2$ and $\ensuremath{\nu}=\frac{2}{(d+2)}$, in agreement with previous estimates.

Scaling and crossover in the one-dimensional true self-avoiding walk

The authors apply scaling and crossover arguments to the one-dimensional true self-avoiding walk, which avoids itself with strength g. The problem is formulated in terms of a grand canonical ensemble, a real-space renormalisation-group analysis shows that for large repulsion, x=e-g is a relevant variable causing crossover from the self-repelling chain (SRC) limit, with crossover exponent phi =1. A physical interpretation of this result is given in terms of competition between the correlation length and the average distance before the walk turns back on itself. The resulting flow suggests the existence of an intermediate attractive fixed point which makes the exponent v different from SRC and random-walk values for all x between 0 and 1, in agreement with recent Monte Carlo results.

Statistics of the true self-avoiding walk in one dimension

Numerical results of a Monte Carlo study for the true self-avoiding walk in one dimension are presented. For any positive value of the strength parameter (0<g< infinity ), the root-mean-square displacement RN and the range SN of the walk are characterised by two universal exponents nu =2/3+or-0.003 and s=2/3+or-0.01 respectively. For negative g (self-attracting walk) both RN and SN exhibit a saturation effect: limN to infinity RN=Rinfinity (g) and limN to infinity SN=Sinfinity (g) with Rinfinity (g) approximately (-g)-1 and Sinfinity (g) approximately (-g)-1 at g<or approximately=0. A simple scaling analysis in N and g is proposed and found to be consistent with the Monte Carlo results.

Asymptotic behavior of the "true" self-avoiding walk

The true self-avoiding random walk is defined as the statistical problem of a traveler who steps randomly, but tries to avoid places he has already visited. We show that this problem is different from the problem of a self-repelling chain (polymer problem). Most striking is perhaps the fact that the upper critical dimensionality of such a walk is 2. Renormalization-group theory is applied to compute logarithmic corrections to ordinary random-walk behavior in two dimensions. The theoretical predictions are confirmed by computer simulations.