The number of free sites next to the end of a self-avoiding walk is known as the atmosphere. The average atmosphere can be related to the number of configurations. Here we study the distribution of atmospheres as a function of length and how the number of walks of fixed atmosphere scale. Certain bounds on these numbers can be proved. We use Monte Carlo estimates to verify our conjectures. Of particular interest are walks that have zero atmosphere, which are known as trapped. We demonstrate that these walks scale in the same way as the full set of self-avoiding walks, barring an overall constant factor.
We calculate the connective constant for self-avoiding walks on the simple cubic lattice to unprecedented accuracy, using a novel application of the pivot algorithm. We estimate that μ= 4.684 039 931(27). Our method also provides accurate estimates of the number of self-avoiding walks, even for walks with millions of steps.
Motivated by recent claims of a proof that the length scale exponent for the end-to-end distance scaling of self-avoiding walks is precisely $7/12=0.5833...$, we present results of large-scale simulations of self-avoiding walks and self-avoiding trails with repulsive contact interactions on the hypercubic lattice. We find no evidence to support this claim; our estimate $ν=0.5874(2)$ is in accord with the best previous results from simulations.
We study terminally attached self-avoiding walks and bridges on the simple cubic lattice, both by series analysis and Monte Carlo methods. We provide strong numerical evidence supporting a scaling relation between self-avoiding walks, bridges, and terminally attached self-avoiding walks, and posit that a corresponding amplitude ratio is a universal quantity.
Self-avoiding walks are a simple and well-known model of long, flexible polymers in a good solvent. Polymers being pulled away from a surface by an external agent can be modelled with self-avoiding walks in a half-space, with a Boltzmann weight $y = e^f$ associated with the pulling force. This model is known to have a critical point at a certain value $y_c$ of this Boltzmann weight, which is the location of a transition between the so-called free and ballistic phases. The value $y_c=1$ has been conjectured by several authors using numerical estimates. We provide a relatively simple proof of th…
We implement a scale-free version of the pivot algorithm and use it to sample pairs of three-dimensional self-avoiding walks, for the purpose of efficiently calculating an observable that corresponds to the probability that pairs of self-avoiding walks remain self-avoiding when they are concatenated. We study the properties of this Markov chain, and then use it to find the critical exponent $γ$ for self-avoiding walks to unprecedented accuracy. Our final estimate for $γ$ is $1.15695300(95)$.
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.
In a recent paper by Beaton et al, it was proved that a model of self-avoiding walks on the honeycomb lattice, interacting with an impenetrable surface, undergoes an adsorption phase transition when the surface fugacity is $1+\sqrt{2}$. Their proof used a generalisation of an identity obtained by Duminil-Copin and Smirnov, and confirmed a conjecture of Batchelor and Yung. We consider a similar model of self-avoiding walk adsorption on the honeycomb lattice, but with the lattice rotated by $π/2$. For this model there also exists a conjecture for the critical surface fugacity, made in 1998 by Ba…
We study self-avoiding walks on the four-dimensional hypercubic lattice via Monte Carlo simulations of walks with up to one billion steps. We study the expected logarithmic corrections to scaling, and find convincing evidence in support the scaling form predicted by the renormalization group, with an estimate for the power of the logarithmic factor of 0.2516(14), which is consistent with the predicted value of 1/4. We also characterize the behaviour of the pivot algorithm for sampling four-dimensional self-avoiding walks, and conjecture that the probability of a pivot move being successful for…
We recently published [J. Phys A: Math. Theor. {\bf 45} 115202 (2012)] a new and more efficient implementation of a transfer-matrix algorithm for exact enumerations of self-avoiding polygons. Here we extend this work to the enumeration of self-avoiding walks on the square lattice. A detailed comparison with our previous best algorithm shows very significant improvement in the running time of the new algorithm. The new algorithm is used to extend the enumeration of self-avoiding walks to length 79 from the previous record of 71 and for metric properties, such as the average end-to-end distance,…
Recently Beaton, de Gier and Guttmann proved a conjecture of Batchelor and Yung that the critical fugacity of self-avoiding walks interacting with (alternate) sites on the surface of the honeycomb lattice is $1+\sqrt{2}$. A key identity used in that proof depends on the existence of a parafermionic observable for self-avoiding walks interacting with a surface on the honeycomb lattice. Despite the absence of a corresponding observable for SAW on the square and triangular lattices, we show that in the limit of large lattices, some of the consequences observed for the honeycomb lattice persist ir…
