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Monte Carlo study of four-dimensional self-avoiding walks of up to one billion steps
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Abstract
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 an N-step walk is .
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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 for self-avoiding walks to unprecedented accuracy. Our final estimate for is 1.15695300(95).
Confirmation bias, as the term is typically used in the psychological literature, connotes the seeking or interpreting of evidence in ways that are partial to existing beliefs, expectations, or a hypothesis in hand. The author reviews evidence of such a bias in a variety of guises and gives examples of its operation in several practical contexts. Possible explanations are considered, and the question of its utility or disutility is discussed.
We examine self-avoiding walks in dimensions 4 to 8 using high-precision Monte Carlo simulations up to length N = 16 384, providing the first such results in dimensions d > 4 on which we concentrate our analysis. We analyse the scaling behaviour of the partition function and the statistics of nearest-neighbour contacts, as well as the average geometric size of the walks, and compare our results to 1/d-expansions and to excellent rigorous bounds that exist. In particular, we obtain precise values for the connective constants, µ5 = 8.838 544(3), µ6 = 10.878 094(4), µ7 = 12.902 817(3), µ8 = 14.919 257(2) and give a revised estimate of µ4 = 6.774 043(5). All of these are by at least one order of magnitude more accurate than those previously given (from other approaches in d > 4 and all approaches in d = 4). Our results are consistent with most theoretical predictions, though in d = 5 we find clear evidence of anomalous N -1/2-corrections for the scaling of the geometric size of the walks, which we understand as a non-analytic correction to scaling of the general form N (4-d)/2 (not present in pure Gaussian random walks).
These lecture notes provide a rapid introduction to a number of rigorous
results on self-avoiding walks, with emphasis on the critical behaviour.
Following an introductory overview of the central problems, an account is given
of the Hammersley--Welsh bound on the number of self-avoiding walks and its
consequences for the growth rates of bridges and self-avoiding polygons. A
detailed proof that the connective constant on the hexagonal lattice equals
is then provided. The lace expansion for self-avoiding
walks is described, and its use in understanding the critical behaviour in
dimensions is discussed. Functional integral representations of the
self-avoiding walk model are discussed and developed, and their use in a
renormalisation group analysis in dimension 4 is sketched. Problems and
solutions from tutorials are included.
We use the lace expansion to study the standard self-avoiding walk in thed-dimensional hypercubic lattice, ford≧5. We prove that the numbercn ofn-step self-avoiding walks satisfiescn~Aμn, where μ is the connective constant (i.e. γ=1), and that the mean square displacement is asymptotically linear in the number of steps (i.e.v=1/2). A bound is obtained forcn(x), the number ofn-step self-avoiding walks ending atx. The correlation length is shown to diverge asymptotically like (μ−−Z)1/2. The critical two-point function is shown to decay at least as fast as ⋎x⋎−2, and its Fourier transform is shown to be asymptotic to a multiple ofk−2 ask→0 (i.e. η=0). We also prove that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. The infinite self-avoiding walk is constructed. In this paper we prove these results assuming convergence of the lace expansion. The convergence of the lace expansion is proved in a companion paper.
We present simulation results for long () self-avoiding walks in four dimensions. We find definite indications of logarithmic corrections, but the data are poorly described by the asymptotically leading terms. Detailed comparisons are presented with renormalization group flow equations derived in direct renormalization and with results of a field theoretic calculation. Comment: NLATEX, 20 pages
The universal asymptotic amplitude ratio between the gyration radius and the hydrodynamic radius of self-avoiding walks is estimated by high-resolution Monte Carlo simulations. By studying chains of length of up to N=225≈34×106 monomers, we find that the ratio takes the value RG/RH=1.5803940(45), which is several orders of magnitude more accurate than the previous state of the art. This is facilitated by a sampling scheme which is quite general and which allows for the efficient estimation of averages of a large class of observables. The competing corrections to scaling for the hydrodynamic radius are clearly discernible. We also find improved estimates for other universal properties that measure the chain dimension. In particular, a method of analysis which eliminates the leading correction to scaling results in a highly accurate estimate for the Flory exponent of ν=0.58759700(40).
We outline a proof, by a rigorous renormalisation group method, that the critical two-point function for continuous-time weakly self-avoiding walk on Z^d decays as |x|^{-(d-2)} in the critical dimension d=4, and also for all d>4. Comment: 25 pages, 1 diagram, lecture at ICM 2010, Hyderabad, in section on probability and statistics, crosslisted with section on mathematical physics
Imagine that you are standing at an intersection in the centre of a large city whose streets are laid out in a square grid. You choose a street at random and begin walking away from your starting point, and at each intersection you reach you choose to continue straight ahead or to turn left or right.
A method for carrying out ‘Monte Carlo’ calculations on chains is described. Close agreement between the present results and those of Wall et al. for chains on a two-choice, plane hexagonal lattice establishes the reliability of the present method. The calculations are extended to the chains whose segments interact in accord with a Lennard-Jones potential.
A two-dimensional n-component spin model with cubic or isotropic symmetry is mapped onto a solid-on-solid model. Subject to some plausible assumptions this leads to an analytic calculation of the critical point and some critical indices for -2
We have computed the number, the mean-square end-to-end distance, the mean-square radius of the gyration, and the mean-square distance of a monomer from the origin, for n-step self-avoiding walks on hypercubic lattices in 4, 5, and 6 dimensions up to 19, 15, and 14 steps, respectively. Two amplitude ratios are estimated; the results for self-avoiding walks in 5 and 6 dimensions agree with the exact values for random walks in arbitrary dimensions.
