Critical Exponents Of Self-avoiding Walks Research Articles

Field theoretic and Monte Carlo analysis of the Domb - Joyce model

We present a field-theoretic analysis of high-precision Monte Carlo data for the Domb - Joyce model on the sc lattice. We vary the repulsion between two segments at the same point from zero (random walk) to infinity (self-avoiding walk). Eventually, we even include a repulsion between segments at neighbour points to increase the excluded volume beyond that of self-avoiding walks. The data for the end-to-end distance, the radius of gyration and the partition function clearly show the existence of two branches of universal behaviour. These two branches can be identified with the weak- and strong-coupling branch of the renormalization group, respectively. A quantitative analysis shows the ability of the standard field theoretic approach to describe the data, including the data for strong coupling, i.e. renormalized coupling u greater than its fixed point value . We conclude, in contrast with some claims in the literature, that the standard formalism of the renormalized field theory can be used even for (strong-coupling branch). In addition, exploiting the fast approach to asymptotic behaviour at the transition between weak and strong coupling, we obtain very precise estimates for the critical exponents of self-avoiding walks.

Critical exponents of self-avoiding walks in three dimensions.

We analyze Monte Carlo data of self-avoiding walks with up to about 8000 steps on a simple cubic lattice with emphasis on the question of the discrepancy between the scaling exponents obtained by renormalization group calculations and Monte Carlo simulations. This discrepancy has been recently investigated by Dayantis and Palierne [Phys. Rev. B 49, 3217 (1994)] for self-avoiding walks of up to 3000 steps, and has been shown to originate from the finite size of chains generated in Monte Carlo simulations. Our analysis demonstrates this conjecture and shows that the exponents decrease for longer chains toward the renormalization group value.

Reply to "Comment on 'Surface critical exponents of self-avoiding walks on a square lattice with an adsorbing linear boundary: A computer simulation study' "

We respond to some statements made in the Comment [Phys. Rev. 51, 2674 (1995)], and argue that exact enumeration results for short chains are not conclusive and that the claimed accuracy of the simulation results is not supported by the data, due to the relatively small samples.

Comment on "Surface critical exponents of self-avoiding walks on a square lattice with an adsorbing linear boundary: A computer simulation study"

Comment on "Surface critical exponents of self-avoiding walks on a square lattice with an adsorbing linear boundary: A computer simulation study"

Surface critical exponents of self-avoiding walks on a square lattice with an adsorbing linear boundary: A computer simulation study.

Using the scanning simulation method, we study a model of a single self-avoiding walk (SAW) terminally attached to an adsorbing impenetrable linear boundary on a square lattice; an interaction energy \ensuremath{\varepsilon} (\ensuremath{\varepsilon}0) is defined between the ``surface'' and a step (bond) that lies on the surface. SAW's of up to N=260 steps are studied from samples generated with different values of the scanning parameter, b=3 and 5. In most cases the different samples lead to the same results, which suggests that they are statistically reliable. At the ordinary point (infinite temperature T) our result for the growth parameter, \ensuremath{\mu}=2.638 16\ifmmode\pm\else\textpm\fi{}0.000 02, is equal, within the error bars, to the best known estimate of Enting and Guttmann [J. Phys. A 18, 1007 (1985)]. Also, our value ${\ensuremath{\gamma}}_{1}$=0.9551\ifmmode\pm\else\textpm\fi{}0.0003 agrees very well with Cardy's value ${\ensuremath{\gamma}}_{1}$=61/64=0.953 . . . , obtained from conformal invariance [Nucl. Phys. B 240, 514 (1984)]. At the special point, we obtain independently the estimates ${\ensuremath{\gamma}}_{1}$=1.478\ifmmode\pm\else\textpm\fi{}0.020 and ${\ensuremath{\gamma}}_{11}$=0.860\ifmmode\pm\else\textpm\fi{}0.026 and, therefore, also two independent estimates for \ensuremath{\mu} that are found to be equal and very close to the Enting-Guttmann value. These results for ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{11}$ satisfy the Barber scaling relation. However, our adsorption critical temperature -\ensuremath{\varepsilon}/${\mathit{k}}_{\mathit{B}}$${\mathit{T}}^{\mathrm{*}}$=${\mathit{K}}^{\mathrm{*}}$=0.722\ifmmode\pm\else\textpm\fi{}0.004 is larger than estimates previously obtained by the transfer-matrix method. Correspondingly, our result for the crossover exponent \ensuremath{\varphi}=0.562\ifmmode\pm\else\textpm\fi{}0.020 is significantly larger than a theoretical value of Burkhardt, Eisenriegler, and Guim [Nucl. Phys. B 316, 559 (1989)], \ensuremath{\varphi}=1/2.