We introduce a self-avoiding walk model for which end-effects are completely eliminated. We enumerate the number of these walks for various lattices in dimensions two and three, and use these enumerations to study the properties of this model. We find that endless self-avoiding walks have the same connective constant as self-avoiding walks, and the same Flory exponent $ν$. However, there is no power law correction to the exponential number growth for this new model, i.e. the critical exponent $γ= 1$ exactly. In addition, we have convincing numerical evidence to support the hypothesis that the …
We consider the model of self-avoiding walks on the $d$-dimensional hypercubic lattice interacting with a $d^*$-dimensional defect, where $1\leq d^*<d$. Such an interaction can be attractive or repulsive, and is controlled by a Boltzmann weight $a$ associated with visits to the defect. When $d=3$ and $d^*=1$ or $2$, this can be seen as a model of long linear polymers in a good solvent, interacting with a linear filament or the interface of two liquids of different density. For all combinations of dimensions, there is a critical value $a_{\rm c}$ which separates the desorbed and adsorbed phases…
We introduce a fast implementation of the pivot algorithm for self-avoiding walks, which we use to obtain large samples of walks on the cubic lattice of up to $33 \times 10^6$ steps. Consequently the critical exponent $ν$ for three-dimensional self-avoiding walks is determined to great accuracy; the final estimate is $ν=0.587597(7)$. The method can be adapted to other models of polymers with short-range interactions, on the lattice or in the continuum.
We study the variable-length ensemble of self-avoiding walks on the complete graph. We obtain the leading order asymptotics of the mean and variance of the walk length, as the number of vertices goes to infinity. Central limit theorems for the walk length are also established, in various regimes of fugacity. Particular attention is given to sequences of fugacities that converge to the critical point, and the effect of the rate of convergence of these fugacity sequences on the limiting walk length is studied in detail. Physically, this corresponds to studying the asymptotic walk length on a gen…
We present a Monte Carlo study of Kinetic Growth Walk on square as well as triangular lattice to show that it is not equivalent to canonical Self Avoiding Walk.
We study various self-avoiding walks (SAWs) which are constrained to lie in the upper half-plane and are subjected to a compressive force. This force is applied to the vertex or vertices of the walk located at the maximum distance above the boundary of the half-space. In the case of bridges, this is the unique end-point. In the case of SAWs or self-avoiding polygons, this corresponds to all vertices of maximal height. We first use the conjectured relation with the Schramm-Loewner evolution to predict the form of the partition function including the values of the exponents, and then we use seri…
The pivot algorithm for self-avoiding walks has been implemented in a manner which is dramatically faster than previous implementations, enabling extremely long walks to be efficiently simulated. We explicitly describe the data structures and algorithms used, and provide a heuristic argument that the mean time per attempted pivot for $N$-step self-avoiding walks is $O(1)$ for the square and simple cubic lattices. Numerical experiments conducted for self-avoiding walks with up to 268 million steps are consistent with $o(\log N)$ behavior for the square lattice and $O(\log N)$ behavior for the s…
We have studied self-avoiding walks contained within an $L \times L$ square whose end-points can lie anywhere within, or on, the boundaries of the square. We prove that such walks behave, asymptotically, as walks crossing a square (WCAS), being those walks whose end-points lie at the south-east and north-west corners of the square. We provide numerical data, enumerating all such walks, and analyse the sequence of coefficients in order to estimate the asymptotic behaviour. We also studied a subset of these walks, those that must contain at least one edge on all four boundaries of the square. We…
The model of self-avoiding lattice walks and the asymptotic analysis of power-series have been two of the major research themes of Tony Guttmann. In this paper we bring the two together and perform a new analysis of the generating functions for the number of square lattice self-avoiding walks and some of their metric properties such as the mean-square end-to-end distance. The critical point $x_c$ for self-avoiding walks is known to a high degree of accuracy and we utilise this knowledge to undertake a new numerical analysis of the series using biased differential approximants. The new method i…
We give an algorithm for counting self-avoiding walks or self-avoiding polygons that runs in time $\exp(C\sqrt{n\log n})$ on 2-dimensional lattices and time $\exp(C_dn^{(d-1)/d}\log n)$ on $d$-dimensional lattices for $d>2$.
