Exponents Of Self-avoiding Walks Research Articles

On the existence of critical exponents for self-avoiding walks

We describe some ideas of John Hammersley for proving the existence of critical exponents for two-dimensional self-avoiding walks and provide numerical evidence for their correctness.

Adsorption of neighbor-avoiding walks on the simple cubic lattice

We investigate neighbor-avoiding walks on the simple cubic lattice in the presence of an adsorbing surface. This class of lattice paths has been less studied using Monte Carlo simulations. Our investigation follows on from our previous results using self-avoiding walks and self-avoiding trails. The connection is that neighbor-avoiding walks are equivalent to the infinitely repulsive limit of self-avoiding walks with monomer-monomer interactions. Such repulsive interactions can be seen to enhance the excluded volume effect. We calculate the critical behavior of the adsorption transition for neighbor-avoiding walks, finding a critical temperature $T_{\text a}=3.274(9)$ and a crossover exponent $\phi=0.482(13)$, which is consistent with the exponent for self-avoiding walks and trails, leading to an overall combined estimate for three dimensions of $\phi_\text{3D}=0.484(7)$. While questions of universality have previously been raised regarding the value of adsorption exponents in three dimensions, our results indicate that the value of $\phi$ in the strongly repulsive regime does not differ from its non-interacting value. However, it is clearly different from the mean-field value of $1/2$ and therefore not super-universal.

"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 <P(B)(r,N)> 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.

Determination of the exponent for SAWs on the two-dimensional Manhattan lattice

We present a high-statistics Monte Carlo determination of the exponent for self-avoiding walks on a Manhattan lattice in two dimensions. A conservative estimate is , in agreement with the universal value on regular lattices, but in conflict with predictions from conformal field theory and with a recent estimate from exact enumerations. We find strong corrections to scaling that seem to indicate the presence of a non-analytic exponent . If we assume we find , where the error is purely statistical.

On the non-universality of a critical exponent for self-avoiding walks

We have extended the enumeration of self-avoiding walks on the Manhattan lattice from 28 to 53 steps and for self-avoiding polygons from 48 to 84 steps. Analysis of this data suggests that the walk generating function exponent γ = 1.3385 ± 0.003, which is different from the corresponding exponent on the square, triangular and honeycomb lattices. This provides numerical support for an argument recently advanced by Cardy, to the effect that excluding walks with parallel nearest-neighbour steps should cause a change in the exponent γ. The lattice topology of the Manhattan lattice precludes such parallel steps.

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.

Diffusion of a self-avoiding polymer.

We study the diffusion of a polymer where we take into account self-avoidance. We introduce a reptation model in which the diffusion of stored length is infinitely rapid. For this model, we find that the diffusion coefficient scales with the length of the polymer as ${\mathit{N}}^{2\ensuremath{\nu}\mathrm{\ensuremath{-}}2}$, where \ensuremath{\nu} is the self-avoiding walk exponent for N ranging over almost two orders of magnitude. This result is used to argue that for physical polymers the diffusion coefficient is proportional to ${\mathit{N}}^{2\ensuremath{\nu}\mathrm{\ensuremath{-}}3}$ when self-avoidance is significant.

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.

Critical exponents for simple non-uniform polymer networks

The authors study a number of non-uniform specified topologies and show rigorously that for certain topologies with cut edges, the critical exponent gamma t is in agreement with a conjecture given by Gaunt et al. (1984) and that the exponent nu t= nu , the exponent for self-avoiding walks. The authors also find that the scaling relations gamma t- gamma t1 and gamma t- gamma t11 are the same as for self-avoiding walks, previously conjectured only for uniform networks. By assigning an interaction energy to a nearest neighbour contact, they prove that the collapse transition for these topologies is the same as that for self-avoiding walks.

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.

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.

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.

Correction-to-scaling exponent for self-avoiding walks.

Correction-to-scaling exponent for self-avoiding walks.

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

Vortex-loop scaling in the three-dimensional XY ferromagnet.

A three-dimensional (3D) generalization of the two-dimensional Kosterlitz real-space scaling procedure yields scaling equations for the 3D XY vortex-loop fugacity ${y}_{l}$ and loop coupling ${K}_{l}$. These agree with the equations obtained by Williams from helium phenomenology. A dominant diverging loop diameter a==${e}^{l}$=${\ensuremath{\xi}}_{\mathrm{\ensuremath{-}}}$\ensuremath{\approxeq}\ensuremath{\Vert}\ensuremath{\epsilon}${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$ [\ensuremath{\epsilon}==(T-${T}_{c}$)/${T}_{c}$] controls the exponents for the helicity modulus ${\ensuremath{\rho}}_{s}$\ensuremath{\approxeq}\ensuremath{\Vert}\ensuremath{\epsilon}${\ensuremath{\Vert}}^{\ensuremath{\nu}}$, spin-spin correlation length \ensuremath{\propto}${\ensuremath{\xi}}_{\mathrm{\ensuremath{-}}}$, specific heat \ensuremath{\alpha}=2-3\ensuremath{\nu}, and magnetic field response \ensuremath{\gamma}=2\ensuremath{\nu}, \ensuremath{\Delta}=(5/2\ensuremath{\nu}, \ensuremath{\delta}=5. \ensuremath{\nu} is found by linearizing about a fixed point. For a loop core size ${a}_{c}$ small compared to the diverging diameter, ${a}_{c}$/${\ensuremath{\xi}}_{\mathrm{\ensuremath{-}}}$\ensuremath{\rightarrow}0, a ``Gaussian'' value \ensuremath{\nu}=(1/2 results. An ansatz ${a}_{c}$(l)\ensuremath{\propto}a is made to represent the onset of screening from crinkled loops: ${a}_{c}$(l)/a\ensuremath{\simeq}(${K}_{l}$${)}^{\ensuremath{\theta}}$ with \ensuremath{\theta} the 3D self-avoiding walk exponent \ensuremath{\approxeq}0.6. This divergent-core ansatz [${a}_{c}$(l-)/${\ensuremath{\xi}}_{\mathrm{\ensuremath{-}}}$=0.57] yields the exponents \ensuremath{\nu}=0.67, \ensuremath{\gamma}=1.34, \ensuremath{\alpha}=-0.015. Monte Carlo results of Kohring et al. on vorticity suppression are discussed.

Nonperturbative tricritical exponents of trails: I. Exact enumeration on a triangular lattice

As the fugacity of intersections in trails (intersecting but non-overlapping lattice walks) is increased, the configurations change from swollen ones, with the scaling exponents of self-avoiding walks, to compact ones. Separating the two regimes is a potentially new tricritical point with no perturbative renormalisation fixed point associated with it. Supporting evidence for the existence of a tricritical point, its likely location and exponents are computed for the first time from exact enumeration of all trails up to length of fifteen lattice constants on the triangular lattice. The divergence of the specific heat indicates the location of the tricritical point. Generalised ratio and Pade methods are used to extract the scaling exponents for the number of configurations and their end-to-end distance.

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.