Surface critical exponents of self-avoiding walks and trails on a square lattice: The universality classes of the theta and theta ' points.

Using the scanning method we carry out for the first time extensive simulations of trails and self-avoiding walks (SAWs) terminally attached to an adsorbing linear boundary on a square lattice. A bulk attraction energy is also defined for a self-intersection of a trail and a pair of nonbonded nearest-neighbor monomers of a SAW. The chains are simulated at the special point. Our critical exponents differ significantly from the exact values of Vanderzande, Stella, and Seno [Phys. Rev. Lett. 67, 2757 (1991)] for the \ensuremath{\theta}' model. Thus, their conjecture, that the \ensuremath{\theta} and \ensuremath{\theta}' points belong to the same universality class, is not supported.

Critical exponents of self-avoiding walks on a family of truncatedn-simplex lattices

The authors have studied self-avoiding walks (SAWs) on a family of truncated n-simplex lattices which provide a family of fractals in which the fractal dimension d can be varied to a wide range while the spectral dimension d is held almost fixed. By means of exact renormalisation group transformations, they have calculated the critical exponents nu , alpha and gamma and the connectivity constant mu of the SAWs for n=5 and 6. They propose an approximate theory for calculating the critical exponents of the SAWs on fractals which is expected to be accurate at large values of n. They show that the theory gives results which are in good agreement with exact values even for small values of n and leads to simple closed relations for the exponents.

Critical exponents of self-avoiding walks on a family of truncated n-simplex lattices

Critical exponents of self-avoiding walks on a family of truncated n-simplex lattices

Critical exponents of self-avoiding walks on fractals with dimension 2-ε

We study critical exponents of self-avoiding walks on a family of finitely ramified Sierpinki-type fractals. The members of the family are characterized by an integer b, 2 ≤ b < ∞. For large b, the fractal dimension of the lattice tends to 2 from below. We use scaling theory to determine the critical exponents for large b. We show that as b → ∞ the susceptibility exponent does not tend to its 2-dimensional value, and determine the leading correction to critical exponents for large but finite b.

A Monte Carlo analysis of self-avoiding walks in three dimensions

The authors present estimates for critical exponents of self-avoiding walks on a cubic lattice. They treat a grand canonical ensemble of walks with free ends in a Monte Carlo approach and make use of real space renormalisation ideas. Their estimate for v is 0.59+or-0.01. The estimate for gamma does not share the same degree of accuracy. However, they are able to pinpoint the source of this discrepancy. In addition, they define a quantity chi L( beta ), the probability that a walk starting at the origin will end outside or at the border of a cube of side L. This quantity turns out to be quite suitable for a real space renormalisation group analysis.

Scanning method with a mean-field parameter: computer simulation study of critical exponents of self-avoiding walks on a square lattice

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTScanning method with a mean-field parameter: computer simulation study of critical exponents of self-avoiding walks on a square latticeHagai MeirovitchCite this: Macromolecules 1985, 18, 3, 563–569Publication Date (Print):March 1, 1985Publication History Published online1 May 2002Published inissue 1 March 1985https://doi.org/10.1021/ma00145a042RIGHTS & PERMISSIONSArticle Views63Altmetric-Citations34LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (849 KB) Get e-Alerts Get e-Alerts