This is a rather personal review of the problem of self-avoiding walks and polygons. After defining the problem, and outlining what is known rigorously and what is merely conjectured, I highlight the major outstanding problems. I then give several applications in which the I have been involved. These include a study of surface adsorption of polymers, counting possible paths in a telecommunication network, hitting probabilities of SAWs in a rectangle, and the modelling of biological experiments on polymers. I hope to show that SAWs are not only of intrinsic mathematical interest, but also have …
The statistics of self-avoiding random walks have been used to model polymer physics for decades. A self-avoiding walk that grows one step at a time on a lattice will eventually trap itself, which occurs after an average of 71 steps on a square lattice. Here, we consider the effect of nearest-neighbor attractive interactions on the growing self-avoiding walk, and examine the effect that self-attraction has both on the statistics of trapping as well as on chain statistics through the transition between expanded and collapsed walks at the theta point. We find the trapping length increases expone…
The decay of directional correlations in self-avoiding random walks on the square lattice is investigated. Analysis of exact enumerations and Monte Carlo data suggest that the correlation between the directions of the first step and the j-th step of the walk decays faster than 1/j, indicating that the persistence length of the walk is finite.
We present an extension of the corner transfer matrix renormalisation group (CTMRG) method to O(n) invariant models, with particular interest in the self-avoiding walk class of models (O(n=0)). The method is illustrated using an interacting self-avoiding walk model. Based on the efficiency and versatility when compared to other available numerical methods, we present CTMRG as the method of choice for two-dimensional self-avoiding walk problems.
We prove some theorems about self-avoiding walks attached to an impenetrable surface (i.e. positive walks) and subject to a force. Specifically we show the force dependence of the free energy is identical when the force is applied at the last vertex or at the top (confining) plane. We discuss the relevance of this result to numerical results and to a recent result about convergence rates when the walk is being pushed towards the surface.
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 nei…
We find the generating function of self-avoiding walks and trails on a semi-regular lattice called the $3.12^2$ lattice in terms of the generating functions of simple graphs, such as self-avoiding walks, polygons and tadpole graphs on the hexagonal lattice. Since the growth constant for these graphs is known on the hexagonal lattice we can find the growth constant for both walks and trails on the $3.12^2$ lattice. A result of Watson then allows us to find the generating function and growth constant of neighbour-avoiding walks on the covering lattice of the $3.12^2$ lattice which is tetra-valen…
Flory-Huggins theory is a mean field theory for modelling the free energy of dense polymer solutions and polymer melts. In this paper we use Flory-Huggins theory as a model of a dense two dimensional self-avoiding walk confined to a square in the square lattice. The theory describes the free energy of the walk well, and we estimate the Flory interaction parameter of the walk to be $χ_{saw} = 0.32(1)$.
We calculate improved lower bounds for the connective constants for self-avoiding walks on the square, hexagonal, triangular, $(4.8^2)$, and $(3.12^2)$ lattices. The bound is found by Kesten's method of irreducible bridges. This involves using transfer-matrix techniques to exactly enumerate the number of bridges of a given span to very many steps. Upper bounds are obtained from recent exact enumeration data for the number of self-avoiding walks and compared to current best available upper bounds from other methods.
We use new algorithms, based on the finite lattice method of series expansion, to extend the enumeration of self-avoiding walks and polygons on the triangular lattice to length 40 and 60, respectively. For self-avoiding walks to length 40 we also calculate series for the metric properties of mean-square end-to-end distance, mean-square radius of gyration and the mean-square distance of a monomer from the end points. For self-avoiding polygons to length 58 we calculate series for the mean-square radius of gyration and the first 10 moments of the area. Analysis of the series yields accurate esti…
We enumerate self-avoiding walks and polygons, counted by perimeter, on the quasiperiodic rhombic Penrose and Ammann-Beenker tilings, thereby considerably extending previous results. In contrast to similar problems on regular lattices, these numbers depend on the chosen start vertex. We compare different ways of counting and demonstrate that suitable averaging improves converge to the asymptotic regime. This leads to improved estimates for critical points and exponents, which support the conjecture that self-avoiding walks on quasiperiodic tilings belong to the same universality class as self-…
A recently introduced extension of the Corner Transfer Matrix Renormalisation Group (CTMRG) method useful for the study of self-avoiding walk type models is presented in detail and applied to a class of interacting self-avoiding walks due to Blöte and Nienhuis. This model displays two different types of collapse transition depending on model parameters. One is the standard $θ$-point transition. The other is found to give rise to a first-order collapse transition despite being known to be in other respects critical.
