Self-avoiding walks (SAW)

introduction

• choose edge in a given lattice
• not allowed to retrace your path
• how many SAWs of length \(n\) are there?
• simple to define, in some ways really easy to study but we are not close to a closed form formula

basics

def

A SAW of length \(n\) is a map \(w:\{0,1,\cdots, n\} \to \mathbb{Z}^d\) such that \(|w(i+1)-w(i)|=1\) and \(w(i)\neq w(j)\) for \(i\neq j\)

• \(W_n\) the set of all SAWs of length \(n\)
• \(C_n(x)=C_n(0,x)\) number of SAW starting at 0 and ending at x
• \(C_n=\sum_{x\in \mathbb{Z}^d}C_n(x)\) number of SAW
• \(R_e^2(w)=|w(n)-w(0)|^2\)
• we have

\[ \begin{align} \langle R_e^2 \rangle&=\frac{1}{C_n}\sum_{w\in W_n}R_e^2(w) \\ &=\frac{1}{C_n}\sum_{w\in W_n}|w(n)|^2 \\ &=\frac{1}{C_n}\sum_{x\in \mathbb{Z}^d}\sum_{w:w(n)=x}|x|^2 \\ &=\frac{1}{C_n}\sum_{x\in \mathbb{Z}^d}|x|^2C_n(x) \end{align} \]

conjecture

We have the following conjecture \[ C_n \sim An^{\gamma-1}\mu^n \label{asymp} \] \[ C_n(x) \sim Bn^{\alpha-2}\mu^n \] \[ \langle R_e^2 \rangle \sim Dn^{2\nu} \]

• critical exponent (universal)
  • \(\alpha\) specific heat
  • \(\gamma\) susceptibility
  • \(\nu\) associated with correlation length

models in the universality class

• Domb-Joyce : weakly avoiding walk (penalty for intersection)
• bead model in the continuum
• polymers

overview of known results

• any solution will not be \(D\)-finite

2d

• Coulomb gas (early 1980's)
• conformal field theory (1980's)
• SLE (since 1998)

3d

• no exact prediction
• numerical method
• renormalization group
• series method
• monte carlo simultation

asymptotics \ref{asymp}

• very little hope of showing this in \(d=3\)
• \(d\geq 5\) has been shown that \(\gamma=1\) via the lace expansion
• \(d=4\) some things proven via exact renormalization group
• \(d=2\), nothing yet, chance of a proof via discrete holomophicity

2d lattice

SAW on 2d square lattice

• \(\{c_n\}_{n \geq 0} : 4,12,36,100,\cdots \)

SAW on 2d honeycomb lattice

conjecture

Let \(c_n\) be the number of SAWs from a fixed starting point on the honeycomb lattice. Then \[ c_n \sim An^{\gamma-1}\mu^n \] as \(n\to \infty\), where \(\mu=\sqrt{2+\sqrt{2}}\) and \(\gamma\) is conjectured to be \(43/32\)

• the fact \(\mu=\sqrt{2+\sqrt{2}}\) was conjectured by Nieuhuis in 1982 and proved in 2012 by Smirnov
• the critical exponent \(\gamma\) is universal
• proof uses discrete holomorphic observables

related items

• non-intersecting paths

computational resource

• https://oeis.org/A001411

expositions

• Slade, Gordon. “Self-Avoiding Walks.” The Mathematical Intelligencer 16, no. 1 (December 1, 1994): 29–35. doi:10.1007/BF03026612.

articles

• Grimmett, Geoffrey R., and Zhongyang Li. “Counting Self-Avoiding Walks.” arXiv:1304.7216 [math-Ph], April 26, 2013. http://arxiv.org/abs/1304.7216.
• Duminil-Copin, Hugo, and Stanislav Smirnov. “The Connective Constant of the Honeycomb Lattice Equals \(\sqrt{2+\sqrt2}\).” arXiv:1007.0575 [math-Ph], July 4, 2010. http://arxiv.org/abs/1007.0575.
• Lawler, Gregory F., Oded Schramm, and Wendelin Werner. “On the Scaling Limit of Planar Self-Avoiding Walk.” arXiv:math/0204277, April 23, 2002. http://arxiv.org/abs/math/0204277.

encyclopedia

• http://en.wikipedia.org/wiki/Self-avoiding_walk
• http://en.wikipedia.org/wiki/Connective_constant

메타데이터

위키데이터

• ID : Q7448025

Spacy 패턴 목록

• [{'LOWER': 'self'}, {'OP': '*'}, {'LOWER': 'avoiding'}, {'LEMMA': 'walk'}]
• [{'LEMMA': 'SAW'}]