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Scale-free Monte Carlo method for calculating the critical exponent γ\gamma of self-avoiding walks

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Journal of Physics A: Mathematical and Theoretical
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Nathan Clisby
Nathan Clisby
Nathan Clisby
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Abstract and Figures

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\gamma for self-avoiding walks to unprecedented accuracy. Our final estimate for γ\gamma is 1.15695300(95).
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1
Journal of Physics A: Mathematical and Theoretical
Scale-free Monte Carlo method for
calculating the critical exponent γ
of self-avoiding walks*
NathanClisby
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010,
Australia
E-mail: nclisby@unimelb.edu.au
Received 2 February 2017, revised 3 May 2017
Accepted for publication 10 May 2017
Published 6 June 2017
Abstract
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 efciently
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 nd the critical exponent
γ for self-avoiding walks to unprecedented accuracy. Our nal estimate for γ
is
1.156 953 00(95)
.
Keywords: self-avoiding walk, critical exponent, Monte Carlo,
pivotalgorithm
(Some guresmay appear in colour only in the online journal)
1. Introduction
An N-step self-avoiding walk (SAW) on the d-dimensional cubic lattice is a mapping
ω:{0, 1,
...
,N}→Zd
with
|ω(i+1)−ω(i)|=1
for each i (
|x|
denotes the Euclidean norm
of x), with
ω(0)
at the origin, and with
ω(i)=ω(j)
for all
i=j
. It is of fundamental interest in
the theory of critical phenomena as the
n→0
limit of the n-vector model, and is the simplest
model which captures the universal behavior of polymers in a good solvent.
The number of self-avoiding walks of length N on
Z3
, which we denote cN, is believed to
be given by
N=ANγ−1µN
1+
N∆1+O
N
(1)
N Clisby
Scale-free Monte Carlo method for calculating the critical exponent γ of self-avoiding walks
Printed in the UK
264003
JPHAC5
© 2017 IOP Publishing Ltd
50
J. Phys. A: Math. Theor.
JPA
1751-8121
10.1088/1751-8121/aa7231
Paper
26
1
13
Journal of Physics A: Mathematical and Theoretical
IOP
* Dedicated to Tony Guttmann on the occasion of his 70th birthday.
2017
1751-8121/17/264003+13$33.00 © 2017 IOP Publishing Ltd Printed in the UK
J. Phys. A: Math. Theor. 50 (2017) 264003 (13pp) https://doi.org/10.1088/1751-8121/aa7231
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