Self-avoiding walks on the body-centered-cubic (BCC) and face-centered-cubic (FCC) lattices are enumerated up to lengths 28 and 24, respectively, using the length-doubling method. Analysis of the enumeration results yields values for the exponents $γ$ and $ν$ which are in agreement with, but less accurate than those obtained earlier from enumeration results on the simple cubic lattice. The non-universal growth constant and amplitudes are accurately determined, yielding for the BCC lattice $μ=6.530520(20)$, $A=1.1785(40)$, and $D=1.0864(50)$, and for the FCC lattice $μ=10.037075(20)$, $A=1.1736…
We compute the exponent gamma for self-avoiding walks in three dimensions. We get gamma = 1.1575 +- 0.0006 in agreement with renormalization-group predictions. Earlier Monte Carlo and exact-enumeration determinations are now seen to be biased by corrections to scaling.
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 gamma = 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 gamma. The lattice topology of the Manhattan lattice preclud…
We introduce an efficient nonreversible Markov chain Monte Carlo algorithm to generate self-avoiding walks with a variable endpoint. In two dimensions, the new algorithm slightly outperforms the two-move nonreversible Berretti-Sokal algorithm introduced by H.~Hu, X.~Chen, and Y.~Deng in \cite{old}, while for three-dimensional walks, it is 3--5 times faster. The new algorithm introduces nonreversible Markov chains that obey global balance and allows for three types of elementary moves on the existing self-avoiding walk: shorten, extend or alter conformation without changing the walk's length.
We refute the claims made by Riera and Chalub [Phys.Rev.E {\bf 58}, 4001 (1998)] by demonstrating that they have not provided enough data (requisite in their series expansion method) to draw reliable conclusions about criticality of self-avoiding walks on the Sierpinski gasket family of fractals.
We consider self-avoiding walks terminally attached to a surface at which they can adsorb. A force is applied, normal to the surface, to desorb the walk and we investigate how the behaviour depends on the vertex of the walk at which the force is applied. We use rigorous arguments to map out some features of the phase diagram, including bounds on the locations of some phase boundaries, and we use Monte Carlo methods to make quantitative predictions about the locations of these boundaries and the nature of the various phase transitions.
The critical behaviour of directed self-avoiding walks is studied on parabolic-like systems with a free boundary at x=\pm Ct^α. Using a scaling argument, 1/C is shown to be a marginal variable when α=ν_\perp/ν_\parallel=1/2, i.e., on a parabola. As a consequence the directed walk may display varying local exponents. Such a behaviour is indeed observed for restricted walks. This generalizes a result of Cardy showing that nonuniversal behaviour occurs at corners for isotropic systems.
The phase diagram for a two-dimensional self-avoiding walk model on the square lattice incorporating attractive short-ranged interactions between parallel sections of walk is derived using numerical transfer matrix techniques. The model displays a collapse transition. In contrast to the standard $θ$-point model, the transition is first order. The phase diagram in the full fugacity-temperature plane displays an additional transition line, when compared to the $θ$-point model, as well as a critical transition at finite temperature in the hamiltonian walk limit.
The nearest neighbor contacts between the two halves of an N-site lattice self-avoiding walk offer an unusual example of scaling random geometry: for N going to infinity they are strictly finite in number but their radius of gyration Rc is power law distributed, ~ Rc^{-τ}, where τ>1 is a novel exponent characterizing universal behavior. A continuum of diverging lengths scales is associated to the Rc distribution. A possibly super-universal τ=2 is also expected for the contacts of a self-avoiding or random walk with a confining wall.
Over the years, problems like percolation and self-avoiding walks have provided important testing grounds for our understanding of the nature of the critical state. I describe some very recent ideas, as well as some older ones, which cast light both on these problems themselves and on the quantum field theories to which they correspond. These ideas come from conformal field theory, Coulomb gas mappings, and stochastic Loewner evolution.
We map self-avoiding random walks with a chemical potential for writhe to the three-dimensional complex O(N) Chern-Simons theory as N -> 0. We argue that at the Wilson-Fisher fixed point which characterizes normal self-avoiding walks (with radius of gyration exponent nu = 0.588) a small chemical potential for writhe is irrelevant and the Chern-Simons field does not modify the monomer- monomer correlation function. For a large chemical potential the polymer collapses.
We introduce several bilocal algorithms for lattice self-avoiding walks that provide reasonable models for the physical kinetics of polymers in the absence of hydrodynamic effects. We discuss their ergodicity in different confined geometries, for instance in strips and in slabs. A short discussion of the dynamical properties in the absence of interactions is given.