This paper is a continuation of the companion paper [14], in which it
was proved that the standard model of self-avoiding walk in five or more
dimensions has the same critical behaviour as the simple random walk,
assuming convergence of the lace expansion. In this paper we prove the
convergence of the lace expansion, an upper and lower infrared bound,
and a number of other estimates that were used in the companion paper.
The proof requires a good upper bound on the critical point (or
equivalently a lower bound on the connective constant). In an appendix,
new upper bounds on the critical point in dimensions higher than two are
obtained, using elementary methods which are independent of the lace
expansion. The proof of convergence of the lace expansion is computer
assisted. Numerical aspects of the proof, including methods for the
numerical evaluation of simple random walk quantities such as the
two-point function (or lattice Green function), are treated in an
appendix.
The method of concatenation (the addition of precomputed shorter chains to the ends of a centrally generated longer chain) has permitted the extension of the exact series for CN-the number of distinct configurations for self-avoiding walks of length N. The authors report on the leading exponent y and xc (the reciprocal of the connectivity constant) for the 2D honeycomb lattice (42 terms) 1.3437, 0.541 1968; the 2D square lattice (30 terms) 1.3436, 0.379 0520; the 3D simple cubic lattice (23 terms) 1.161 932, 0.213 4987; the 4D hypercubic (18 terms) y approximately=1, 0.147 60 and the 5D hypercubic lattice (13 terms) y<or=1.025, 0.113 05. In addition they have also evaluated the leading correction terms: honeycomb Delta approximately=1, square Delta approximately=0.85, simple cubic Delta approximately=1.0 and the 4D hypercubic logarithmic correction with delta approximately=0.25.
We introduce a new method for the enumeration of self-avoiding walks based on the lace expansion. We also introduce an algorithmic improvement, called the two-step method, for self-avoiding walk enumeration problems. We obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d 3: we enumerate 32-step self-avoiding polygons in d = 3, 26-step self-avoiding polygons in d = 4, 30-step self-avoiding walks in d = 3, and 24-step self-avoiding walks and polygons in all dimensions d 4. We analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 d 8. We also provide major extensions of 1/d expansions for the connective constant and for two critical amplitudes.
The pivot algorithm is a dynamic Monte Carlo algorithm, first invented by Lal, which generates self-avoiding walks (SAWs) in a canonical (fixed-N) ensemble with free endpoints (hereN is the number of steps in the walk). We find that the pivot algorithm is extraordinarily efficient: one effectively independent sample can be produced in a computer time of orderN. This paper is a comprehensive study of the pivot algorithm, including: a heuristic and numerical analysis of the acceptance fraction and autocorrelation time; an exact analysis of the pivot algorithm for ordinary random walk; a discussion of data structures and computational complexity; a rigorous proof of ergodicity; and numerical results on self-avoiding walks in two and three dimensions. Our estimates for critical exponents are=0.74960.0007 ind=2 and= 0.5920.003 ind=3 (95% confidence limits), based on SAWs of lengths 200N10000 and 200N 3000, respectively.
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 simple cubic lattice. Our method can be adapted to other models of polymers with short-range interactions,
on the lattice or in the continuum, and hence promises to be widely useful.
KeywordsSelf-avoiding walk-Polymer-Monte Carlo-Pivot algorithm
We investigate analytically the asymptotic logarithmic behavior in four dimensions (d = 4) of very long continuous polymer chains with excluded volume, described by the two-parameter Edwards model. Using renormalization theory, we calculate directly all the connected partition functions of polymer chains in this model, and obtain their logarithmic expression, including the subleading order. To first order, they are found in the universal form , and where S is the Brownian area of the chains, given in terms of their Brownian radius °R, by °R2 = dS. The partition functions 1, 2, 3, 4, and the averageend-to-end square distance R2, and square radius of gyration RG2 are calculated up to logarithmic sub-subleading order. The virial expansion in 4d of the osmotic pressure is obtained. For semi-dilute solutions, the logarithmic correction to Edwards' tree approximation for the osmotic pressure is calculated, and we find the new result Πβ ∝ C2/√lnC, where C is the monomer concentration. We give in 4d a set of geometrical quantities, characterizing a single chain: the form factor, the average 2nth power R[2n] of the end-to-end distance, the average 2nth radius of gyration RG[2n], and their universal ratio. We also consider in 4d the probability distribution for the internal distances inside a chain.
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\ifmmode\times\else\texttimes\fi{}{10}^{6} steps. Consequently the critical exponent \nu for three-dimensional self-avoiding walks is determined to great accuracy; the final estimate is \nu. The method can be adapted to other models of polymers with short-range interactions, on the lattice or in the continuum.
The pivot algorithm is a Markov Chain Monte Carlo algorithm for simulating the selfavoiding walk. At each iteration a pivot which produces a global change in the walk is proposed. If the resulting walk is self-avoiding, the new walk is accepted; otherwise, it is rejected. Past implementations of the algorithm required a time O(N) per accepted pivot, where N is the number of steps in the walk. We show how to implement the algorithm so that the time required per accepted pivot is O(N q ) with q < 1. We estimate that q is less than 0:57 in two dimensions, and less than 0:85 in three dimensions. Corrections to the O(N q ) make an accurate estimate of q impossible. They also imply that the asymptotic behavior of O(N q ) cannot be seen for walk lengths which can be simulated. In simulations the eective q is around 0:7 in two dimensions and 0:9 in three dimensions. Comparisons with simulations that use the standard implementation of the pivot algorithm using a hash table indicate that our implementation is faster by as much as a factor of 80 in two dimensions and as much as a factor of 7 in three dimensions. Our method does not require the use of a hash table and should also be applicable to the pivot algorithm for o-lattice models. Key words: self-avoiding walk, pivot algorithm, polymer. 1 1