Self-avoiding walk enumeration via the lace expansion

Clisby, Nathan; Liang, Richard; Slade, Gordon

doi:10.1088/1751-8113/40/36/003

Cited by 77 publications

(162 citation statements)

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“…The length of the path, ℓ, is given by the number of steps made until the walk is terminated. A large number of studies of SAWs on regular lattices were devoted to the enumeration of paths as a function of their length [26,27,28,29,30,31]. These studies provided much insight on the structure and thermodynamics of polymers [6,7].…”

Section: Introductionmentioning

confidence: 99%

The distribution of path lengths of self avoiding walks on Erdős–Rényi networks

Tishby¹,

Biham²,

Katzav³

2016

J. Phys. A: Math. Theor.

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Abstract. We present an analytical and numerical study of the paths of self avoiding walks (SAWs) on random networks. Since these walks do not retrace their paths, they effectively delete the nodes they visit, together with their links, thus pruning the network. The walkers hop between neighboring nodes, until they reach a deadend node from which they cannot proceed. Focusing on Erdős-Rényi networks we show that the pruned networks maintain a Poisson degree distribution, p t (k), with an average degree, k t , that decreases linearly in time. We enumerate the SAW paths of any given length and find that the number of paths, n T (ℓ), increases dramatically as a function of ℓ. We also obtain analytical results for the path-length distribution, P (ℓ), of the SAW paths which are actually pursued, starting from a random initial node. It turns out that P (ℓ) follows the Gompertz distribution, which means that the termination probability of an SAW path increases with its length.

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Section: Introductionmentioning

confidence: 99%

The distribution of path lengths of self avoiding walks on Erdős–Rényi networks

Tishby¹,

Biham²,

Katzav³

2016

J. Phys. A: Math. Theor.

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“…Finally, using the RG derived values for ν = 0.5876 and γ = 1.1568 7/6, [23,34,35] we obtain χ = 3ν + γ − 1 = 1.9196 .…”

Section: Computation Of Power-law Exponentsmentioning

confidence: 79%

Self-avoiding wormlike chain model for double-stranded-DNA loop formation

2014

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We compute for the first time the effects of excluded volume on the probability for double-stranded DNA to form a loop. We utilize a Monte-Carlo algorithm for generation of large ensembles of selfavoiding worm-like chains, which are used to compute the J-factor for varying lengthscales. In the entropic regime, we confirm the scaling-theory prediction of a power-law drop off of −1.92, which is significantly stronger than the −1.5 power-law predicted by the non-self-avoiding worm-like chain model. In the elastic regime, we find that the angle-independent end-to-end chain distribution is highly anisotropic. This anisotropy, combined with the excluded volume constraints, lead to an increase in the J-factor of the self-avoiding worm-like chain by about half an order of magnitude relative to its non-self-avoiding counterpart. This increase could partially explain the anomalous results of recent cyclization experiments, in which short dsDNA molecules were found to have an increased propensity to form a loop.

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“…About the same time, the formula µ(d) = 2d−1−(2d) −1 +O((2d) −2 ) was proved in [140]. In [101,102], the lace expansion was used to prove the existence of an asymptotic expansion to all orders , and also that µ(d) = 2d−1−(2d) [53]. It seems likely that the asymptotic expansion has radius of convergence zero, though there is no proof of this.…”

Section: (2d)mentioning

confidence: 99%

“…The strategy in this exercise is based on that used in [53,122,123], and is simpler than that used in [101]. Equation (5.70) was first proved in [140], using completely different methods.…”

Section: Convergence Of the Expansionmentioning

confidence: 99%

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The Lace Expansion and its Applications

Saint-Flour

Slade

Picard³

2006

Lecture Notes in Mathematics

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ii The Lace Expansion and its ApplicationsThe Lace Expansion and its Applications iii Preface Several superficially simple mathematical models, such as the self-avoiding walk and percolation, are paradigms for the study of critical phenomena in statistical mechanics. Although these models have been studied by mathematicians for about half a century, exciting new developments continue to occur and the subject is flourishing. Much progress has been made, but it remains a major challenge for mathematical physics and probability theory to obtain a complete and mathematically rigorous understanding of the scaling theory of these models at criticality.These lecture notes concern the lace expansion, which is a powerful tool for the analysis of the critical scaling of several models above their upper critical dimensions, namely:• the self-avoiding walk on Z d for d > 4, • lattice trees and lattice animals onResults include proofs of existence of critical exponents, with mean-field values, and construction of scaling limits. Often, the scaling limit is described in terms of super-Brownian motion. There are two distinct goals for these notes. The first goal is to provide a written accompaniment to my lectures at the Saint-Flour summer school in 2004, and at the Pacific Institute for the Mathematical Sciences -University of British Columbia summer school in 2005. The notes contain an introduction to the lace expansion and several of its applications, with sufficient background and depth to prepare a newcomer to do research using the lace expansion. Basic graduate level probability theory will be used, but no previous knowledge of the lace expansion or super-Brownian motion is assumed. The second goal is to provide a survey of the field, so that an interested reader can follow up by consulting the original literature. In pursuit of the second goal, these notes include more material than can be covered during a summer school course.Following a brief initial section concerning random walk, the notes can be divided into four parts, whose contents are summarized as follows.Part I, which concerns the self-avoiding walk, consists of Sections 2-6. A complete and self-contained proof is given of the convergence of the lace expansion for the nearest-neighbour model in dimensions d 4, and for the spread-out model of self-avoiding walks which take steps of length at most L, with L 1, in dimensions d > 4. The convergence proof presented here seems simpler than all previous lace expansion convergence proofs. As a consequence of convergence, it is shown that the critical exponent γ for the generating function of the number of n-step self-avoiding walks exists and is equal to 1. A survey is then given of the many extensions of this result that have been obtained using the lace expansion. Part II, which concerns lattice trees and lattice animals, consists of Sections 7-8. It is shown how a minor modification of the expansion for the self-avoiding walk can be applied to give expansions for lattice trees and lattice animals, and an indicatio...

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Self-avoiding walk enumeration via the lace expansion

Cited by 77 publications

References 54 publications

The distribution of path lengths of self avoiding walks on Erdős–Rényi networks

The distribution of path lengths of self avoiding walks on Erdős–Rényi networks

Self-avoiding wormlike chain model for double-stranded-DNA loop formation

The Lace Expansion and its Applications

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