The coil-globule transition of an isolated polymer has been well established to be a second-order phase transition described by a standard tricritical O(0) field theory. We present Monte-Carlo simulations of interacting self-avoiding walks and interacting self-avoiding trails in four dimensions which provide compelling evidence that the approach to this (tri)critical point is dominated by the build-up of first-order-like singularities masking the second-order nature of the coil-globule transition.
We consider a self-avoiding walk on the $d$-dimensional hypercubic lattice, terminally attached to an impenetrable hyperplane at which it can adsorb. When a force is applied the walk can be pulled off the surface and we consider the situation where the force is applied at the middle vertex of the walk. We show that the temperature dependence of the critical force required for desorption differs from the corresponding value when the force is applied at the end-point of the walk. This is of interest in single molecule pulling experiments since it shows that the required force can depend on where…
The phase diagram and critical behaviour of a simple toy model for DNA zipping/unzipping is examined in the framework of the Bethe approximation. The effects of solvent quality are included, and found to lead to a variety of different thermodynamic behaviours.
A global picture of a random particle movement is given by the convex hull of the visited points. We obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of self-avoiding random walks, namely the classical Self-Avoiding Walk, the Smart-Kinetic Self-Avoiding Walk, and the Loop-Erased Random Walk. To obtain a comprehensive description of the measured random quantities, we applied sophisticated large-deviation techniques, which allowed us to obtain the distributions over a large range of the support down to probabilities fa…
We study three-dimensional self-avoiding walks in presence of a one-dimensional excluded region. We show the appearance of a universal sub-leading exponent which is independent of the particular shape and symmetries of the excluded region. A classical argument provides the estimate: $Δ= 2 ν- 1 \approx 0.175(1)$. The numerical simulation gives $Δ= 0.18(2)$.
We study the self-avoiding walk on complex fractal networks called the (u,v)-flower by mapping it to the N-vector model in a generating function formalism. First, we analytically calculate the critical exponent ν and the connective constant by a renormalization-group analysis in arbitrary fractal dimensions. We find that the exponent ν is equal to the displacement exponent, which describes the speed of diffusion in terms of the shortest distance. Second, by obtaining an exact solution for the (u,u)-flower, we provide an example which supports the conjecture that the universality class of the s…
A square-lattice model for the formation of secondary structures in proteins, the hydrogen-bonding model, extended to include the effects of solvent quality, is examined in the framework of the Bethe approximation.
A two-dimensional square-lattice model for the formation of secondary structures in proteins, the hydrogen-bonding model, is extended to include the effects of solvent quality. This is achieved by allowing configuration-dependent nearest-neighbour interactions. The phase diagram is presented, and found to have a much richer variety of phases than either the pure hydrogen-bonding self-avoiding walk model or the standard $Θ$-point model.
We have explained in detail why the canonical partition function of Interacting Self Avoiding Walk (ISAW), is exactly equivalent to the configurational average of the weights associated with growth walks, such as the Interacting Growth Walk (IGW), if the average is taken over the entire genealogical tree of the walk. In this context, we have shown that it is not always possible to factor the the density of states out of the canonical partition function if the local growth rule is temperature-dependent. We have presented Monte Carlo results for IGWs on a diamond lattice in order to demonstrate …
We study self-avoiding walks on three-dimensional critical percolation clusters using a new exact enumeration method. It overcomes the exponential increase in computation time by exploiting the clusters' fractal nature. We enumerate walks of over $10^4$ steps, far more than has ever been possible. The scaling exponent $ν$ for the end-to-end distance turns out to be smaller than previously thought and appears to be the same on the backbones as on full clusters. We find strong evidence against the widely assumed scaling law for the number of conformations and propose an alternative, which perfec…
We discuss possible sources of systematic errors in the computation of critical exponents by renormalization-group methods, extrapolations from exact enumerations and Monte Carlo simulations. A careful Monte Carlo determination of the susceptibility exponent gamma for three-dimensional self-avoiding walks has been used to test the claimed accuracy of the various methods.
The phase diagram and surface critical behaviour of the vertex-interacting self-avoiding walk are examined using transfer matrix methods extended using DMRG and coupled with finite-size scaling. Particular attention is paid to the critical exponents at the ordinary and special points along the collapse transition line. The question of the bulk exponents ($ν$ and $γ$) is addressed, and the results found are at variance with previously conjectured exact values.
